math.d

// Written in the D programming language.

/**
 * Elementary mathematical functions
 *
 * Contains the elementary mathematical functions (powers, roots,
 * and trignometric functions), and low-level floating-point operations.
 * Mathematical special functions are available in std.mathspecial.
 *
 * The functionality closely follows the IEEE754-2008 standard for
 * floating-point arithmetic, including the use of camelCase names rather
 * than C99-style lower case names. All of these functions behave correctly
 * when presented with an infinity or NaN.
 *
 * Unlike C, there is no global 'errno' variable. Consequently, almost all of
 * these functions are pure nothrow.
 *
 * Status:
 * The semantics and names of feqrel and approxEqual will be revised.
 *
 * Macros:
 *      WIKI = Phobos/StdMath
 *
 *      TABLE_SV = <table border=1 cellpadding=4 cellspacing=0>
 *              <caption>Special Values</caption>
 *              $0</table>
 *      SVH = $(TR $(TH $1) $(TH $2))
 *      SV  = $(TR $(TD $1) $(TD $2))
 *
 *      NAN = $(RED NAN)
 *      SUP = <span style="vertical-align:super;font-size:smaller">$0</span>
 *      GAMMA = &#915;
 *      THETA = &theta;
 *      INTEGRAL = &#8747;
 *      INTEGRATE = $(BIG &#8747;<sub>$(SMALL $1)</sub><sup>$2</sup>)
 *      POWER = $1<sup>$2</sup>
 *      SUB = $1<sub>$2</sub>
 *      BIGSUM = $(BIG &Sigma; <sup>$2</sup><sub>$(SMALL $1)</sub>)
 *      CHOOSE = $(BIG &#40;) <sup>$(SMALL $1)</sup><sub>$(SMALL $2)</sub> $(BIG &#41;)
 *      PLUSMN = &plusmn;
 *      INFIN = &infin;
 *      PLUSMNINF = &plusmn;&infin;
 *      PI = &pi;
 *      LT = &lt;
 *      GT = &gt;
 *      SQRT = &radic;
 *      HALF = &frac12;
 *
 * Copyright: Copyright Digital Mars 2000 - 2011.
 * License:   <a href="http://www.boost.org/LICENSE_1_0.txt">Boost License 1.0</a>.
 * Authors:   $(WEB digitalmars.com, Walter Bright),
 *                        Don Clugston
 * Source: $(PHOBOSSRC std/_math.d)
 */
module std.math;

import core.stdc.math;
import std.range, std.traits;

version(unittest) {
    import std.typetuple;
}

version(LDC) {
    import ldc.intrinsics;
}

version(DigitalMars){
    version = INLINE_YL2X;        // x87 has opcodes for these
}

version (X86){
    version = X86_Any;
}

version (X86_64){
    version = X86_Any;
}

version(D_InlineAsm_X86){
    version = InlineAsm_X86_Any;
}
else version(D_InlineAsm_X86_64){
    version = InlineAsm_X86_Any;
}




private:
/*
 * The following IEEE 'real' formats are currently supported:
 * 64 bit Big-endian  'double' (eg PowerPC)
 * 128 bit Big-endian 'quadruple' (eg SPARC)
 * 64 bit Little-endian 'double' (eg x86-SSE2)
 * 80 bit Little-endian, with implied bit 'real80' (eg x87, Itanium).
 * 128 bit Little-endian 'quadruple' (not implemented on any known processor!)
 *
 * Non-IEEE 128 bit Big-endian 'doubledouble' (eg PowerPC) has partial support
 */
version(LittleEndian) {
    static assert(real.mant_dig == 53 || real.mant_dig==64
               || real.mant_dig == 113,
      "Only 64-bit, 80-bit, and 128-bit reals"
      " are supported for LittleEndian CPUs");
} else {
    static assert(real.mant_dig == 53 || real.mant_dig==106
               || real.mant_dig == 113,
    "Only 64-bit and 128-bit reals are supported for BigEndian CPUs."
    " double-double reals have partial support");
}

// Constants used for extracting the components of the representation.
// They supplement the built-in floating point properties.
template floatTraits(T) {
    // EXPMASK is a ushort mask to select the exponent portion (without sign)
    // EXPPOS_SHORT is the index of the exponent when represented as a ushort array.
    // SIGNPOS_BYTE is the index of the sign when represented as a ubyte array.
    // RECIP_EPSILON is the value such that (smallest_subnormal) * RECIP_EPSILON == T.min_normal
    enum T RECIP_EPSILON = (1/T.epsilon);
    static if (T.mant_dig == 24)
    { // float
        enum ushort EXPMASK = 0x7F80;
        enum ushort EXPBIAS = 0x3F00;
        enum uint EXPMASK_INT = 0x7F80_0000;
        enum uint MANTISSAMASK_INT = 0x007F_FFFF;
        version(LittleEndian) {
            enum EXPPOS_SHORT = 1;
        } else {
            enum EXPPOS_SHORT = 0;
        }
    }
    else static if (T.mant_dig == 53) // double, or real==double
    {
        enum ushort EXPMASK = 0x7FF0;
        enum ushort EXPBIAS = 0x3FE0;
        enum uint EXPMASK_INT = 0x7FF0_0000;
        enum uint MANTISSAMASK_INT = 0x000F_FFFF; // for the MSB only
        version(LittleEndian) {
            enum EXPPOS_SHORT = 3;
            enum SIGNPOS_BYTE = 7;
        } else {
            enum EXPPOS_SHORT = 0;
            enum SIGNPOS_BYTE = 0;
        }
    }
    else static if (T.mant_dig == 64) // real80
    {
        enum ushort EXPMASK = 0x7FFF;
        enum ushort EXPBIAS = 0x3FFE;
        version(LittleEndian)
        {
            enum EXPPOS_SHORT = 4;
            enum SIGNPOS_BYTE = 9;
        }
        else
        {
            enum EXPPOS_SHORT = 0;
            enum SIGNPOS_BYTE = 0;
        }
    } else static if (T.mant_dig == 113){ // quadruple
        enum ushort EXPMASK = 0x7FFF;
        version(LittleEndian)
        {
            enum EXPPOS_SHORT = 7;
            enum SIGNPOS_BYTE = 15;
        }
        else
        {
            enum EXPPOS_SHORT = 0;
            enum SIGNPOS_BYTE = 0;
        }
    } else static if (T.mant_dig == 106) { // doubledouble
        enum ushort EXPMASK = 0x7FF0;
        // the exponent byte is not unique
        version(LittleEndian)
        {
            enum EXPPOS_SHORT = 7; // [3] is also an exp short
            enum SIGNPOS_BYTE = 15;
        }
        else
        {
            enum EXPPOS_SHORT = 0; // [4] is also an exp short
            enum SIGNPOS_BYTE = 0;
        }
    }
}

// These apply to all floating-point types
version(LittleEndian)
{
    enum MANTISSA_LSB = 0;
    enum MANTISSA_MSB = 1;
}
else
{
    enum MANTISSA_LSB = 1;
    enum MANTISSA_MSB = 0;
}

public:

// Values obtained from Wolfram Alpha. 116 bits ought to be enough for anybody.
// Wolfram Alpha LLC. 2011. Wolfram|Alpha. http://www.wolframalpha.com/input/?i=e+in+base+16 (access July 6, 2011).
enum real E =          0x1.5bf0a8b1457695355fb8ac404e7a8p+1L; /** e = 2.718281... */
enum real LOG2T =      0x1.a934f0979a3715fc9257edfe9b5fbp+1L; /** $(SUB log, 2)10 = 3.321928... */
enum real LOG2E =      0x1.71547652b82fe1777d0ffda0d23a8p+0L; /** $(SUB log, 2)e = 1.442695... */
enum real LOG2 =       0x1.34413509f79fef311f12b35816f92p-2L; /** $(SUB log, 10)2 = 0.301029... */
enum real LOG10E =     0x1.bcb7b1526e50e32a6ab7555f5a67cp-2L; /** $(SUB log, 10)e = 0.434294... */
enum real LN2 =        0x1.62e42fefa39ef35793c7673007e5fp-1L; /** ln 2  = 0.693147... */
enum real LN10 =       0x1.26bb1bbb5551582dd4adac5705a61p+1L; /** ln 10 = 2.302585... */
enum real PI =         0x1.921fb54442d18469898cc51701b84p+1L; /** $(_PI) = 3.141592... */
enum real PI_2 =       PI/2;                                  /** $(PI) / 2 = 1.570796... */
enum real PI_4 =       PI/4;                                  /** $(PI) / 4 = 0.785398... */
enum real M_1_PI =     0x1.45f306dc9c882a53f84eafa3ea69cp-2L; /** 1 / $(PI) = 0.318309... */
enum real M_2_PI =     2*M_1_PI;                              /** 2 / $(PI) = 0.636619... */
enum real M_2_SQRTPI = 0x1.20dd750429b6d11ae3a914fed7fd8p+0L; /** 2 / $(SQRT)$(PI) = 1.128379... */
enum real SQRT2 =      0x1.6a09e667f3bcc908b2fb1366ea958p+0L; /** $(SQRT)2 = 1.414213... */
enum real SQRT1_2 =    SQRT2/2;                               /** $(SQRT)$(HALF) = 0.707106... */
// Note: Make sure the magic numbers in compiler backend for x87 match these.

/*
        Octal versions:
        PI/64800        0.00001 45530 36176 77347 02143 15351 61441 26767
        PI/180          0.01073 72152 11224 72344 25603 54276 63351 22056
        PI/8            0.31103 75524 21026 43021 51423 06305 05600 67016
        SQRT(1/PI)      0.44067 27240 41233 33210 65616 51051 77327 77303
        2/PI            0.50574 60333 44710 40522 47741 16537 21752 32335
        PI/4            0.62207 73250 42055 06043 23046 14612 13401 56034
        SQRT(2/PI)      0.63041 05147 52066 24106 41762 63612 00272 56161

        PI              3.11037 55242 10264 30215 14230 63050 56006 70163
        LOG2            0.23210 11520 47674 77674 61076 11263 26013 37111
 */

/***********************************
 * Calculates the absolute value
 *
 * For complex numbers, abs(z) = sqrt( $(POWER z.re, 2) + $(POWER z.im, 2) )
 * = hypot(z.re, z.im).
 */
Num abs(Num)(Num x) @safe pure nothrow
    if (is(typeof(Num.init >= 0)) && is(typeof(-Num.init)) &&
            !(is(Num* : const(ifloat*)) || is(Num* : const(idouble*))
                    || is(Num* : const(ireal*))))
{
    static if (isFloatingPoint!(Num))
        return fabs(x);
    else
        return x>=0 ? x : -x;
}

auto abs(Num)(Num z) @safe pure nothrow
    if (is(Num* : const(cfloat*)) || is(Num* : const(cdouble*))
            || is(Num* : const(creal*)))
{
    return hypot(z.re, z.im);
}

/** ditto */
real abs(Num)(Num y) @safe pure nothrow
    if (is(Num* : const(ifloat*)) || is(Num* : const(idouble*))
            || is(Num* : const(ireal*)))
{
    return fabs(y.im);
}


unittest
{
    assert(isIdentical(abs(-0.0L), 0.0L));
    assert(isNaN(abs(real.nan)));
    assert(abs(-real.infinity) == real.infinity);
    assert(abs(-3.2Li) == 3.2L);
    assert(abs(71.6Li) == 71.6L);
    assert(abs(-56) == 56);
    assert(abs(2321312L)  == 2321312L);
    assert(abs(-1+1i) == sqrt(2.0));
}

/***********************************
 * Complex conjugate
 *
 *  conj(x + iy) = x - iy
 *
 * Note that z * conj(z) = $(POWER z.re, 2) - $(POWER z.im, 2)
 * is always a real number
 */
creal conj(creal z) @safe pure nothrow
{
    return z.re - z.im*1i;
}

/** ditto */
ireal conj(ireal y) @safe pure nothrow
{
    return -y;
}


unittest
{
    assert(conj(7 + 3i) == 7-3i);
    ireal z = -3.2Li;
    assert(conj(z) == -z);
}

/***********************************
 * Returns cosine of x. x is in radians.
 *
 *      $(TABLE_SV
 *      $(TR $(TH x)                 $(TH cos(x)) $(TH invalid?))
 *      $(TR $(TD $(NAN))            $(TD $(NAN)) $(TD yes)     )
 *      $(TR $(TD $(PLUSMN)$(INFIN)) $(TD $(NAN)) $(TD yes)     )
 *      )
 * Bugs:
 *      Results are undefined if |x| >= $(POWER 2,64).
 */

real cos(real x) @safe pure nothrow;       /* intrinsic */

/***********************************
 * Returns sine of x. x is in radians.
 *
 *      $(TABLE_SV
 *      $(TR $(TH x)               $(TH sin(x))      $(TH invalid?))
 *      $(TR $(TD $(NAN))          $(TD $(NAN))      $(TD yes))
 *      $(TR $(TD $(PLUSMN)0.0)    $(TD $(PLUSMN)0.0) $(TD no))
 *      $(TR $(TD $(PLUSMNINF))    $(TD $(NAN))      $(TD yes))
 *      )
 * Bugs:
 *      Results are undefined if |x| >= $(POWER 2,64).
 */

real sin(real x) @safe pure nothrow;       /* intrinsic */


/***********************************
 *  sine, complex and imaginary
 *
 *  sin(z) = sin(z.re)*cosh(z.im) + cos(z.re)*sinh(z.im)i
 *
 * If both sin($(THETA)) and cos($(THETA)) are required,
 * it is most efficient to use expi($(THETA)).
 */
creal sin(creal z) @safe pure nothrow
{
    creal cs = expi(z.re);
    creal csh = coshisinh(z.im);
    return cs.im * csh.re + cs.re * csh.im * 1i;
}

/** ditto */
ireal sin(ireal y) @safe pure nothrow
{
    return cosh(y.im)*1i;
}

unittest
{
  assert(sin(0.0+0.0i) == 0.0);
  assert(sin(2.0+0.0i) == sin(2.0L) );
}

/***********************************
 *  cosine, complex and imaginary
 *
 *  cos(z) = cos(z.re)*cosh(z.im) - sin(z.re)*sinh(z.im)i
 */
creal cos(creal z) @safe pure nothrow
{
    creal cs = expi(z.re);
    creal csh = coshisinh(z.im);
    return cs.re * csh.re - cs.im * csh.im * 1i;
}

/** ditto */
real cos(ireal y) @safe pure nothrow
{
    return cosh(y.im);
}

unittest{
    assert(cos(0.0+0.0i)==1.0);
    assert(cos(1.3L+0.0i)==cos(1.3L));
    assert(cos(5.2Li)== cosh(5.2L));
}

/****************************************************************************
 * Returns tangent of x. x is in radians.
 *
 *      $(TABLE_SV
 *      $(TR $(TH x)             $(TH tan(x))       $(TH invalid?))
 *      $(TR $(TD $(NAN))        $(TD $(NAN))       $(TD yes))
 *      $(TR $(TD $(PLUSMN)0.0)  $(TD $(PLUSMN)0.0) $(TD no))
 *      $(TR $(TD $(PLUSMNINF))  $(TD $(NAN))       $(TD yes))
 *      )
 */

real tan(real x) @trusted pure nothrow
{
    version(D_InlineAsm_X86)
    {
    asm
    {
        fld     x[EBP]                  ; // load theta
        fxam                            ; // test for oddball values
        fstsw   AX                      ;
        sahf                            ;
        jc      trigerr                 ; // x is NAN, infinity, or empty
                                          // 387's can handle subnormals
SC18:   fptan                           ;
        fstp    ST(0)                   ; // dump X, which is always 1
        fstsw   AX                      ;
        sahf                            ;
        jnp     Lret                    ; // C2 = 1 (x is out of range)

        // Do argument reduction to bring x into range
        fldpi                           ;
        fxch                            ;
SC17:   fprem1                          ;
        fstsw   AX                      ;
        sahf                            ;
        jp      SC17                    ;
        fstp    ST(1)                   ; // remove pi from stack
        jmp     SC18                    ;

trigerr:
        jnp     Lret                    ; // if theta is NAN, return theta
        fstp    ST(0)                   ; // dump theta
    }
    return real.nan;

Lret:
    ;
    }
    else version(D_InlineAsm_X86_64)
    {
    asm
    {
        fld     x[RBP]                  ; // load theta
        fxam                            ; // test for oddball values
        fstsw   AX                      ;
        test    AH,1                    ;
        jnz     trigerr                 ; // x is NAN, infinity, or empty
                                          // 387's can handle subnormals
SC18:   fptan                           ;
        fstp    ST(0)                   ; // dump X, which is always 1
        fstsw   AX                      ;
        test    AH,4                    ;
        jz      Lret                    ; // C2 = 1 (x is out of range)

        // Do argument reduction to bring x into range
        fldpi                           ;
        fxch                            ;
SC17:   fprem1                          ;
        fstsw   AX                      ;
        test    AH,4                    ;
        jnz     SC17                    ;
        fstp    ST(1)                   ; // remove pi from stack
        jmp     SC18                    ;

trigerr:
        test    AH,4                    ;
        jz      Lret                    ; // if theta is NAN, return theta
        fstp    ST(0)                   ; // dump theta
    }
    return real.nan;

Lret:
    ;
    } else {
        return core.stdc.math.tanl(x);
    }
}

unittest
{
    static real vals[][2] =     // angle,tan
        [
         [   0,   0],
         [   .5,  .5463024898],
         [   1,   1.557407725],
         [   1.5, 14.10141995],
         [   2,  -2.185039863],
         [   2.5,-.7470222972],
         [   3,  -.1425465431],
         [   3.5, .3745856402],
         [   4,   1.157821282],
         [   4.5, 4.637332055],
         [   5,  -3.380515006],
         [   5.5,-.9955840522],
         [   6,  -.2910061914],
         [   6.5, .2202772003],
         [   10,  .6483608275],

         // special angles
         [   PI_4,   1],
         //[   PI_2,   real.infinity], // PI_2 is not _exactly_ pi/2.
         [   3*PI_4, -1],
         [   PI,     0],
         [   5*PI_4, 1],
         //[   3*PI_2, -real.infinity],
         [   7*PI_4, -1],
         [   2*PI,   0],
         ];
    int i;

    for (i = 0; i < vals.length; i++)
    {
        real x = vals[i][0];
        real r = vals[i][1];
        real t = tan(x);

        //printf("tan(%Lg) = %Lg, should be %Lg\n", x, t, r);
        if (!isIdentical(r, t)) assert(fabs(r-t) <= .0000001);

        x = -x;
        r = -r;
        t = tan(x);
        //printf("tan(%Lg) = %Lg, should be %Lg\n", x, t, r);
        if (!isIdentical(r, t) && !(r!<>=0 && t!<>=0)) assert(fabs(r-t) <= .0000001);
    }
    // overflow
    assert(isNaN(tan(real.infinity)));
    assert(isNaN(tan(-real.infinity)));
    // NaN propagation
    assert(isIdentical( tan(NaN(0x0123L)), NaN(0x0123L) ));
}

/***************
 * Calculates the arc cosine of x,
 * returning a value ranging from 0 to $(PI).
 *
 *      $(TABLE_SV
 *      $(TR $(TH x)         $(TH acos(x)) $(TH invalid?))
 *      $(TR $(TD $(GT)1.0)  $(TD $(NAN))  $(TD yes))
 *      $(TR $(TD $(LT)-1.0) $(TD $(NAN))  $(TD yes))
 *      $(TR $(TD $(NAN))    $(TD $(NAN))  $(TD yes))
 *  )
 */
real acos(real x) @safe pure nothrow
{
    return atan2(sqrt(1-x*x), x);
}

/// ditto
double acos(double x) @safe pure nothrow { return acos(cast(real)x); }
/// ditto
float acos(float x) @safe pure nothrow  { return acos(cast(real)x); }

/***************
 * Calculates the arc sine of x,
 * returning a value ranging from -$(PI)/2 to $(PI)/2.
 *
 *      $(TABLE_SV
 *      $(TR $(TH x)            $(TH asin(x))      $(TH invalid?))
 *      $(TR $(TD $(PLUSMN)0.0) $(TD $(PLUSMN)0.0) $(TD no))
 *      $(TR $(TD $(GT)1.0)     $(TD $(NAN))       $(TD yes))
 *      $(TR $(TD $(LT)-1.0)    $(TD $(NAN))       $(TD yes))
 *  )
 */
real asin(real x) @safe pure nothrow
{
    return atan2(x, sqrt(1-x*x));
}
/// ditto
double asin(double x) @safe pure nothrow { return asin(cast(real)x); }
/// ditto
float asin(float x) @safe pure nothrow  { return asin(cast(real)x); }

/***************
 * Calculates the arc tangent of x,
 * returning a value ranging from -$(PI)/2 to $(PI)/2.
 *
 *      $(TABLE_SV
 *      $(TR $(TH x)                 $(TH atan(x))      $(TH invalid?))
 *      $(TR $(TD $(PLUSMN)0.0)      $(TD $(PLUSMN)0.0) $(TD no))
 *      $(TR $(TD $(PLUSMN)$(INFIN)) $(TD $(NAN))       $(TD yes))
 *  )
 */
real atan(real x) @safe pure nothrow { return atan2(x, 1.0L); }
/// ditto
double atan(double x) @safe pure nothrow { return atan(cast(real)x); }
/// ditto
float atan(float x)  @safe pure nothrow { return atan(cast(real)x); }

/***************
 * Calculates the arc tangent of y / x,
 * returning a value ranging from -$(PI) to $(PI).
 *
 *      $(TABLE_SV
 *      $(TR $(TH y)                 $(TH x)            $(TH atan(y, x)))
 *      $(TR $(TD $(NAN))            $(TD anything)     $(TD $(NAN)) )
 *      $(TR $(TD anything)          $(TD $(NAN))       $(TD $(NAN)) )
 *      $(TR $(TD $(PLUSMN)0.0)      $(TD $(GT)0.0)     $(TD $(PLUSMN)0.0) )
 *      $(TR $(TD $(PLUSMN)0.0)      $(TD +0.0)         $(TD $(PLUSMN)0.0) )
 *      $(TR $(TD $(PLUSMN)0.0)      $(TD $(LT)0.0)     $(TD $(PLUSMN)$(PI)))
 *      $(TR $(TD $(PLUSMN)0.0)      $(TD -0.0)         $(TD $(PLUSMN)$(PI)))
 *      $(TR $(TD $(GT)0.0)          $(TD $(PLUSMN)0.0) $(TD $(PI)/2) )
 *      $(TR $(TD $(LT)0.0)          $(TD $(PLUSMN)0.0) $(TD -$(PI)/2) )
 *      $(TR $(TD $(GT)0.0)          $(TD $(INFIN))     $(TD $(PLUSMN)0.0) )
 *      $(TR $(TD $(PLUSMN)$(INFIN)) $(TD anything)     $(TD $(PLUSMN)$(PI)/2))
 *      $(TR $(TD $(GT)0.0)          $(TD -$(INFIN))    $(TD $(PLUSMN)$(PI)) )
 *      $(TR $(TD $(PLUSMN)$(INFIN)) $(TD $(INFIN))     $(TD $(PLUSMN)$(PI)/4))
 *      $(TR $(TD $(PLUSMN)$(INFIN)) $(TD -$(INFIN))    $(TD $(PLUSMN)3$(PI)/4))
 *      )
 */
real atan2(real y, real x) @trusted pure nothrow
{
    version(InlineAsm_X86_Any)
    {
        asm {
            fld y;
            fld x;
            fpatan;
        }
    }
    else
    {
        return core.stdc.math.atan2l(y,x);
    }
}

/// ditto
double atan2(double y, double x) @safe pure nothrow
{
    return atan2(cast(real)y, cast(real)x);
}

/// ditto
float atan2(float y, float x) @safe pure nothrow
{
    return atan2(cast(real)y, cast(real)x);
}

/***********************************
 * Calculates the hyperbolic cosine of x.
 *
 *      $(TABLE_SV
 *      $(TR $(TH x)                 $(TH cosh(x))      $(TH invalid?))
 *      $(TR $(TD $(PLUSMN)$(INFIN)) $(TD $(PLUSMN)0.0) $(TD no) )
 *      )
 */
real cosh(real x) @safe pure nothrow
{
    //  cosh = (exp(x)+exp(-x))/2.
    // The naive implementation works correctly.
    real y = exp(x);
    return (y + 1.0/y) * 0.5;
}
/// ditto
double cosh(double x) @safe pure nothrow { return cosh(cast(real)x); }
/// ditto
float cosh(float x) @safe pure nothrow  { return cosh(cast(real)x); }


/***********************************
 * Calculates the hyperbolic sine of x.
 *
 *      $(TABLE_SV
 *      $(TR $(TH x)                 $(TH sinh(x))           $(TH invalid?))
 *      $(TR $(TD $(PLUSMN)0.0)      $(TD $(PLUSMN)0.0)      $(TD no))
 *      $(TR $(TD $(PLUSMN)$(INFIN)) $(TD $(PLUSMN)$(INFIN)) $(TD no))
 *      )
 */
real sinh(real x) @safe pure nothrow
{
    //  sinh(x) =  (exp(x)-exp(-x))/2;
    // Very large arguments could cause an overflow, but
    // the maximum value of x for which exp(x) + exp(-x)) != exp(x)
    // is x = 0.5 * (real.mant_dig) * LN2. // = 22.1807 for real80.
    if (fabs(x) > real.mant_dig * LN2) {
        return copysign(0.5 * exp(fabs(x)), x);
    }
    real y = expm1(x);
    return 0.5 * y / (y+1) * (y+2);
}
/// ditto
double sinh(double x) @safe pure nothrow { return sinh(cast(real)x); }
/// ditto
float sinh(float x) @safe pure nothrow  { return sinh(cast(real)x); }


/***********************************
 * Calculates the hyperbolic tangent of x.
 *
 *      $(TABLE_SV
 *      $(TR $(TH x)                 $(TH tanh(x))      $(TH invalid?))
 *      $(TR $(TD $(PLUSMN)0.0)      $(TD $(PLUSMN)0.0) $(TD no) )
 *      $(TR $(TD $(PLUSMN)$(INFIN)) $(TD $(PLUSMN)1.0) $(TD no))
 *      )
 */
real tanh(real x) @safe pure nothrow
{
    //  tanh(x) = (exp(x) - exp(-x))/(exp(x)+exp(-x))
    if (fabs(x) > real.mant_dig * LN2) {
        return copysign(1, x);
    }
    real y = expm1(2*x);
    return y / (y + 2);
}
/// ditto
double tanh(double x) @safe pure nothrow { return tanh(cast(real)x); }
/// ditto
float tanh(float x) @safe pure nothrow { return tanh(cast(real)x); }

package:
/* Returns cosh(x) + I * sinh(x)
 * Only one call to exp() is performed.
 */
creal coshisinh(real x) @safe pure nothrow
{
    // See comments for cosh, sinh.
    if (fabs(x) > real.mant_dig * LN2) {
        real y = exp(fabs(x));
        return y * 0.5 + 0.5i * copysign(y, x);
    } else {
        real y = expm1(x);
        return (y + 1.0 + 1.0/(y + 1.0)) * 0.5 + 0.5i * y / (y+1) * (y+2);
    }
}

unittest {
    creal c = coshisinh(3.0L);
    assert(c.re == cosh(3.0L));
    assert(c.im == sinh(3.0L));
}

public:

/***********************************
 * Calculates the inverse hyperbolic cosine of x.
 *
 *  Mathematically, acosh(x) = log(x + sqrt( x*x - 1))
 *
 * $(TABLE_DOMRG
 *  $(DOMAIN 1..$(INFIN))
 *  $(RANGE  1..log(real.max), $(INFIN)) )
 *      $(TABLE_SV
 *    $(SVH  x,     acosh(x) )
 *    $(SV  $(NAN), $(NAN) )
 *    $(SV  $(LT)1,     $(NAN) )
 *    $(SV  1,      0       )
 *    $(SV  +$(INFIN),+$(INFIN))
 *  )
 */
real acosh(real x) @safe pure nothrow
{
    if (x > 1/real.epsilon)
        return LN2 + log(x);
    else
        return log(x + sqrt(x*x - 1));
}
/// ditto
double acosh(double x) @safe pure nothrow { return acosh(cast(real)x); }
/// ditto
float acosh(float x) @safe pure nothrow  { return acosh(cast(real)x); }


unittest
{
    assert(isNaN(acosh(0.9)));
    assert(isNaN(acosh(real.nan)));
    assert(acosh(1.0)==0.0);
    assert(acosh(real.infinity) == real.infinity);
}

/***********************************
 * Calculates the inverse hyperbolic sine of x.
 *
 *  Mathematically,
 *  ---------------
 *  asinh(x) =  log( x + sqrt( x*x + 1 )) // if x >= +0
 *  asinh(x) = -log(-x + sqrt( x*x + 1 )) // if x <= -0
 *  -------------
 *
 *    $(TABLE_SV
 *    $(SVH x,                asinh(x)       )
 *    $(SV  $(NAN),           $(NAN)         )
 *    $(SV  $(PLUSMN)0,       $(PLUSMN)0      )
 *    $(SV  $(PLUSMN)$(INFIN),$(PLUSMN)$(INFIN))
 *    )
 */
real asinh(real x) @safe pure nothrow
{
    return (fabs(x) > 1 / real.epsilon)
       // beyond this point, x*x + 1 == x*x
       ?  copysign(LN2 + log(fabs(x)), x)
       // sqrt(x*x + 1) ==  1 + x * x / ( 1 + sqrt(x*x + 1) )
       : copysign(log1p(fabs(x) + x*x / (1 + sqrt(x*x + 1)) ), x);
}
/// ditto
double asinh(double x) @safe pure nothrow { return asinh(cast(real)x); }
/// ditto
float asinh(float x) @safe pure nothrow { return asinh(cast(real)x); }

unittest
{
    assert(isIdentical(asinh(0.0), 0.0));
    assert(isIdentical(asinh(-0.0), -0.0));
    assert(asinh(real.infinity) == real.infinity);
    assert(asinh(-real.infinity) == -real.infinity);
    assert(isNaN(asinh(real.nan)));
}

/***********************************
 * Calculates the inverse hyperbolic tangent of x,
 * returning a value from ranging from -1 to 1.
 *
 * Mathematically, atanh(x) = log( (1+x)/(1-x) ) / 2
 *
 *
 * $(TABLE_DOMRG
 *  $(DOMAIN -$(INFIN)..$(INFIN))
 *  $(RANGE  -1..1) )
 * $(TABLE_SV
 *    $(SVH  x,     acosh(x) )
 *    $(SV  $(NAN), $(NAN) )
 *    $(SV  $(PLUSMN)0, $(PLUSMN)0)
 *    $(SV  -$(INFIN), -0)
 * )
 */
real atanh(real x) @safe pure nothrow
{
    // log( (1+x)/(1-x) ) == log ( 1 + (2*x)/(1-x) )
    return  0.5 * log1p( 2 * x / (1 - x) );
}
/// ditto
double atanh(double x) @safe pure nothrow { return atanh(cast(real)x); }
/// ditto
float atanh(float x) @safe pure nothrow { return atanh(cast(real)x); }


unittest
{
    assert(isIdentical(atanh(0.0), 0.0));
    assert(isIdentical(atanh(-0.0),-0.0));
    assert(isNaN(atanh(real.nan)));
    assert(isNaN(atanh(-real.infinity)));
}

/*****************************************
 * Returns x rounded to a long value using the current rounding mode.
 * If the integer value of x is
 * greater than long.max, the result is
 * indeterminate.
 */
long rndtol(real x) @safe pure nothrow;    /* intrinsic */


/*****************************************
 * Returns x rounded to a long value using the FE_TONEAREST rounding mode.
 * If the integer value of x is
 * greater than long.max, the result is
 * indeterminate.
 */
extern (C) real rndtonl(real x);

/***************************************
 * Compute square root of x.
 *
 *      $(TABLE_SV
 *      $(TR $(TH x)         $(TH sqrt(x))   $(TH invalid?))
 *      $(TR $(TD -0.0)      $(TD -0.0)      $(TD no))
 *      $(TR $(TD $(LT)0.0)  $(TD $(NAN))    $(TD yes))
 *      $(TR $(TD +$(INFIN)) $(TD +$(INFIN)) $(TD no))
 *      )
 */
float sqrt(float x) @safe pure nothrow;    /* intrinsic */
double sqrt(double x) @safe pure nothrow;  /* intrinsic */ /// ditto
real sqrt(real x) @safe pure nothrow;      /* intrinsic */ /// ditto

creal sqrt(creal z) @safe pure nothrow
{
    creal c;
    real x,y,w,r;

    if (z == 0)
    {
        c = 0 + 0i;
    }
    else
    {
        real z_re = z.re;
        real z_im = z.im;

        x = fabs(z_re);
        y = fabs(z_im);
        if (x >= y)
        {
            r = y / x;
            w = sqrt(x) * sqrt(0.5 * (1 + sqrt(1 + r * r)));
        }
        else
        {
            r = x / y;
            w = sqrt(y) * sqrt(0.5 * (r + sqrt(1 + r * r)));
        }

        if (z_re >= 0)
        {
            c = w + (z_im / (w + w)) * 1.0i;
        }
        else
        {
            if (z_im < 0)
                w = -w;
            c = z_im / (w + w) + w * 1.0i;
        }
    }
    return c;
}

/**
 * Calculates e$(SUP x).
 *
 *  $(TABLE_SV
 *    $(TR $(TH x)             $(TH e$(SUP x)) )
 *    $(TR $(TD +$(INFIN))     $(TD +$(INFIN)) )
 *    $(TR $(TD -$(INFIN))     $(TD +0.0)      )
 *    $(TR $(TD $(NAN))        $(TD $(NAN))    )
 *  )
 */
real exp(real x) @trusted pure nothrow
{
    version(D_InlineAsm_X86)
    {
        //  e^^x = 2^^(LOG2E*x)
        // (This is valid because the overflow & underflow limits for exp
        // and exp2 are so similar).
        return exp2(LOG2E*x);
    }
    else version(D_InlineAsm_X86_64)
    {
        //  e^^x = 2^^(LOG2E*x)
        // (This is valid because the overflow & underflow limits for exp
        // and exp2 are so similar).
        return exp2(LOG2E*x);
    } else {
        return core.stdc.math.expl(x);
    }
}
/// ditto
double exp(double x) @safe pure nothrow  { return exp(cast(real)x); }
/// ditto
float exp(float x)  @safe pure nothrow   { return exp(cast(real)x); }


/**
 * Calculates the value of the natural logarithm base (e)
 * raised to the power of x, minus 1.
 *
 * For very small x, expm1(x) is more accurate
 * than exp(x)-1.
 *
 *  $(TABLE_SV
 *    $(TR $(TH x)             $(TH e$(SUP x)-1)  )
 *    $(TR $(TD $(PLUSMN)0.0)  $(TD $(PLUSMN)0.0) )
 *    $(TR $(TD +$(INFIN))     $(TD +$(INFIN))    )
 *    $(TR $(TD -$(INFIN))     $(TD -1.0)         )
 *    $(TR $(TD $(NAN))        $(TD $(NAN))       )
 *  )
 */
real expm1(real x) @trusted pure nothrow
{
    version(D_InlineAsm_X86) {
        enum { PARAMSIZE = (real.sizeof+3)&(0xFFFF_FFFC) } // always a multiple of 4
        asm {
            /*  expm1() for x87 80-bit reals, IEEE754-2008 conformant.
             * Author: Don Clugston.
             *
             *    expm1(x) = 2^^(rndint(y))* 2^^(y-rndint(y)) - 1 where y = LN2*x.
             *    = 2rndy * 2ym1 + 2rndy - 1, where 2rndy = 2^^(rndint(y))
             *     and 2ym1 = (2^^(y-rndint(y))-1).
             *    If 2rndy  < 0.5*real.epsilon, result is -1.
             *    Implementation is otherwise the same as for exp2()
             */
            naked;
            fld real ptr [ESP+4] ; // x
            mov AX, [ESP+4+8]; // AX = exponent and sign
            sub ESP, 12+8; // Create scratch space on the stack
            // [ESP,ESP+2] = scratchint
            // [ESP+4..+6, +8..+10, +10] = scratchreal
            // set scratchreal mantissa = 1.0
            mov dword ptr [ESP+8], 0;
            mov dword ptr [ESP+8+4], 0x80000000;
            and AX, 0x7FFF; // drop sign bit
            cmp AX, 0x401D; // avoid InvalidException in fist
            jae L_extreme;
            fldl2e;
            fmulp ST(1), ST; // y = x*log2(e)
            fist dword ptr [ESP]; // scratchint = rndint(y)
            fisub dword ptr [ESP]; // y - rndint(y)
            // and now set scratchreal exponent
            mov EAX, [ESP];
            add EAX, 0x3fff;
            jle short L_largenegative;
            cmp EAX,0x8000;
            jge short L_largepositive;
            mov [ESP+8+8],AX;
            f2xm1; // 2ym1 = 2^^(y-rndint(y)) -1
            fld real ptr [ESP+8] ; // 2rndy = 2^^rndint(y)
            fmul ST(1), ST;  // ST=2rndy, ST(1)=2rndy*2ym1
            fld1;
            fsubp ST(1), ST; // ST = 2rndy-1, ST(1) = 2rndy * 2ym1 - 1
            faddp ST(1), ST; // ST = 2rndy * 2ym1 + 2rndy - 1
            add ESP,12+8;
            ret PARAMSIZE;

L_extreme:  // Extreme exponent. X is very large positive, very
            // large negative, infinity, or NaN.
            fxam;
            fstsw AX;
            test AX, 0x0400; // NaN_or_zero, but we already know x!=0
            jz L_was_nan;  // if x is NaN, returns x
            test AX, 0x0200;
            jnz L_largenegative;
L_largepositive:
            // Set scratchreal = real.max.
            // squaring it will create infinity, and set overflow flag.
            mov word  ptr [ESP+8+8], 0x7FFE;
            fstp ST(0);
            fld real ptr [ESP+8];  // load scratchreal
            fmul ST(0), ST;        // square it, to create havoc!
L_was_nan:
            add ESP,12+8;
            ret PARAMSIZE;
L_largenegative:
            fstp ST(0);
            fld1;
            fchs; // return -1. Underflow flag is not set.
            add ESP,12+8;
            ret PARAMSIZE;
        }
    } else version(D_InlineAsm_X86_64) {
        asm
        {
            /*  expm1() for x87 80-bit reals, IEEE754-2008 conformant.
             * Author: Don Clugston.
             *
             *    expm1(x) = 2^(rndint(y))* 2^(y-rndint(y)) - 1 where y = LN2*x.
             *    = 2rndy * 2ym1 + 2rndy - 1, where 2rndy = 2^(rndint(y))
             *     and 2ym1 = (2^(y-rndint(y))-1).
             *    If 2rndy  < 0.5*real.epsilon, result is -1.
             *    Implementation is otherwise the same as for exp2()
             */
            naked;
            fld real ptr [RSP+8] ; // x
            mov AX, [RSP+8+8]; // AX = exponent and sign
            sub RSP, 24;       // Create scratch space on the stack
            // [RSP,RSP+2] = scratchint
            // [RSP+4..+6, +8..+10, +10] = scratchreal
            // set scratchreal mantissa = 1.0
            mov dword ptr [RSP+8], 0;
            mov dword ptr [RSP+8+4], 0x80000000;
            and AX, 0x7FFF; // drop sign bit
            cmp AX, 0x401D; // avoid InvalidException in fist
            jae L_extreme;
            fldl2e;
            fmul ; // y = x*log2(e)
            fist dword ptr [RSP]; // scratchint = rndint(y)
            fisub dword ptr [RSP]; // y - rndint(y)
            // and now set scratchreal exponent
            mov EAX, [RSP];
            add EAX, 0x3fff;
            jle short L_largenegative;
            cmp EAX,0x8000;
            jge short L_largepositive;
            mov [RSP+8+8],AX;
            f2xm1; // 2^(y-rndint(y)) -1
            fld real ptr [RSP+8] ; // 2^rndint(y)
            fmul ST(1), ST;
            fld1;
            fsubp ST(1), ST;
            fadd;
            add RSP,24;
            ret;

L_extreme: // Extreme exponent. X is very large positive, very
            // large negative, infinity, or NaN.
            fxam;
            fstsw AX;
            test AX, 0x0400; // NaN_or_zero, but we already know x!=0
            jz L_was_nan;  // if x is NaN, returns x
            test AX, 0x0200;
            jnz L_largenegative;
L_largepositive:
            // Set scratchreal = real.max.
            // squaring it will create infinity, and set overflow flag.
            mov word  ptr [RSP+8+8], 0x7FFE;
            fstp ST(0);
            fld real ptr [RSP+8];  // load scratchreal
            fmul ST(0), ST;        // square it, to create havoc!
L_was_nan:
            add RSP,24;
            ret;

L_largenegative:
            fstp ST(0);
            fld1;
            fchs; // return -1. Underflow flag is not set.
            add RSP,24;
            ret;
        }
    } else {
        return core.stdc.math.expm1l(x);
    }
}



/**
 * Calculates 2$(SUP x).
 *
 *  $(TABLE_SV
 *    $(TR $(TH x)             $(TH exp2(x))   )
 *    $(TR $(TD +$(INFIN))     $(TD +$(INFIN)) )
 *    $(TR $(TD -$(INFIN))     $(TD +0.0)      )
 *    $(TR $(TD $(NAN))        $(TD $(NAN))    )
 *  )
 */
real exp2(real x) @trusted pure nothrow
{
    version(D_InlineAsm_X86) {
        enum { PARAMSIZE = (real.sizeof+3)&(0xFFFF_FFFC) } // always a multiple of 4
        asm {
            /*  exp2() for x87 80-bit reals, IEEE754-2008 conformant.
             * Author: Don Clugston.
             *
             * exp2(x) = 2^^(rndint(x))* 2^^(y-rndint(x))
             * The trick for high performance is to avoid the fscale(28cycles on core2),
             * frndint(19 cycles), leaving f2xm1(19 cycles) as the only slow instruction.
             *
             * We can do frndint by using fist. BUT we can't use it for huge numbers,
             * because it will set the Invalid Operation flag if overflow or NaN occurs.
             * Fortunately, whenever this happens the result would be zero or infinity.
             *
             * We can perform fscale by directly poking into the exponent. BUT this doesn't
             * work for the (very rare) cases where the result is subnormal. So we fall back
             * to the slow method in that case.
             */
            naked;
            fld real ptr [ESP+4] ; // x
            mov AX, [ESP+4+8]; // AX = exponent and sign
            sub ESP, 12+8; // Create scratch space on the stack
            // [ESP,ESP+2] = scratchint
            // [ESP+4..+6, +8..+10, +10] = scratchreal
            // set scratchreal mantissa = 1.0
            mov dword ptr [ESP+8], 0;
            mov dword ptr [ESP+8+4], 0x80000000;
            and AX, 0x7FFF; // drop sign bit
            cmp AX, 0x401D; // avoid InvalidException in fist
            jae L_extreme;
            fist dword ptr [ESP]; // scratchint = rndint(x)
            fisub dword ptr [ESP]; // x - rndint(x)
            // and now set scratchreal exponent
            mov EAX, [ESP];
            add EAX, 0x3fff;
            jle short L_subnormal;
            cmp EAX,0x8000;
            jge short L_overflow;
            mov [ESP+8+8],AX;
L_normal:
            f2xm1;
            fld1;
            faddp ST(1), ST; // 2^^(x-rndint(x))
            fld real ptr [ESP+8] ; // 2^^rndint(x)
            add ESP,12+8;
            fmulp ST(1), ST;
            ret PARAMSIZE;

L_subnormal:
            // Result will be subnormal.
            // In this rare case, the simple poking method doesn't work.
            // The speed doesn't matter, so use the slow fscale method.
            fild dword ptr [ESP];  // scratchint
            fld1;
            fscale;
            fstp real ptr [ESP+8]; // scratchreal = 2^^scratchint
            fstp ST(0);         // drop scratchint
            jmp L_normal;

L_extreme:  // Extreme exponent. X is very large positive, very
            // large negative, infinity, or NaN.
            fxam;
            fstsw AX;
            test AX, 0x0400; // NaN_or_zero, but we already know x!=0
            jz L_was_nan;  // if x is NaN, returns x
            // set scratchreal = real.min_normal
            // squaring it will return 0, setting underflow flag
            mov word  ptr [ESP+8+8], 1;
            test AX, 0x0200;
            jnz L_waslargenegative;
L_overflow:
            // Set scratchreal = real.max.
            // squaring it will create infinity, and set overflow flag.
            mov word  ptr [ESP+8+8], 0x7FFE;
L_waslargenegative:
            fstp ST(0);
            fld real ptr [ESP+8];  // load scratchreal
            fmul ST(0), ST;        // square it, to create havoc!
L_was_nan:
            add ESP,12+8;
            ret PARAMSIZE;
        }
    } else version(D_InlineAsm_X86_64) {
        asm {
            /*  exp2() for x87 80-bit reals, IEEE754-2008 conformant.
             * Author: Don Clugston.
             *
             * exp2(x) = 2^(rndint(x))* 2^(y-rndint(x))
             * The trick for high performance is to avoid the fscale(28cycles on core2),
             * frndint(19 cycles), leaving f2xm1(19 cycles) as the only slow instruction.
             *
             * We can do frndint by using fist. BUT we can't use it for huge numbers,
             * because it will set the Invalid Operation flag is overflow or NaN occurs.
             * Fortunately, whenever this happens the result would be zero or infinity.
             *
             * We can perform fscale by directly poking into the exponent. BUT this doesn't
             * work for the (very rare) cases where the result is subnormal. So we fall back
             * to the slow method in that case.
             */
            naked;
            fld real ptr [RSP+8] ; // x
            mov AX, [RSP+8+8]; // AX = exponent and sign
            sub RSP, 24; // Create scratch space on the stack
            // [RSP,RSP+2] = scratchint
            // [RSP+4..+6, +8..+10, +10] = scratchreal
            // set scratchreal mantissa = 1.0
            mov dword ptr [RSP+8], 0;
            mov dword ptr [RSP+8+4], 0x80000000;
            and AX, 0x7FFF; // drop sign bit
            cmp AX, 0x401D; // avoid InvalidException in fist
            jae L_extreme;
            fist dword ptr [RSP]; // scratchint = rndint(x)
            fisub dword ptr [RSP]; // x - rndint(x)
            // and now set scratchreal exponent
            mov EAX, [RSP];
            add EAX, 0x3fff;
            jle short L_subnormal;
            cmp EAX,0x8000;
            jge short L_overflow;
            mov [RSP+8+8],AX;
L_normal:
            f2xm1;
            fld1;
            fadd; // 2^(x-rndint(x))
            fld real ptr [RSP+8] ; // 2^rndint(x)
            add RSP,24;
            fmulp ST(1), ST;
            ret;

L_subnormal:
            // Result will be subnormal.
            // In this rare case, the simple poking method doesn't work.
            // The speed doesn't matter, so use the slow fscale method.
            fild dword ptr [RSP];  // scratchint
            fld1;
            fscale;
            fstp real ptr [RSP+8]; // scratchreal = 2^scratchint
            fstp ST(0);         // drop scratchint
            jmp L_normal;

L_extreme: // Extreme exponent. X is very large positive, very
            // large negative, infinity, or NaN.
            fxam;
            fstsw AX;
            test AX, 0x0400; // NaN_or_zero, but we already know x!=0
            jz L_was_nan;  // if x is NaN, returns x
            // set scratchreal = real.min
            // squaring it will return 0, setting underflow flag
            mov word  ptr [RSP+8+8], 1;
            test AX, 0x0200;
            jnz L_waslargenegative;
L_overflow:
            // Set scratchreal = real.max.
            // squaring it will create infinity, and set overflow flag.
            mov word  ptr [RSP+8+8], 0x7FFE;
L_waslargenegative:
            fstp ST(0);
            fld real ptr [RSP+8];  // load scratchreal
            fmul ST(0), ST;        // square it, to create havoc!
L_was_nan:
            add RSP,24;
            ret;
        }
    } else {
        return core.stdc.math.exp2l(x);
    }
}

unittest{
    assert(exp2(0.5L)== SQRT2);
    assert(exp2(8.0L) == 256.0);
    assert(exp2(-9.0L)== 1.0L/512.0);
    assert(exp(3.0L) == E*E*E);
}

unittest
{
    FloatingPointControl ctrl;
    ctrl.disableExceptions(FloatingPointControl.allExceptions);
    ctrl.rounding = FloatingPointControl.roundToNearest;

    // @@BUG@@: Non-immutable array literals are ridiculous.
    // Note that these are only valid for 80-bit reals: overflow will be different for 64-bit reals.
    static const real [2][] exptestpoints =
    [ // x,            exp(x)
        [1.0L,           E                           ],
        [0.5L,           0x1.A612_98E1_E069_BC97p+0L ],
        [3.0L,           E*E*E                       ],
        [0x1.1p13L,      0x1.29aeffefc8ec645p+12557L ], // near overflow
        [-0x1.18p13L,    0x1.5e4bf54b4806db9p-12927L ], // near underflow
        [-0x1.625p13L,   0x1.a6bd68a39d11f35cp-16358L],
        [-0x1p30L,       0                           ], // underflow - subnormal
        [-0x1.62DAFp13L, 0x1.96c53d30277021dp-16383L ],
        [-0x1.643p13L,   0x1p-16444L                 ],
        [-0x1.645p13L,   0                           ], // underflow to zero
        [0x1p80L,        real.infinity               ], // far overflow
        [real.infinity,  real.infinity               ],
        [0x1.7p13L,      real.infinity               ]  // close overflow
    ];
    real x;
    IeeeFlags f;
    for (int i=0; i<exptestpoints.length;++i) {
        resetIeeeFlags();
        x = exp(exptestpoints[i][0]);
        f = ieeeFlags;
        assert(x == exptestpoints[i][1]);
        // Check the overflow bit
        assert(f.overflow() == (fabs(x) == real.infinity));
        // Check the underflow bit
        assert(f.underflow() == (fabs(x) < real.min_normal));
        // Invalid and div by zero shouldn't be affected.
        assert(!f.invalid);
        assert(!f.divByZero);
    }
    // Ideally, exp(0) would not set the inexact flag.
    // Unfortunately, fldl2e sets it!
    // So it's not realistic to avoid setting it.
    assert(exp(0.0L) == 1.0);

    // NaN propagation. Doesn't set flags, bcos was already NaN.
    resetIeeeFlags();
    x = exp(real.nan);
    f = ieeeFlags;
    assert(isIdentical(x,real.nan));
    assert(f.flags == 0);

    resetIeeeFlags();
    x = exp(-real.nan);
    f = ieeeFlags;
    assert(isIdentical(x, -real.nan));
    assert(f.flags == 0);

    x = exp(NaN(0x123));
    assert(isIdentical(x, NaN(0x123)));

    // High resolution test
    assert(exp(0.5L) == 0x1.A612_98E1_E069_BC97_2DFE_FAB6D_33Fp+0L);

}


/**
 * Calculate cos(y) + i sin(y).
 *
 * On many CPUs (such as x86), this is a very efficient operation;
 * almost twice as fast as calculating sin(y) and cos(y) separately,
 * and is the preferred method when both are required.
 */
creal expi(real y) @trusted pure nothrow
{
    version(InlineAsm_X86_Any)
    {
        asm
        {
            fld y;
            fsincos;
            fxch ST(1), ST(0);
        }
    }
    else
    {
        return cos(y) + sin(y)*1i;
    }
}

unittest
{
    assert(expi(1.3e5L) == cos(1.3e5L) + sin(1.3e5L) * 1i);
    assert(expi(0.0L) == 1L + 0.0Li);
}

/*********************************************************************
 * Separate floating point value into significand and exponent.
 *
 * Returns:
 *      Calculate and return $(I x) and $(I exp) such that
 *      value =$(I x)*2$(SUP exp) and
 *      .5 $(LT)= |$(I x)| $(LT) 1.0
 *
 *      $(I x) has same sign as value.
 *
 *      $(TABLE_SV
 *      $(TR $(TH value)           $(TH returns)         $(TH exp))
 *      $(TR $(TD $(PLUSMN)0.0)    $(TD $(PLUSMN)0.0)    $(TD 0))
 *      $(TR $(TD +$(INFIN))       $(TD +$(INFIN))       $(TD int.max))
 *      $(TR $(TD -$(INFIN))       $(TD -$(INFIN))       $(TD int.min))
 *      $(TR $(TD $(PLUSMN)$(NAN)) $(TD $(PLUSMN)$(NAN)) $(TD int.min))
 *      )
 */
real frexp(real value, out int exp) @trusted pure nothrow
{
    ushort* vu = cast(ushort*)&value;
    long* vl = cast(long*)&value;
    uint ex;
    alias floatTraits!(real) F;

    ex = vu[F.EXPPOS_SHORT] & F.EXPMASK;
    static if (real.mant_dig == 64) { // real80
        if (ex) { // If exponent is non-zero
            if (ex == F.EXPMASK) {   // infinity or NaN
                if (*vl &  0x7FFF_FFFF_FFFF_FFFF) {  // NaN
                    *vl |= 0xC000_0000_0000_0000;  // convert NaNS to NaNQ
                    exp = int.min;
                } else if (vu[F.EXPPOS_SHORT] & 0x8000) {   // negative infinity
                    exp = int.min;
                } else {   // positive infinity
                    exp = int.max;
                }
            } else {
                exp = ex - F.EXPBIAS;
                vu[F.EXPPOS_SHORT] = (0x8000 & vu[F.EXPPOS_SHORT]) | 0x3FFE;
            }
        } else if (!*vl) {
            // value is +-0.0
            exp = 0;
        } else {
            // subnormal
            value *= F.RECIP_EPSILON;
            ex = vu[F.EXPPOS_SHORT] & F.EXPMASK;
            exp = ex - F.EXPBIAS - real.mant_dig + 1;
            vu[F.EXPPOS_SHORT] = (0x8000 & vu[F.EXPPOS_SHORT]) | 0x3FFE;
        }
    } else static if (real.mant_dig == 113) { // quadruple
        if (ex) { // If exponent is non-zero
            if (ex == F.EXPMASK) {   // infinity or NaN
                if (vl[MANTISSA_LSB] |
                    ( vl[MANTISSA_MSB] & 0x0000_FFFF_FFFF_FFFF)) {  // NaN
                    // convert NaNS to NaNQ
                    vl[MANTISSA_MSB] |= 0x0000_8000_0000_0000;
                    exp = int.min;
                } else if (vu[F.EXPPOS_SHORT] & 0x8000) {   // negative infinity
                    exp = int.min;
                } else {   // positive infinity
                    exp = int.max;
                }
            } else {
                exp = ex - F.EXPBIAS;
                vu[F.EXPPOS_SHORT] =
                    cast(ushort)((0x8000 & vu[F.EXPPOS_SHORT]) | 0x3FFE);
            }
        } else if ((vl[MANTISSA_LSB]
                       |(vl[MANTISSA_MSB] & 0x0000_FFFF_FFFF_FFFF)) == 0) {
            // value is +-0.0
            exp = 0;
        } else {
            // subnormal
            value *= F.RECIP_EPSILON;
            ex = vu[F.EXPPOS_SHORT] & F.EXPMASK;
            exp = ex - F.EXPBIAS - real.mant_dig + 1;
            vu[F.EXPPOS_SHORT] =
                cast(ushort)((0x8000 & vu[F.EXPPOS_SHORT]) | 0x3FFE);
        }
    } else static if (real.mant_dig==53) { // real is double
        if (ex) { // If exponent is non-zero
            if (ex == F.EXPMASK) {   // infinity or NaN
                if (*vl == 0x7FF0_0000_0000_0000) {  // positive infinity
                    exp = int.max;
                } else if (*vl == 0xFFF0_0000_0000_0000) { // negative infinity
                    exp = int.min;
                } else { // NaN
                    *vl |= 0x0008_0000_0000_0000;  // convert NaNS to NaNQ
                    exp = int.min;
                }
            } else {
                exp = (ex - F.EXPBIAS) >> 4;
                vu[F.EXPPOS_SHORT] = cast(ushort)((0x8000 & vu[F.EXPPOS_SHORT]) | 0x3FE0);
            }
        } else if (!(*vl & 0x7FFF_FFFF_FFFF_FFFF)) {
            // value is +-0.0
            exp = 0;
        } else {
            // subnormal
            value *= F.RECIP_EPSILON;
            ex = vu[F.EXPPOS_SHORT] & F.EXPMASK;
            exp = ((ex - F.EXPBIAS)>> 4) - real.mant_dig + 1;
            vu[F.EXPPOS_SHORT] =
                cast(ushort)((0x8000 & vu[F.EXPPOS_SHORT]) | 0x3FE0);
        }
    } else { //static if(real.mant_dig==106) // doubledouble
        assert (0, "frexp not implemented");
    }
    return value;
}


unittest
{
    static real vals[][3] =     // x,frexp,exp
        [
         [0.0,   0.0,    0],
         [-0.0,  -0.0,   0],
         [1.0,   .5,     1],
         [-1.0,  -.5,    1],
         [2.0,   .5,     2],
         [double.min_normal/2.0, .5, -1022],
         [real.infinity,real.infinity,int.max],
         [-real.infinity,-real.infinity,int.min],
         [real.nan,real.nan,int.min],
         [-real.nan,-real.nan,int.min],
         ];

    int i;

    for (i = 0; i < vals.length; i++) {
        real x = vals[i][0];
        real e = vals[i][1];
        int exp = cast(int)vals[i][2];
        int eptr;
        real v = frexp(x, eptr);
        assert(isIdentical(e, v));
        assert(exp == eptr);

    }
    static if (real.mant_dig == 64) {
        static real extendedvals[][3] = [ // x,frexp,exp
                                          [0x1.a5f1c2eb3fe4efp+73L, 0x1.A5F1C2EB3FE4EFp-1L,   74],    // normal
                                          [0x1.fa01712e8f0471ap-1064L,  0x1.fa01712e8f0471ap-1L,     -1063],
                                          [real.min_normal,  .5,     -16381],
                                          [real.min_normal/2.0L, .5,     -16382]    // subnormal
                                           ];

        for (i = 0; i < extendedvals.length; i++) {
            real x = extendedvals[i][0];
            real e = extendedvals[i][1];
            int exp = cast(int)extendedvals[i][2];
            int eptr;
            real v = frexp(x, eptr);
            assert(isIdentical(e, v));
            assert(exp == eptr);

        }
    }
}

/******************************************
 * Extracts the exponent of x as a signed integral value.
 *
 * If x is not a special value, the result is the same as
 * $(D cast(int)logb(x)).
 *
 *      $(TABLE_SV
 *      $(TR $(TH x)                $(TH ilogb(x))     $(TH Range error?))
 *      $(TR $(TD 0)                 $(TD FP_ILOGB0)   $(TD yes))
 *      $(TR $(TD $(PLUSMN)$(INFIN)) $(TD int.max)     $(TD no))
 *      $(TR $(TD $(NAN))            $(TD FP_ILOGBNAN) $(TD no))
 *      )
 */
int ilogb(real x)  @trusted nothrow    { return core.stdc.math.ilogbl(x); }

alias core.stdc.math.FP_ILOGB0   FP_ILOGB0;
alias core.stdc.math.FP_ILOGBNAN FP_ILOGBNAN;


/*******************************************
 * Compute n * 2$(SUP exp)
 * References: frexp
 */

real ldexp(real n, int exp) @safe pure nothrow;    /* intrinsic */

unittest {
    assert(ldexp(1, -16384) == 0x1p-16384L);
    assert(ldexp(1, -16382) == 0x1p-16382L);
    int x;
    real n = frexp(0x1p-16384L, x);
    assert(n==0.5L);
    assert(x==-16383);
    assert(ldexp(n, x)==0x1p-16384L);

}

/**************************************
 * Calculate the natural logarithm of x.
 *
 *    $(TABLE_SV
 *    $(TR $(TH x)            $(TH log(x))    $(TH divide by 0?) $(TH invalid?))
 *    $(TR $(TD $(PLUSMN)0.0) $(TD -$(INFIN)) $(TD yes)          $(TD no))
 *    $(TR $(TD $(LT)0.0)     $(TD $(NAN))    $(TD no)           $(TD yes))
 *    $(TR $(TD +$(INFIN))    $(TD +$(INFIN)) $(TD no)           $(TD no))
 *    )
 */

real log(real x) @safe pure nothrow
{
    version (INLINE_YL2X)
        return yl2x(x, LN2);
    else
        return core.stdc.math.logl(x);
}

unittest
{
    assert(log(E) == 1);
}

/**************************************
 * Calculate the base-10 logarithm of x.
 *
 *      $(TABLE_SV
 *      $(TR $(TH x)            $(TH log10(x))  $(TH divide by 0?) $(TH invalid?))
 *      $(TR $(TD $(PLUSMN)0.0) $(TD -$(INFIN)) $(TD yes)          $(TD no))
 *      $(TR $(TD $(LT)0.0)     $(TD $(NAN))    $(TD no)           $(TD yes))
 *      $(TR $(TD +$(INFIN))    $(TD +$(INFIN)) $(TD no)           $(TD no))
 *      )
 */

real log10(real x) @safe pure nothrow
{
    version (INLINE_YL2X)
        return yl2x(x, LOG2);
    else
        return core.stdc.math.log10l(x);
}

unittest
{
    //printf("%Lg\n", log10(1000) - 3);
    assert(fabs(log10(1000) - 3) < .000001);
}

/******************************************
 *      Calculates the natural logarithm of 1 + x.
 *
 *      For very small x, log1p(x) will be more accurate than
 *      log(1 + x).
 *
 *  $(TABLE_SV
 *  $(TR $(TH x)            $(TH log1p(x))     $(TH divide by 0?) $(TH invalid?))
 *  $(TR $(TD $(PLUSMN)0.0) $(TD $(PLUSMN)0.0) $(TD no)           $(TD no))
 *  $(TR $(TD -1.0)         $(TD -$(INFIN))    $(TD yes)          $(TD no))
 *  $(TR $(TD $(LT)-1.0)    $(TD $(NAN))       $(TD no)           $(TD yes))
 *  $(TR $(TD +$(INFIN))    $(TD -$(INFIN))    $(TD no)           $(TD no))
 *  )
 */

real log1p(real x) @safe pure nothrow
{
    version(INLINE_YL2X)
    {
        // On x87, yl2xp1 is valid if and only if -0.5 <= lg(x) <= 0.5,
        //    ie if -0.29<=x<=0.414
        return (fabs(x) <= 0.25)  ? yl2xp1(x, LN2) : yl2x(x+1, LN2);
    }
    else
    {
        return core.stdc.math.log1pl(x);
    }
}

/***************************************
 * Calculates the base-2 logarithm of x:
 * $(SUB log, 2)x
 *
 *  $(TABLE_SV
 *  $(TR $(TH x)            $(TH log2(x))   $(TH divide by 0?) $(TH invalid?))
 *  $(TR $(TD $(PLUSMN)0.0) $(TD -$(INFIN)) $(TD yes)          $(TD no) )
 *  $(TR $(TD $(LT)0.0)     $(TD $(NAN))    $(TD no)           $(TD yes) )
 *  $(TR $(TD +$(INFIN))    $(TD +$(INFIN)) $(TD no)           $(TD no) )
 *  )
 */
real log2(real x) @safe pure nothrow
{
    version (INLINE_YL2X)
        return yl2x(x, 1);
    else
        return core.stdc.math.log2l(x);
}

/*****************************************
 * Extracts the exponent of x as a signed integral value.
 *
 * If x is subnormal, it is treated as if it were normalized.
 * For a positive, finite x:
 *
 * 1 $(LT)= $(I x) * FLT_RADIX$(SUP -logb(x)) $(LT) FLT_RADIX
 *
 *      $(TABLE_SV
 *      $(TR $(TH x)                 $(TH logb(x))   $(TH divide by 0?) )
 *      $(TR $(TD $(PLUSMN)$(INFIN)) $(TD +$(INFIN)) $(TD no))
 *      $(TR $(TD $(PLUSMN)0.0)      $(TD -$(INFIN)) $(TD yes) )
 *      )
 */
real logb(real x) @trusted nothrow    { return core.stdc.math.logbl(x); }

/************************************
 * Calculates the remainder from the calculation x/y.
 * Returns:
 * The value of x - i * y, where i is the number of times that y can
 * be completely subtracted from x. The result has the same sign as x.
 *
 * $(TABLE_SV
 *  $(TR $(TH x)              $(TH y)             $(TH fmod(x, y))   $(TH invalid?))
 *  $(TR $(TD $(PLUSMN)0.0)   $(TD not 0.0)       $(TD $(PLUSMN)0.0) $(TD no))
 *  $(TR $(TD $(PLUSMNINF))   $(TD anything)      $(TD $(NAN))       $(TD yes))
 *  $(TR $(TD anything)       $(TD $(PLUSMN)0.0)  $(TD $(NAN))       $(TD yes))
 *  $(TR $(TD !=$(PLUSMNINF)) $(TD $(PLUSMNINF))  $(TD x)            $(TD no))
 * )
 */
real fmod(real x, real y) @trusted nothrow { return core.stdc.math.fmodl(x, y); }

/************************************
 * Breaks x into an integral part and a fractional part, each of which has
 * the same sign as x. The integral part is stored in i.
 * Returns:
 * The fractional part of x.
 *
 * $(TABLE_SV
 *  $(TR $(TH x)              $(TH i (on input))  $(TH modf(x, i))   $(TH i (on return)))
 *  $(TR $(TD $(PLUSMNINF))   $(TD anything)      $(TD $(PLUSMN)0.0) $(TD $(PLUSMNINF)))
 * )
 */
real modf(real x, ref real i) @trusted nothrow { return core.stdc.math.modfl(x,&i); }

/*************************************
 * Efficiently calculates x * 2$(SUP n).
 *
 * scalbn handles underflow and overflow in
 * the same fashion as the basic arithmetic operators.
 *
 *      $(TABLE_SV
 *      $(TR $(TH x)                 $(TH scalb(x)))
 *      $(TR $(TD $(PLUSMNINF))      $(TD $(PLUSMNINF)) )
 *      $(TR $(TD $(PLUSMN)0.0)      $(TD $(PLUSMN)0.0) )
 *      )
 */
real scalbn(real x, int n) @trusted nothrow
{
    version(InlineAsm_X86_Any) {
        // scalbnl is not supported on DMD-Windows, so use asm.
        asm {
            fild n;
            fld x;
            fscale;
            fstp ST(1);
        }
    } else {
        return core.stdc.math.scalbnl(x, n);
    }
}

unittest {
    assert(scalbn(-real.infinity, 5) == -real.infinity);
}

/***************
 * Calculates the cube root of x.
 *
 *      $(TABLE_SV
 *      $(TR $(TH $(I x))            $(TH cbrt(x))           $(TH invalid?))
 *      $(TR $(TD $(PLUSMN)0.0)      $(TD $(PLUSMN)0.0)      $(TD no) )
 *      $(TR $(TD $(NAN))            $(TD $(NAN))            $(TD yes) )
 *      $(TR $(TD $(PLUSMN)$(INFIN)) $(TD $(PLUSMN)$(INFIN)) $(TD no) )
 *      )
 */
real cbrt(real x) @trusted nothrow    { return core.stdc.math.cbrtl(x); }


/*******************************
 * Returns |x|
 *
 *      $(TABLE_SV
 *      $(TR $(TH x)                 $(TH fabs(x)))
 *      $(TR $(TD $(PLUSMN)0.0)      $(TD +0.0) )
 *      $(TR $(TD $(PLUSMN)$(INFIN)) $(TD +$(INFIN)) )
 *      )
 */
real fabs(real x) @safe pure nothrow;      /* intrinsic */


/***********************************************************************
 * Calculates the length of the
 * hypotenuse of a right-angled triangle with sides of length x and y.
 * The hypotenuse is the value of the square root of
 * the sums of the squares of x and y:
 *
 *      sqrt($(POWER x, 2) + $(POWER y, 2))
 *
 * Note that hypot(x, y), hypot(y, x) and
 * hypot(x, -y) are equivalent.
 *
 *  $(TABLE_SV
 *  $(TR $(TH x)            $(TH y)            $(TH hypot(x, y)) $(TH invalid?))
 *  $(TR $(TD x)            $(TD $(PLUSMN)0.0) $(TD |x|)         $(TD no))
 *  $(TR $(TD $(PLUSMNINF)) $(TD y)            $(TD +$(INFIN))   $(TD no))
 *  $(TR $(TD $(PLUSMNINF)) $(TD $(NAN))       $(TD +$(INFIN))   $(TD no))
 *  )
 */

real hypot(real x, real y) @safe pure nothrow
{
    // Scale x and y to avoid underflow and overflow.
    // If one is huge and the other tiny, return the larger.
    // If both are huge, avoid overflow by scaling by 1/sqrt(real.max/2).
    // If both are tiny, avoid underflow by scaling by sqrt(real.min_normal*real.epsilon).

    enum real SQRTMIN = 0.5*sqrt(real.min_normal); // This is a power of 2.
    enum real SQRTMAX = 1.0L/SQRTMIN; // 2^^((max_exp)/2) = nextUp(sqrt(real.max))

    static assert(2*(SQRTMAX/2)*(SQRTMAX/2) <= real.max);
    static assert(real.min_normal*real.max>2 && real.min_normal*real.max<=4); // Proves that sqrt(real.max) ~~  0.5/sqrt(real.min_normal)

    real u = fabs(x);
    real v = fabs(y);
    if (u !>= v)  // check for NaN as well.
    {
        v = u;
        u = fabs(y);
        if (u == real.infinity) return u; // hypot(inf, nan) == inf
        if (v == real.infinity) return v; // hypot(nan, inf) == inf
    }
    // Now u >= v, or else one is NaN.
    if (v >= SQRTMAX*0.5)
    {
            // hypot(huge, huge) -- avoid overflow
        u *= SQRTMIN*0.5;
        v *= SQRTMIN*0.5;
        return sqrt(u*u + v*v) * SQRTMAX * 2.0;
    }
    if (u <= SQRTMIN)
    {
        // hypot (tiny, tiny) -- avoid underflow
        // This is only necessary to avoid setting the underflow
        // flag.
        u *= SQRTMAX / real.epsilon;
        v *= SQRTMAX / real.epsilon;
        return sqrt(u*u + v*v) * SQRTMIN * real.epsilon;
    }
    if (u * real.epsilon > v)
    {
        // hypot (huge, tiny) = huge
        return u;
    }

    // both are in the normal range
    return sqrt(u*u + v*v);
}

unittest
{
    static real vals[][3] =     // x,y,hypot
        [
            [ 0.0,     0.0,   0.0],
            [ 0.0,    -0.0,   0.0],
            [ -0.0,   -0.0,   0.0],
            [ 3.0,     4.0,   5.0],
            [ -300,   -400,   500],
            [0.0,      7.0,   7.0],
            [9.0,   9*real.epsilon,   9.0],
            [88/(64*sqrt(real.min_normal)), 105/(64*sqrt(real.min_normal)), 137/(64*sqrt(real.min_normal))],
            [88/(128*sqrt(real.min_normal)), 105/(128*sqrt(real.min_normal)), 137/(128*sqrt(real.min_normal))],
            [3*real.min_normal*real.epsilon, 4*real.min_normal*real.epsilon, 5*real.min_normal*real.epsilon],
            [ real.min_normal, real.min_normal, sqrt(2.0L)*real.min_normal],
            [ real.max/sqrt(2.0L), real.max/sqrt(2.0L), real.max],
            [ real.infinity, real.nan, real.infinity],
            [ real.nan, real.infinity, real.infinity],
            [ real.nan, real.nan, real.nan],
            [ real.nan, real.max, real.nan],
            [ real.max, real.nan, real.nan],
        ];
        for (int i = 0; i < vals.length; i++)
        {
            real x = vals[i][0];
            real y = vals[i][1];
            real z = vals[i][2];
            real h = hypot(x, y);
            assert(isIdentical(z, h));
        }
}

/**************************************
 * Returns the value of x rounded upward to the next integer
 * (toward positive infinity).
 */
real ceil(real x)  @trusted nothrow    { return core.stdc.math.ceill(x); }

/**************************************
 * Returns the value of x rounded downward to the next integer
 * (toward negative infinity).
 */
real floor(real x) @trusted nothrow    { return core.stdc.math.floorl(x); }

/******************************************
 * Rounds x to the nearest integer value, using the current rounding
 * mode.
 *
 * Unlike the rint functions, nearbyint does not raise the
 * FE_INEXACT exception.
 */
real nearbyint(real x) @trusted nothrow { return core.stdc.math.nearbyintl(x); }

/**********************************
 * Rounds x to the nearest integer value, using the current rounding
 * mode.
 * If the return value is not equal to x, the FE_INEXACT
 * exception is raised.
 * $(B nearbyint) performs
 * the same operation, but does not set the FE_INEXACT exception.
 */
real rint(real x) @safe pure nothrow;      /* intrinsic */

/***************************************
 * Rounds x to the nearest integer value, using the current rounding
 * mode.
 *
 * This is generally the fastest method to convert a floating-point number
 * to an integer. Note that the results from this function
 * depend on the rounding mode, if the fractional part of x is exactly 0.5.
 * If using the default rounding mode (ties round to even integers)
 * lrint(4.5) == 4, lrint(5.5)==6.
 */
long lrint(real x) @trusted pure nothrow
{
    version(InlineAsm_X86_Any)
    {
        long n;
        asm
        {
            fld x;
            fistp n;
        }
        return n;
    } else {
        return core.stdc.math.llrintl(x);
    }
}

/*******************************************
 * Return the value of x rounded to the nearest integer.
 * If the fractional part of x is exactly 0.5, the return value is rounded to
 * the even integer.
 */
real round(real x) @trusted nothrow { return core.stdc.math.roundl(x); }

/**********************************************
 * Return the value of x rounded to the nearest integer.
 *
 * If the fractional part of x is exactly 0.5, the return value is rounded
 * away from zero.
 */
long lround(real x) @trusted nothrow
{
    version (Posix)
        return core.stdc.math.llroundl(x);
    else
        assert (0, "lround not implemented");
}

version(Posix)
{
    unittest
    {
        assert(lround(0.49) == 0);
        assert(lround(0.5) == 1);
        assert(lround(1.5) == 2);
    }
}

/****************************************************
 * Returns the integer portion of x, dropping the fractional portion.
 *
 * This is also known as "chop" rounding.
 */
real trunc(real x) @trusted nothrow { return core.stdc.math.truncl(x); }

/****************************************************
 * Calculate the remainder x REM y, following IEC 60559.
 *
 * REM is the value of x - y * n, where n is the integer nearest the exact
 * value of x / y.
 * If |n - x / y| == 0.5, n is even.
 * If the result is zero, it has the same sign as x.
 * Otherwise, the sign of the result is the sign of x / y.
 * Precision mode has no effect on the remainder functions.
 *
 * remquo returns n in the parameter n.
 *
 * $(TABLE_SV
 *  $(TR $(TH x)               $(TH y)            $(TH remainder(x, y)) $(TH n)   $(TH invalid?))
 *  $(TR $(TD $(PLUSMN)0.0)    $(TD not 0.0)      $(TD $(PLUSMN)0.0)    $(TD 0.0) $(TD no))
 *  $(TR $(TD $(PLUSMNINF))    $(TD anything)     $(TD $(NAN))          $(TD ?)   $(TD yes))
 *  $(TR $(TD anything)        $(TD $(PLUSMN)0.0) $(TD $(NAN))          $(TD ?)   $(TD yes))
 *  $(TR $(TD != $(PLUSMNINF)) $(TD $(PLUSMNINF)) $(TD x)               $(TD ?)   $(TD no))
 * )
 *
 * Note: remquo not supported on windows
 */
real remainder(real x, real y) @trusted nothrow { return core.stdc.math.remainderl(x, y); }

real remquo(real x, real y, out int n) @trusted nothrow  /// ditto
{
    version (Posix)
        return core.stdc.math.remquol(x, y, &n);
    else
        assert (0, "remquo not implemented");
}

/** IEEE exception status flags ('sticky bits')

 These flags indicate that an exceptional floating-point condition has occurred.
 They indicate that a NaN or an infinity has been generated, that a result
 is inexact, or that a signalling NaN has been encountered. If floating-point
 exceptions are enabled (unmasked), a hardware exception will be generated
 instead of setting these flags.

 Example:
 ----
    real a=3.5;
    // Set all the flags to zero
    resetIeeeFlags();
    assert(!ieeeFlags.divByZero);
    // Perform a division by zero.
    a/=0.0L;
    assert(a==real.infinity);
    assert(ieeeFlags.divByZero);
    // Create a NaN
    a*=0.0L;
    assert(ieeeFlags.invalid);
    assert(isNaN(a));

    // Check that calling func() has no effect on the
    // status flags.
    IeeeFlags f = ieeeFlags;
    func();
    assert(ieeeFlags == f);

 ----
 */
struct IeeeFlags
{
private:
    // The x87 FPU status register is 16 bits.
    // The Pentium SSE2 status register is 32 bits.
    uint flags;
    version (X86_Any) {
        // Applies to both x87 status word (16 bits) and SSE2 status word(32 bits).
        enum : int {
            INEXACT_MASK   = 0x20,
            UNDERFLOW_MASK = 0x10,
            OVERFLOW_MASK  = 0x08,
            DIVBYZERO_MASK = 0x04,
            INVALID_MASK   = 0x01
        }
        // Don't bother about subnormals, they are not supported on most CPUs.
        //  SUBNORMAL_MASK = 0x02;
    } else version (PPC) {
        // PowerPC FPSCR is a 32-bit register.
        enum : int {
            INEXACT_MASK   = 0x600,
            UNDERFLOW_MASK = 0x010,
            OVERFLOW_MASK  = 0x008,
            DIVBYZERO_MASK = 0x020,
            INVALID_MASK   = 0xF80 // PowerPC has five types of invalid exceptions.
        }
    } else version (ARM) {
        // TODO: Fill this in for VFP.
    } else version(SPARC) { // SPARC FSR is a 32bit register
             //(64 bits for Sparc 7 & 8, but high 32 bits are uninteresting).
        enum : int {
            INEXACT_MASK   = 0x020,
            UNDERFLOW_MASK = 0x080,
            OVERFLOW_MASK  = 0x100,
            DIVBYZERO_MASK = 0x040,
            INVALID_MASK   = 0x200
        }
    } else
        static assert(0, "Not implemented");
private:
    static uint getIeeeFlags()
    {
        version(D_InlineAsm_X86) {
            asm {
                 fstsw AX;
                 // NOTE: If compiler supports SSE2, need to OR the result with
                 // the SSE2 status register.
                 // Clear all irrelevant bits
                 and EAX, 0x03D;
            }
        } else version(D_InlineAsm_X86_64) {
            asm {
                 fstsw AX;
                 // NOTE: If compiler supports SSE2, need to OR the result with
                 // the SSE2 status register.
                 // Clear all irrelevant bits
                 and RAX, 0x03D;
            }
        } else version (SPARC) {
           /*
               int retval;
               asm { st %fsr, retval; }
               return retval;
            */
           assert(0, "Not yet supported");
        } else version (ARM) {
            assert(false, "Not yet supported.");
        } else
            assert(0, "Not yet supported");
    }
    static void resetIeeeFlags()
    {
        version(InlineAsm_X86_Any) {
            asm {
                fnclex;
            }
        } else {
            /* SPARC:
              int tmpval;
              asm { st %fsr, tmpval; }
              tmpval &=0xFFFF_FC00;
              asm { ld tmpval, %fsr; }
            */
           assert(0, "Not yet supported");
        }
    }
public:
     version (X86_Any) { // TODO: Lift this version condition when we support !x86.
     /// The result cannot be represented exactly, so rounding occured.
     /// (example: x = sin(0.1); )
     @property bool inexact() { return (flags & INEXACT_MASK) != 0; }
     /// A zero was generated by underflow (example: x = real.min*real.epsilon/2;)
     @property bool underflow() { return (flags & UNDERFLOW_MASK) != 0; }
     /// An infinity was generated by overflow (example: x = real.max*2;)
     @property bool overflow() { return (flags & OVERFLOW_MASK) != 0; }
     /// An infinity was generated by division by zero (example: x = 3/0.0; )
     @property bool divByZero() { return (flags & DIVBYZERO_MASK) != 0; }
     /// A machine NaN was generated. (example: x = real.infinity * 0.0; )
     @property bool invalid() { return (flags & INVALID_MASK) != 0; }
     }
}


/// Set all of the floating-point status flags to false.
void resetIeeeFlags() { IeeeFlags.resetIeeeFlags(); }

/// Return a snapshot of the current state of the floating-point status flags.
@property IeeeFlags ieeeFlags()
{
   return IeeeFlags(IeeeFlags.getIeeeFlags());
}

/** Control the Floating point hardware

  Change the IEEE754 floating-point rounding mode and the floating-point
  hardware exceptions.

  By default, the rounding mode is roundToNearest and all hardware exceptions
  are disabled. For most applications, debugging is easier if the $(I division
  by zero), $(I overflow), and $(I invalid operation) exceptions are enabled.
  These three are combined into a $(I severeExceptions) value for convenience.
  Note in particular that if $(I invalidException) is enabled, a hardware trap
  will be generated whenever an uninitialized floating-point variable is used.

  All changes are temporary. The previous state is restored at the
  end of the scope.


Example:
 ----
  {
    // Enable hardware exceptions for division by zero, overflow to infinity,
    // invalid operations, and uninitialized floating-point variables.

    FloatingPointControl fpctrl;
    fpctrl.enableExceptions(FloatingPointControl.severeExceptions);

    double y = x*3.0; // will generate a hardware exception, if x is uninitialized.
    //
    fpctrl.rounding = FloatingPointControl.roundUp;

    // The hardware exceptions will be disabled when leaving this scope.
    // The original rounding mode will also be restored.
  }

 ----

 */
struct FloatingPointControl
{
    alias uint RoundingMode;

    /** IEEE rounding modes.
     * The default mode is roundToNearest.
     */
    enum : RoundingMode
    {
        roundToNearest = 0x0000,
        roundDown      = 0x0400,
        roundUp        = 0x0800,
        roundToZero    = 0x0C00
    };
    /** IEEE hardware exceptions.
     *  By default, all exceptions are masked (disabled).
     */
    enum : uint
    {
        inexactException      = 0x20,
        underflowException    = 0x10,
        overflowException     = 0x08,
        divByZeroException    = 0x04,
        subnormalException    = 0x02,
        invalidException      = 0x01,
        /// Severe = The overflow, division by zero, and invalid exceptions.
        severeExceptions   = overflowException | divByZeroException
                             | invalidException,
        allExceptions      = severeExceptions | underflowException
                             | inexactException | subnormalException,
    };
private:
    enum ushort EXCEPTION_MASK = 0x3F;
    enum ushort ROUNDING_MASK = 0xC00;
public:
    /// Enable (unmask) specific hardware exceptions. Multiple exceptions may be ORed together.
    void enableExceptions(uint exceptions)
    {
        initialize();
        setControlState(getControlState() & ~(exceptions & EXCEPTION_MASK));
    }
    /// Disable (mask) specific hardware exceptions. Multiple exceptions may be ORed together.
    void disableExceptions(uint exceptions)
    {
        initialize();
        setControlState(getControlState() | (exceptions & EXCEPTION_MASK));
    }
    //// Change the floating-point hardware rounding mode
    @property void rounding(RoundingMode newMode)
    {
        ushort old = getControlState();
        setControlState((old & ~ROUNDING_MASK) | (newMode & ROUNDING_MASK));
    }
    /// Return the exceptions which are currently enabled (unmasked)
    @property static uint enabledExceptions()
    {
        return (getControlState() & EXCEPTION_MASK) ^ EXCEPTION_MASK;
    }
    /// Return the currently active rounding mode
    @property static RoundingMode rounding()
    {
        return cast(RoundingMode)(getControlState() & ROUNDING_MASK);
    }
    ///  Clear all pending exceptions, then restore the original exception state and rounding mode.
    ~this()
    {
        clearExceptions();
        setControlState(savedState);
    }
private:
    ushort savedState;

    bool initialized=false;
    void initialize()
    {
        // BUG: This works around the absence of this() constructors.
        if (initialized) return;
        clearExceptions();
        savedState = getControlState();
        initialized=true;
    }
    // Clear all pending exceptions
    static void clearExceptions()
    {
        version (InlineAsm_X86_Any)
        {
            asm
            {
                fclex;
            }
        }
        else
            assert(0, "Not yet supported");
    }
    // Read from the control register
    static ushort getControlState()
    {
        version (D_InlineAsm_X86)
        {
            short cont;
            asm
            {
                xor EAX, EAX;
                fstcw cont;
            }
            return cont;
        }
        else
        version (D_InlineAsm_X86_64)
        {
            short cont;
            asm
            {
                xor RAX, RAX;
                fstcw cont;
            }
            return cont;
        }
        else
            assert(0, "Not yet supported");
    }
    // Set the control register
    static void setControlState(ushort newState)
    {
        version (InlineAsm_X86_Any)
        {
            asm
            {
                 fclex;
                 fldcw newState;
            }
        }
        else
            assert(0, "Not yet supported");
    }
}

unittest
{
   {
        FloatingPointControl ctrl;
        ctrl.enableExceptions(FloatingPointControl.divByZeroException
                           | FloatingPointControl.overflowException);
        assert(ctrl.enabledExceptions() ==
            (FloatingPointControl.divByZeroException
          | FloatingPointControl.overflowException));

        ctrl.rounding = FloatingPointControl.roundUp;
        assert(FloatingPointControl.rounding == FloatingPointControl.roundUp);
    }
    assert(FloatingPointControl.rounding
       == FloatingPointControl.roundToNearest);
    assert(FloatingPointControl.enabledExceptions() ==0);
}


/*********************************
 * Returns !=0 if e is a NaN.
 */

bool isNaN(real x) @trusted pure nothrow
{
    alias floatTraits!(real) F;
    static if (real.mant_dig==53) { // double
        ulong*  p = cast(ulong *)&x;
        return ((*p & 0x7FF0_0000_0000_0000) == 0x7FF0_0000_0000_0000)
        && *p & 0x000F_FFFF_FFFF_FFFF;
    } else static if (real.mant_dig==64) {     // real80
        ushort e = F.EXPMASK & (cast(ushort *)&x)[F.EXPPOS_SHORT];
        ulong*  ps = cast(ulong *)&x;
        return e == F.EXPMASK &&
        *ps & 0x7FFF_FFFF_FFFF_FFFF; // not infinity
    } else static if (real.mant_dig==113) {  // quadruple
        ushort e = F.EXPMASK & (cast(ushort *)&x)[F.EXPPOS_SHORT];
        ulong*  ps = cast(ulong *)&x;
        return e == F.EXPMASK &&
        (ps[MANTISSA_LSB] | (ps[MANTISSA_MSB]& 0x0000_FFFF_FFFF_FFFF))!=0;
    } else {
        return x!=x;
    }
}


unittest
{
    assert(isNaN(float.nan));
    assert(isNaN(-double.nan));
    assert(isNaN(real.nan));

    assert(!isNaN(53.6));
    assert(!isNaN(float.infinity));
}

/*********************************
 * Returns !=0 if e is finite (not infinite or $(NAN)).
 */

int isFinite(real e) @trusted pure nothrow
{
    alias floatTraits!(real) F;
    ushort* pe = cast(ushort *)&e;
    return (pe[F.EXPPOS_SHORT] & F.EXPMASK) != F.EXPMASK;
}

unittest
{
    assert(isFinite(1.23));
    assert(!isFinite(double.infinity));
    assert(!isFinite(float.nan));
}


/*********************************
 * Returns !=0 if x is normalized (not zero, subnormal, infinite, or $(NAN)).
 */

/* Need one for each format because subnormal floats might
 * be converted to normal reals.
 */

int isNormal(X)(X x) @trusted pure nothrow
{
    alias floatTraits!(X) F;

    static if(real.mant_dig==106) { // doubledouble
        // doubledouble is normal if the least significant part is normal.
        return isNormal((cast(double*)&x)[MANTISSA_LSB]);
    } else {
        ushort e = F.EXPMASK & (cast(ushort *)&x)[F.EXPPOS_SHORT];
        return (e != F.EXPMASK && e!=0);
    }
}


unittest
{
    float f = 3;
    double d = 500;
    real e = 10e+48;

    assert(isNormal(f));
    assert(isNormal(d));
    assert(isNormal(e));
    f = d = e = 0;
    assert(!isNormal(f));
    assert(!isNormal(d));
    assert(!isNormal(e));
    assert(!isNormal(real.infinity));
    assert(isNormal(-real.max));
    assert(!isNormal(real.min_normal/4));

}

/*********************************
 * Is number subnormal? (Also called "denormal".)
 * Subnormals have a 0 exponent and a 0 most significant mantissa bit.
 */

/* Need one for each format because subnormal floats might
 * be converted to normal reals.
 */

int isSubnormal(float f) @trusted pure nothrow
{
    uint *p = cast(uint *)&f;
    return (*p & 0x7F80_0000) == 0 && *p & 0x007F_FFFF;
}

unittest
{
    float f = 3.0;

    for (f = 1.0; !isSubnormal(f); f /= 2)
        assert(f != 0);
}

/// ditto

int isSubnormal(double d) @trusted pure nothrow
{
    uint *p = cast(uint *)&d;
    return (p[MANTISSA_MSB] & 0x7FF0_0000) == 0
        && (p[MANTISSA_LSB] || p[MANTISSA_MSB] & 0x000F_FFFF);
}

unittest
{
    double f;

    for (f = 1; !isSubnormal(f); f /= 2)
        assert(f != 0);
}

/// ditto

int isSubnormal(real x) @trusted pure nothrow
{
    alias floatTraits!(real) F;
    static if (real.mant_dig == 53) { // double
        return isSubnormal(cast(double)x);
    } else static if (real.mant_dig == 113) { // quadruple
        ushort e = F.EXPMASK & (cast(ushort *)&x)[F.EXPPOS_SHORT];
        long*   ps = cast(long *)&x;
        return (e == 0 &&
          (((ps[MANTISSA_LSB]|(ps[MANTISSA_MSB]& 0x0000_FFFF_FFFF_FFFF))) !=0));
    } else static if (real.mant_dig==64) { // real80
        ushort* pe = cast(ushort *)&x;
        long*   ps = cast(long *)&x;

        return (pe[F.EXPPOS_SHORT] & F.EXPMASK) == 0 && *ps > 0;
    } else { // double double
        return isSubnormal((cast(double*)&x)[MANTISSA_MSB]);
    }
}

unittest
{
    real f;

    for (f = 1; !isSubnormal(f); f /= 2)
        assert(f != 0);
}

/*********************************
 * Return !=0 if e is $(PLUSMN)$(INFIN).
 */

bool isInfinity(real x) @trusted pure nothrow
{
    alias floatTraits!(real) F;
    static if (real.mant_dig == 53) { // double
        return ((*cast(ulong *)&x) & 0x7FFF_FFFF_FFFF_FFFF)
            == 0x7FF8_0000_0000_0000;
    } else static if(real.mant_dig == 106) { //doubledouble
        return (((cast(ulong *)&x)[MANTISSA_MSB]) & 0x7FFF_FFFF_FFFF_FFFF)
            == 0x7FF8_0000_0000_0000;
    } else static if (real.mant_dig == 113) { // quadruple
        long*   ps = cast(long *)&x;
        return (ps[MANTISSA_LSB] == 0)
            && (ps[MANTISSA_MSB] & 0x7FFF_FFFF_FFFF_FFFF) == 0x7FFF_0000_0000_0000;
    } else { // real80
        ushort e = cast(ushort)(F.EXPMASK & (cast(ushort *)&x)[F.EXPPOS_SHORT]);
        ulong*  ps = cast(ulong *)&x;
        // On Motorola 68K, infinity can have hidden bit=1 or 0. On x86, it is always 1.
        return e == F.EXPMASK && (*ps & 0x7FFF_FFFF_FFFF_FFFF) == 0;
    }
}

unittest
{
    assert(isInfinity(float.infinity));
    assert(!isInfinity(float.nan));
    assert(isInfinity(double.infinity));
    assert(isInfinity(-real.infinity));

    assert(isInfinity(-1.0 / 0.0));
}

/*********************************
 * Is the binary representation of x identical to y?
 *
 * Same as ==, except that positive and negative zero are not identical,
 * and two $(NAN)s are identical if they have the same 'payload'.
 */

bool isIdentical(real x, real y) @trusted pure nothrow
{
    // We're doing a bitwise comparison so the endianness is irrelevant.
    long*   pxs = cast(long *)&x;
    long*   pys = cast(long *)&y;
    static if (real.mant_dig == 53)
    { //double
        return pxs[0] == pys[0];
    }
    else static if (real.mant_dig == 113 || real.mant_dig==106)
    {
        // quadruple or doubledouble
        return pxs[0] == pys[0] && pxs[1] == pys[1];
    }
    else
    { // real80
        ushort* pxe = cast(ushort *)&x;
        ushort* pye = cast(ushort *)&y;
        return pxe[4] == pye[4] && pxs[0] == pys[0];
    }
}

/*********************************
 * Return 1 if sign bit of e is set, 0 if not.
 */

int signbit(real x) @trusted pure nothrow
{
    return ((cast(ubyte *)&x)[floatTraits!(real).SIGNPOS_BYTE] & 0x80) != 0;
}

unittest
{
    debug (math) printf("math.signbit.unittest\n");
    assert(!signbit(float.nan));
    assert(signbit(-float.nan));
    assert(!signbit(168.1234));
    assert(signbit(-168.1234));
    assert(!signbit(0.0));
    assert(signbit(-0.0));
    assert(signbit(-double.max));
    assert(!signbit(double.max));
}

/*********************************
 * Return a value composed of to with from's sign bit.
 */

real copysign(real to, real from) @trusted pure nothrow
{
    ubyte* pto   = cast(ubyte *)&to;
    const ubyte* pfrom = cast(ubyte *)&from;

    alias floatTraits!(real) F;
    pto[F.SIGNPOS_BYTE] &= 0x7F;
    pto[F.SIGNPOS_BYTE] |= pfrom[F.SIGNPOS_BYTE] & 0x80;
    return to;
}

unittest
{
    real e;

    e = copysign(21, 23.8);
    assert(e == 21);

    e = copysign(-21, 23.8);
    assert(e == 21);

    e = copysign(21, -23.8);
    assert(e == -21);

    e = copysign(-21, -23.8);
    assert(e == -21);

    e = copysign(real.nan, -23.8);
    assert(isNaN(e) && signbit(e));
}

/*********************************
Returns $(D -1) if $(D x < 0), $(D x) if $(D x == 0), $(D 1) if
$(D x > 0), and $(NAN) if x==$(NAN).
 */
F sgn(F)(F x) @safe pure nothrow
{
    // @@@TODO@@@: make this faster
    return x > 0 ? 1 : x < 0 ? -1 : x;
}

unittest
{
    debug (math) printf("math.sgn.unittest\n");
    assert(sgn(168.1234) == 1);
    assert(sgn(-168.1234) == -1);
    assert(sgn(0.0) == 0);
    assert(sgn(-0.0) == 0);
}

// Functions for NaN payloads
/*
 * A 'payload' can be stored in the significand of a $(NAN). One bit is required
 * to distinguish between a quiet and a signalling $(NAN). This leaves 22 bits
 * of payload for a float; 51 bits for a double; 62 bits for an 80-bit real;
 * and 111 bits for a 128-bit quad.
*/
/**
 * Create a quiet $(NAN), storing an integer inside the payload.
 *
 * For floats, the largest possible payload is 0x3F_FFFF.
 * For doubles, it is 0x3_FFFF_FFFF_FFFF.
 * For 80-bit or 128-bit reals, it is 0x3FFF_FFFF_FFFF_FFFF.
 */
real NaN(ulong payload) @trusted pure nothrow
{
    static if (real.mant_dig == 64) { //real80
        ulong v = 3; // implied bit = 1, quiet bit = 1
    } else {
        ulong v = 2; // no implied bit. quiet bit = 1
    }

    ulong a = payload;

    // 22 Float bits
    ulong w = a & 0x3F_FFFF;
    a -= w;

    v <<=22;
    v |= w;
    a >>=22;

    // 29 Double bits
    v <<=29;
    w = a & 0xFFF_FFFF;
    v |= w;
    a -= w;
    a >>=29;

    static if (real.mant_dig == 53) { // double
        v |=0x7FF0_0000_0000_0000;
        real x;
        * cast(ulong *)(&x) = v;
        return x;
    } else {
        v <<=11;
        a &= 0x7FF;
        v |= a;
        real x = real.nan;
        // Extended real bits
        static if (real.mant_dig==113) { //quadruple
            v<<=1; // there's no implicit bit
            version(LittleEndian) {
                *cast(ulong*)(6+cast(ubyte*)(&x)) = v;
            } else {
                *cast(ulong*)(2+cast(ubyte*)(&x)) = v;
            }
        } else { // real80
            * cast(ulong *)(&x) = v;
        }
        return x;
    }
}

/**
 * Extract an integral payload from a $(NAN).
 *
 * Returns:
 * the integer payload as a ulong.
 *
 * For floats, the largest possible payload is 0x3F_FFFF.
 * For doubles, it is 0x3_FFFF_FFFF_FFFF.
 * For 80-bit or 128-bit reals, it is 0x3FFF_FFFF_FFFF_FFFF.
 */
ulong getNaNPayload(real x) @trusted pure nothrow
{
    //  assert(isNaN(x));
    static if (real.mant_dig == 53) {
        ulong m = *cast(ulong *)(&x);
        // Make it look like an 80-bit significand.
        // Skip exponent, and quiet bit
        m &= 0x0007_FFFF_FFFF_FFFF;
        m <<= 10;
    } else static if (real.mant_dig==113) { // quadruple
        version(LittleEndian) {
            ulong m = *cast(ulong*)(6+cast(ubyte*)(&x));
        } else {
            ulong m = *cast(ulong*)(2+cast(ubyte*)(&x));
        }
        m>>=1; // there's no implicit bit
    } else {
        ulong m = *cast(ulong *)(&x);
    }
    // ignore implicit bit and quiet bit
    ulong f = m & 0x3FFF_FF00_0000_0000L;
    ulong w = f >>> 40;
            w |= (m & 0x00FF_FFFF_F800L) << (22 - 11);
            w |= (m & 0x7FF) << 51;
            return w;
}

debug(UnitTest) {
    unittest {
        real nan4 = NaN(0x789_ABCD_EF12_3456);
        static if (real.mant_dig == 64 || real.mant_dig==113) {
            assert (getNaNPayload(nan4) == 0x789_ABCD_EF12_3456);
        } else {
            assert (getNaNPayload(nan4) == 0x1_ABCD_EF12_3456);
        }
        double nan5 = nan4;
        assert (getNaNPayload(nan5) == 0x1_ABCD_EF12_3456);
        float nan6 = nan4;
        assert (getNaNPayload(nan6) == 0x12_3456);
        nan4 = NaN(0xFABCD);
        assert (getNaNPayload(nan4) == 0xFABCD);
        nan6 = nan4;
        assert (getNaNPayload(nan6) == 0xFABCD);
        nan5 = NaN(0x100_0000_0000_3456);
        assert(getNaNPayload(nan5) == 0x0000_0000_3456);
    }
}

/**
 * Calculate the next largest floating point value after x.
 *
 * Return the least number greater than x that is representable as a real;
 * thus, it gives the next point on the IEEE number line.
 *
 *  $(TABLE_SV
 *    $(SVH x,            nextUp(x)   )
 *    $(SV  -$(INFIN),    -real.max   )
 *    $(SV  $(PLUSMN)0.0, real.min_normal*real.epsilon )
 *    $(SV  real.max,     $(INFIN) )
 *    $(SV  $(INFIN),     $(INFIN) )
 *    $(SV  $(NAN),       $(NAN)   )
 * )
 */
real nextUp(real x) @trusted pure nothrow
{
    alias floatTraits!(real) F;
    static if (real.mant_dig == 53) { // double
        return nextUp(cast(double)x);
    } else static if(real.mant_dig==113) {  // quadruple
        ushort e = F.EXPMASK & (cast(ushort *)&x)[F.EXPPOS_SHORT];
        if (e == F.EXPMASK) { // NaN or Infinity
            if (x == -real.infinity) return -real.max;
            return x; // +Inf and NaN are unchanged.
        }
        ulong*   ps = cast(ulong *)&e;
        if (ps[MANTISSA_LSB] & 0x8000_0000_0000_0000)  { // Negative number
            if (ps[MANTISSA_LSB] == 0
                && ps[MANTISSA_MSB] == 0x8000_0000_0000_0000) {
                // it was negative zero, change to smallest subnormal
                ps[MANTISSA_LSB] = 0x0000_0000_0000_0001;
                ps[MANTISSA_MSB] = 0;
                return x;
            }
            --*ps;
            if (ps[MANTISSA_LSB]==0) --ps[MANTISSA_MSB];
        } else { // Positive number
            ++ps[MANTISSA_LSB];
            if (ps[MANTISSA_LSB]==0) ++ps[MANTISSA_MSB];
        }
        return x;

    } else static if(real.mant_dig==64){ // real80
        // For 80-bit reals, the "implied bit" is a nuisance...
        ushort *pe = cast(ushort *)&x;
        ulong  *ps = cast(ulong  *)&x;

        if ((pe[F.EXPPOS_SHORT] & F.EXPMASK) == F.EXPMASK) {
            // First, deal with NANs and infinity
            if (x == -real.infinity) return -real.max;
            return x; // +Inf and NaN are unchanged.
        }
        if (pe[F.EXPPOS_SHORT] & 0x8000)  {
            // Negative number -- need to decrease the significand
            --*ps;
            // Need to mask with 0x7FFF... so subnormals are treated correctly.
            if ((*ps & 0x7FFF_FFFF_FFFF_FFFF) == 0x7FFF_FFFF_FFFF_FFFF) {
                if (pe[F.EXPPOS_SHORT] == 0x8000) { // it was negative zero
                    *ps = 1;
                    pe[F.EXPPOS_SHORT] = 0; // smallest subnormal.
                    return x;
                }
                --pe[F.EXPPOS_SHORT];
                if (pe[F.EXPPOS_SHORT] == 0x8000) {
                    return x; // it's become a subnormal, implied bit stays low.
                }
                *ps = 0xFFFF_FFFF_FFFF_FFFF; // set the implied bit
                return x;
            }
            return x;
        } else {
            // Positive number -- need to increase the significand.
            // Works automatically for positive zero.
            ++*ps;
            if ((*ps & 0x7FFF_FFFF_FFFF_FFFF) == 0) {
                // change in exponent
                ++pe[F.EXPPOS_SHORT];
                *ps = 0x8000_0000_0000_0000; // set the high bit
            }
        }
        return x;
    } // doubledouble is not supported
}

/** ditto */
double nextUp(double x) @trusted pure nothrow
{
    ulong *ps = cast(ulong *)&x;

    if ((*ps & 0x7FF0_0000_0000_0000) == 0x7FF0_0000_0000_0000) {
        // First, deal with NANs and infinity
        if (x == -x.infinity) return -x.max;
        return x; // +INF and NAN are unchanged.
    }
    if (*ps & 0x8000_0000_0000_0000)  { // Negative number
        if (*ps == 0x8000_0000_0000_0000) { // it was negative zero
            *ps = 0x0000_0000_0000_0001; // change to smallest subnormal
            return x;
        }
        --*ps;
    } else { // Positive number
        ++*ps;
    }
    return x;
}

/** ditto */
float nextUp(float x) @trusted pure nothrow
{
    uint *ps = cast(uint *)&x;

    if ((*ps & 0x7F80_0000) == 0x7F80_0000) {
        // First, deal with NANs and infinity
        if (x == -x.infinity) return -x.max;
        return x; // +INF and NAN are unchanged.
    }
    if (*ps & 0x8000_0000)  { // Negative number
        if (*ps == 0x8000_0000) { // it was negative zero
            *ps = 0x0000_0001; // change to smallest subnormal
            return x;
        }
        --*ps;
    } else { // Positive number
        ++*ps;
    }
    return x;
}

/**
 * Calculate the next smallest floating point value before x.
 *
 * Return the greatest number less than x that is representable as a real;
 * thus, it gives the previous point on the IEEE number line.
 *
 *  $(TABLE_SV
 *    $(SVH x,            nextDown(x)   )
 *    $(SV  $(INFIN),     real.max  )
 *    $(SV  $(PLUSMN)0.0, -real.min_normal*real.epsilon )
 *    $(SV  -real.max,    -$(INFIN) )
 *    $(SV  -$(INFIN),    -$(INFIN) )
 *    $(SV  $(NAN),       $(NAN)    )
 * )
 */
real nextDown(real x) @safe pure nothrow
{
    return -nextUp(-x);
}

/** ditto */
double nextDown(double x) @safe pure nothrow
{
    return -nextUp(-x);
}

/** ditto */
float nextDown(float x) @safe pure nothrow
{
    return -nextUp(-x);
}

unittest {
    assert( nextDown(1.0 + real.epsilon) == 1.0);
}

unittest {
    static if (real.mant_dig == 64) {

        // Tests for 80-bit reals
        assert(isIdentical(nextUp(NaN(0xABC)), NaN(0xABC)));
        // negative numbers
        assert( nextUp(-real.infinity) == -real.max );
        assert( nextUp(-1.0L-real.epsilon) == -1.0 );
        assert( nextUp(-2.0L) == -2.0 + real.epsilon);
        // subnormals and zero
        assert( nextUp(-real.min_normal) == -real.min_normal*(1-real.epsilon) );
        assert( nextUp(-real.min_normal*(1-real.epsilon)) == -real.min_normal*(1-2*real.epsilon) );
        assert( isIdentical(-0.0L, nextUp(-real.min_normal*real.epsilon)) );
        assert( nextUp(-0.0L) == real.min_normal*real.epsilon );
        assert( nextUp(0.0L) == real.min_normal*real.epsilon );
        assert( nextUp(real.min_normal*(1-real.epsilon)) == real.min_normal );
        assert( nextUp(real.min_normal) == real.min_normal*(1+real.epsilon) );
        // positive numbers
        assert( nextUp(1.0L) == 1.0 + real.epsilon );
        assert( nextUp(2.0L-real.epsilon) == 2.0 );
        assert( nextUp(real.max) == real.infinity );
        assert( nextUp(real.infinity)==real.infinity );
    }

    double n = NaN(0xABC);
    assert(isIdentical(nextUp(n), n));
    // negative numbers
    assert( nextUp(-double.infinity) == -double.max );
    assert( nextUp(-1-double.epsilon) == -1.0 );
    assert( nextUp(-2.0) == -2.0 + double.epsilon);
    // subnormals and zero

    assert( nextUp(-double.min_normal) == -double.min_normal*(1-double.epsilon) );
    assert( nextUp(-double.min_normal*(1-double.epsilon)) == -double.min_normal*(1-2*double.epsilon) );
    assert( isIdentical(-0.0, nextUp(-double.min_normal*double.epsilon)) );
    assert( nextUp(0.0) == double.min_normal*double.epsilon );
    assert( nextUp(-0.0) == double.min_normal*double.epsilon );
    assert( nextUp(double.min_normal*(1-double.epsilon)) == double.min_normal );
    assert( nextUp(double.min_normal) == double.min_normal*(1+double.epsilon) );
    // positive numbers
    assert( nextUp(1.0) == 1.0 + double.epsilon );
    assert( nextUp(2.0-double.epsilon) == 2.0 );
    assert( nextUp(double.max) == double.infinity );

    float fn = NaN(0xABC);
    assert(isIdentical(nextUp(fn), fn));
    float f = -float.min_normal*(1-float.epsilon);
    float f1 = -float.min_normal;
    assert( nextUp(f1) ==  f);
    f = 1.0f+float.epsilon;
    f1 = 1.0f;
    assert( nextUp(f1) == f );
    f1 = -0.0f;
    assert( nextUp(f1) == float.min_normal*float.epsilon);
    assert( nextUp(float.infinity)==float.infinity );

    assert(nextDown(1.0L+real.epsilon)==1.0);
    assert(nextDown(1.0+double.epsilon)==1.0);
    f = 1.0f+float.epsilon;
    assert(nextDown(f)==1.0);
    assert(nextafter(1.0+real.epsilon, -real.infinity)==1.0);
}



/******************************************
 * Calculates the next representable value after x in the direction of y.
 *
 * If y > x, the result will be the next largest floating-point value;
 * if y < x, the result will be the next smallest value.
 * If x == y, the result is y.
 *
 * Remarks:
 * This function is not generally very useful; it's almost always better to use
 * the faster functions nextUp() or nextDown() instead.
 *
 * The FE_INEXACT and FE_OVERFLOW exceptions will be raised if x is finite and
 * the function result is infinite. The FE_INEXACT and FE_UNDERFLOW
 * exceptions will be raised if the function value is subnormal, and x is
 * not equal to y.
 */
T nextafter(T)(T x, T y) @safe pure nothrow
{
    if (x==y) return y;
    return ((y>x) ? nextUp(x) :  nextDown(x));
}

unittest
{
    float a = 1;
    assert(is(typeof(nextafter(a, a)) == float));
    assert(nextafter(a, a.infinity) > a);

    double b = 2;
    assert(is(typeof(nextafter(b, b)) == double));
    assert(nextafter(b, b.infinity) > b);

    real c = 3;
    assert(is(typeof(nextafter(c, c)) == real));
    assert(nextafter(c, c.infinity) > c);
}

//real nexttoward(real x, real y) { return core.stdc.math.nexttowardl(x, y); }

/*******************************************
 * Returns the positive difference between x and y.
 * Returns:
 *      $(TABLE_SV
 *      $(TR $(TH x, y)       $(TH fdim(x, y)))
 *      $(TR $(TD x $(GT) y)  $(TD x - y))
 *      $(TR $(TD x $(LT)= y) $(TD +0.0))
 *      )
 */
real fdim(real x, real y) @safe pure nothrow { return (x > y) ? x - y : +0.0; }

/****************************************
 * Returns the larger of x and y.
 */
real fmax(real x, real y) @safe pure nothrow { return x > y ? x : y; }

/****************************************
 * Returns the smaller of x and y.
 */
real fmin(real x, real y) @safe pure nothrow { return x < y ? x : y; }

/**************************************
 * Returns (x * y) + z, rounding only once according to the
 * current rounding mode.
 *
 * BUGS: Not currently implemented - rounds twice.
 */
real fma(real x, real y, real z) @safe pure nothrow { return (x * y) + z; }

/*******************************************************************
 * Compute the value of x $(SUP n), where n is an integer
 */
Unqual!F pow(F, G)(F x, G n) @trusted pure nothrow
    if (isFloatingPoint!(F) && isIntegral!(G))
{
    real p = 1.0, v = void;
    Unsigned!(Unqual!G) m = n;
    if (n < 0)
    {
        switch (n)
        {
        case -1:
            return 1 / x;
        case -2:
            return 1 / (x * x);
        default:
        }

        m = -n;
        v = p / x;
    }
    else
    {
        switch (n)
        {
        case 0:
            return 1.0;
        case 1:
            return x;
        case 2:
            return x * x;
        default:
        }

        v = x;
    }

    while (1)
    {
        if (m & 1)
            p *= v;
        m >>= 1;
        if (!m)
            break;
        v *= v;
    }
    return p;
}

unittest
{
    // Make sure it instantiates and works properly on immutable values and
    // with various integer and float types.
    immutable real x = 46;
    immutable float xf = x;
    immutable double xd = x;
    immutable uint one = 1;
    immutable ushort two = 2;
    immutable ubyte three = 3;
    immutable ulong eight = 8;

    immutable int neg1 = -1;
    immutable short neg2 = -2;
    immutable byte neg3 = -3;
    immutable long neg8 = -8;


    assert(pow(x,0) == 1.0);
    assert(pow(xd,one) == x);
    assert(pow(xf,two) == x * x);
    assert(pow(x,three) == x * x * x);
    assert(pow(x,eight) == (x * x) * (x * x) * (x * x) * (x * x));

    assert(pow(x, neg1) == 1 / x);

    version(X86_64)
    {
        pragma(msg, "test disabled on x86_64, see bug 5628");
    }
    else
    {
        assert(pow(xd, neg2) == 1 / (x * x));
        assert(pow(xf, neg8) == 1 / ((x * x) * (x * x) * (x * x) * (x * x)));
    }

    assert(pow(x, neg3) == 1 / (x * x * x));
}

/** Compute the value of an integer x, raised to the power of a positive
 * integer n.
 *
 *  If both x and n are 0, the result is 1.
 *  If n is negative, an integer divide error will occur at runtime,
 * regardless of the value of x.
 */

typeof(Unqual!(F).init * Unqual!(G).init) pow(F, G)(F x, G n) @trusted pure nothrow
if (isIntegral!(F) && isIntegral!(G))
{
    if (n<0) return x/0; // Only support positive powers
    typeof(return) p, v = void;
    Unqual!G m = n;

    switch (m)
    {
    case 0:
        p = 1;
        break;

    case 1:
        p = x;
        break;

    case 2:
        p = x * x;
        break;

    default:
        v = x;
        p = 1;
        while (1){
            if (m & 1)
                p *= v;
            m >>= 1;
            if (!m)
                break;
            v *= v;
        }
        break;
    }
    return p;
}

unittest
{
    immutable int one = 1;
    immutable byte two = 2;
    immutable ubyte three = 3;
    immutable short four = 4;
    immutable long ten = 10;

    assert(pow(two, three) == 8);
    assert(pow(two, ten) == 1024);
    assert(pow(one, ten) == 1);
    assert(pow(ten, four) == 10_000);
    assert(pow(four, 10) == 1_048_576);
    assert(pow(three, four) == 81);

}

/**Computes integer to floating point powers.*/
real pow(I, F)(I x, F y) @trusted pure nothrow
    if(isIntegral!I && isFloatingPoint!F)
{
    return pow(cast(real) x, cast(Unqual!F) y);
}

/*********************************************
 * Calculates x$(SUP y).
 *
 * $(TABLE_SV
 * $(TR $(TH x) $(TH y) $(TH pow(x, y))
 *      $(TH div 0) $(TH invalid?))
 * $(TR $(TD anything)      $(TD $(PLUSMN)0.0)                $(TD 1.0)
 *      $(TD no)        $(TD no) )
 * $(TR $(TD |x| $(GT) 1)    $(TD +$(INFIN))                  $(TD +$(INFIN))
 *      $(TD no)        $(TD no) )
 * $(TR $(TD |x| $(LT) 1)    $(TD +$(INFIN))                  $(TD +0.0)
 *      $(TD no)        $(TD no) )
 * $(TR $(TD |x| $(GT) 1)    $(TD -$(INFIN))                  $(TD +0.0)
 *      $(TD no)        $(TD no) )
 * $(TR $(TD |x| $(LT) 1)    $(TD -$(INFIN))                  $(TD +$(INFIN))
 *      $(TD no)        $(TD no) )
 * $(TR $(TD +$(INFIN))      $(TD $(GT) 0.0)                  $(TD +$(INFIN))
 *      $(TD no)        $(TD no) )
 * $(TR $(TD +$(INFIN))      $(TD $(LT) 0.0)                  $(TD +0.0)
 *      $(TD no)        $(TD no) )
 * $(TR $(TD -$(INFIN))      $(TD odd integer $(GT) 0.0)      $(TD -$(INFIN))
 *      $(TD no)        $(TD no) )
 * $(TR $(TD -$(INFIN))      $(TD $(GT) 0.0, not odd integer) $(TD +$(INFIN))
 *      $(TD no)        $(TD no))
 * $(TR $(TD -$(INFIN))      $(TD odd integer $(LT) 0.0)      $(TD -0.0)
 *      $(TD no)        $(TD no) )
 * $(TR $(TD -$(INFIN))      $(TD $(LT) 0.0, not odd integer) $(TD +0.0)
 *      $(TD no)        $(TD no) )
 * $(TR $(TD $(PLUSMN)1.0)   $(TD $(PLUSMN)$(INFIN))          $(TD $(NAN))
 *      $(TD no)        $(TD yes) )
 * $(TR $(TD $(LT) 0.0)      $(TD finite, nonintegral)        $(TD $(NAN))
 *      $(TD no)        $(TD yes))
 * $(TR $(TD $(PLUSMN)0.0)   $(TD odd integer $(LT) 0.0)      $(TD $(PLUSMNINF))
 *      $(TD yes)       $(TD no) )
 * $(TR $(TD $(PLUSMN)0.0)   $(TD $(LT) 0.0, not odd integer) $(TD +$(INFIN))
 *      $(TD yes)       $(TD no))
 * $(TR $(TD $(PLUSMN)0.0)   $(TD odd integer $(GT) 0.0)      $(TD $(PLUSMN)0.0)
 *      $(TD no)        $(TD no) )
 * $(TR $(TD $(PLUSMN)0.0)   $(TD $(GT) 0.0, not odd integer) $(TD +0.0)
 *      $(TD no)        $(TD no) )
 * )
 */

Unqual!(Largest!(F, G)) pow(F, G)(F x, G y) @trusted pure nothrow
    if (isFloatingPoint!(F) && isFloatingPoint!(G))
{
    alias typeof(return) Float;

    static real impl(real x, real y) pure nothrow
    {
        if (isNaN(y))
            return y;

        if (y == 0)
            return 1;           // even if x is $(NAN)
        if (isNaN(x) && y != 0)
            return x;
        if (isInfinity(y))
        {
            if (fabs(x) > 1)
            {
                if (signbit(y))
                    return +0.0;
                else
                    return F.infinity;
            }
            else if (fabs(x) == 1)
            {
                return y * 0; // generate NaN.
            }
            else // < 1
            {
                if (signbit(y))
                    return F.infinity;
                else
                    return +0.0;
            }
        }
        if (isInfinity(x))
        {
            if (signbit(x))
            {   long i;

                i = cast(long)y;
                if (y > 0)
                {
                    if (i == y && i & 1)
                        return -F.infinity;
                    else
                        return F.infinity;
                }
                else if (y < 0)
                {
                    if (i == y && i & 1)
                        return -0.0;
                    else
                        return +0.0;
                }
            }
            else
            {
                if (y > 0)
                    return F.infinity;
                else if (y < 0)
                    return +0.0;
            }
        }

        if (x == 0.0)
        {
            if (signbit(x))
            {   long i;

                i = cast(long)y;
                if (y > 0)
                {
                    if (i == y && i & 1)
                        return -0.0;
                    else
                        return +0.0;
                }
                else if (y < 0)
                {
                    if (i == y && i & 1)
                        return -F.infinity;
                    else
                        return F.infinity;
                }
            }
            else
            {
                if (y > 0)
                    return +0.0;
                else if (y < 0)
                    return F.infinity;
            }
        }
        double sign = 1.0;
        if (x < 0) {
            // Result is real only if y is an integer
            // Check for a non-zero fractional part
            if (y > -1.0 / real.epsilon && y < 1.0 / real.epsilon)
            {
                long w = cast(long)y;
                if (w != y)
                    return sqrt(x); // Complex result -- create a NaN
                if (w & 1) sign = -1.0;
            }
            x = -x;
        }
        version(INLINE_YL2X) {
            // If x > 0, x ^^ y == 2 ^^ ( y * log2(x) )
            // TODO: This is not accurate in practice. A fast and accurate
            // (though complicated) method is described in:
            // "An efficient rounding boundary test for pow(x, y)
            // in double precision", C.Q. Lauter and V. Lefèvre, INRIA (2007).
            return sign * exp2( yl2x(x, y) );
        } else {
            return sign * core.stdc.math.powl(x, y);
        }
    }
    return impl(x, y);
}

unittest
{
    // Test all the special values.  These unittests can be run on Windows
    // by temporarily changing the version(linux) to version(all).
    immutable float zero = 0;
    immutable real one = 1;
    immutable double two = 2;
    immutable float three = 3;
    immutable float fnan = float.nan;
    immutable double dnan = double.nan;
    immutable real rnan = real.nan;
    immutable dinf = double.infinity;
    immutable rninf = -real.infinity;

    assert(pow(fnan, zero) == 1);
    assert(pow(dnan, zero) == 1);
    assert(pow(rnan, zero) == 1);

    assert(pow(two, dinf) == double.infinity);
    assert(isIdentical(pow(0.2f, dinf), +0.0));
    assert(pow(0.99999999L, rninf) == real.infinity);
    assert(isIdentical(pow(1.000000001, rninf), +0.0));
    assert(pow(dinf, 0.001) == dinf);
    assert(isIdentical(pow(dinf, -0.001), +0.0));
    assert(pow(rninf, 3.0L) == rninf);
    assert(pow(rninf, 2.0L) == real.infinity);
    assert(isIdentical(pow(rninf, -3.0), -0.0));
    assert(isIdentical(pow(rninf, -2.0), +0.0));

    // @@@BUG@@@ somewhere
    version(OSX) {} else assert(isNaN(pow(one, dinf)));
    version(OSX) {} else assert(isNaN(pow(-one, dinf)));
    assert(isNaN(pow(-0.2, PI)));
    // boundary cases. Note that epsilon == 2^^-n for some n,
    // so 1/epsilon == 2^^n is always even.
    assert(pow(-1.0L, 1/real.epsilon - 1.0L) == -1.0L);
    assert(pow(-1.0L, 1/real.epsilon) == 1.0L);
    assert(isNaN(pow(-1.0L, 1/real.epsilon-0.5L)));
    assert(isNaN(pow(-1.0L, -1/real.epsilon+0.5L)));

    assert(pow(0.0, -3.0) == double.infinity);
    assert(pow(-0.0, -3.0) == -double.infinity);
    assert(pow(0.0, -PI) == double.infinity);
    assert(pow(-0.0, -PI) == double.infinity);
    assert(isIdentical(pow(0.0, 5.0), 0.0));
    assert(isIdentical(pow(-0.0, 5.0), -0.0));
    assert(isIdentical(pow(0.0, 6.0), 0.0));
    assert(isIdentical(pow(-0.0, 6.0), 0.0));

    // Now, actual numbers.
    assert(approxEqual(pow(two, three), 8.0));
    assert(approxEqual(pow(two, -2.5), 0.1767767));

    // Test integer to float power.
    immutable uint twoI = 2;
    assert(approxEqual(pow(twoI, three), 8.0));
}

/**************************************
 * To what precision is x equal to y?
 *
 * Returns: the number of mantissa bits which are equal in x and y.
 * eg, 0x1.F8p+60 and 0x1.F1p+60 are equal to 5 bits of precision.
 *
 *      $(TABLE_SV
 *      $(TR $(TH x)      $(TH y)          $(TH feqrel(x, y)))
 *      $(TR $(TD x)      $(TD x)          $(TD real.mant_dig))
 *      $(TR $(TD x)      $(TD $(GT)= 2*x) $(TD 0))
 *      $(TR $(TD x)      $(TD $(LT)= x/2) $(TD 0))
 *      $(TR $(TD $(NAN)) $(TD any)        $(TD 0))
 *      $(TR $(TD any)    $(TD $(NAN))     $(TD 0))
 *      )
 */
int feqrel(X)(X x, X y) @trusted pure nothrow
    if (isFloatingPoint!(X))
{
    /* Public Domain. Author: Don Clugston, 18 Aug 2005.
     */
    static if (X.mant_dig == 106)   // doubledouble
    {
        if (cast(double*)(&x)[MANTISSA_MSB] == cast(double*)(&y)[MANTISSA_MSB])
        {
            return double.mant_dig
            + feqrel(cast(double*)(&x)[MANTISSA_LSB],
                    cast(double*)(&y)[MANTISSA_LSB]);
        }
        else
        {
            return feqrel(cast(double*)(&x)[MANTISSA_MSB],
                    cast(double*)(&y)[MANTISSA_MSB]);
        }
    }
    else
    {
        static assert( X.mant_dig == 64 || X.mant_dig == 113
            || X.mant_dig == double.mant_dig || X.mant_dig == float.mant_dig);

        if (x == y)
            return X.mant_dig; // ensure diff!=0, cope with INF.

        X diff = fabs(x - y);

        ushort *pa = cast(ushort *)(&x);
        ushort *pb = cast(ushort *)(&y);
        ushort *pd = cast(ushort *)(&diff);

        alias floatTraits!(X) F;

        // The difference in abs(exponent) between x or y and abs(x-y)
        // is equal to the number of significand bits of x which are
        // equal to y. If negative, x and y have different exponents.
        // If positive, x and y are equal to 'bitsdiff' bits.
        // AND with 0x7FFF to form the absolute value.
        // To avoid out-by-1 errors, we subtract 1 so it rounds down
        // if the exponents were different. This means 'bitsdiff' is
        // always 1 lower than we want, except that if bitsdiff==0,
        // they could have 0 or 1 bits in common.

        static if (X.mant_dig == 64 || X.mant_dig == 113)
        {   // real80 or quadruple
            int bitsdiff = ( ((pa[F.EXPPOS_SHORT] & F.EXPMASK)
                              + (pb[F.EXPPOS_SHORT] & F.EXPMASK) - 1) >> 1)
                              - pd[F.EXPPOS_SHORT];
        }
        else static if (X.mant_dig == double.mant_dig)
        {   // double
            int bitsdiff = (( ((pa[F.EXPPOS_SHORT]&0x7FF0)
                               + (pb[F.EXPPOS_SHORT]&0x7FF0)-0x10)>>1)
                               - (pd[F.EXPPOS_SHORT]&0x7FF0))>>4;
        }
        else static if (X.mant_dig == float.mant_dig)
        {   // float
            int bitsdiff = (( ((pa[F.EXPPOS_SHORT]&0x7F80)
                               + (pb[F.EXPPOS_SHORT]&0x7F80)-0x80)>>1)
                               - (pd[F.EXPPOS_SHORT]&0x7F80))>>7;
        }
        if ( (pd[F.EXPPOS_SHORT] & F.EXPMASK) == 0)
        {   // Difference is subnormal
            // For subnormals, we need to add the number of zeros that
            // lie at the start of diff's significand.
            // We do this by multiplying by 2^^real.mant_dig
            diff *= F.RECIP_EPSILON;
            return bitsdiff + X.mant_dig - pd[F.EXPPOS_SHORT];
        }

        if (bitsdiff > 0)
            return bitsdiff + 1; // add the 1 we subtracted before

        // Avoid out-by-1 errors when factor is almost 2.
        static if (X.mant_dig == 64 || X.mant_dig == 113)
        {   // real80 or quadruple
            return (bitsdiff == 0) ? (pa[F.EXPPOS_SHORT] == pb[F.EXPPOS_SHORT]) : 0;
        }
        else static if (X.mant_dig == double.mant_dig || X.mant_dig == float.mant_dig)
        {
            if (bitsdiff == 0
                && !((pa[F.EXPPOS_SHORT] ^ pb[F.EXPPOS_SHORT]) & F.EXPMASK))
            {
                return 1;
            } else return 0;
        }
    }
}

unittest
{
    void testFeqrel(F)()
    {
       // Exact equality
       assert(feqrel(F.max, F.max) == F.mant_dig);
       assert(feqrel!(F)(0.0, 0.0) == F.mant_dig);
       assert(feqrel(F.infinity, F.infinity) == F.mant_dig);

       // a few bits away from exact equality
       F w=1;
       for (int i = 1; i < F.mant_dig - 1; ++i)
       {
          assert(feqrel!(F)(1.0 + w * F.epsilon, 1.0) == F.mant_dig-i);
          assert(feqrel!(F)(1.0 - w * F.epsilon, 1.0) == F.mant_dig-i);
          assert(feqrel!(F)(1.0, 1 + (w-1) * F.epsilon) == F.mant_dig - i + 1);
          w*=2;
       }

       assert(feqrel!(F)(1.5+F.epsilon, 1.5) == F.mant_dig-1);
       assert(feqrel!(F)(1.5-F.epsilon, 1.5) == F.mant_dig-1);
       assert(feqrel!(F)(1.5-F.epsilon, 1.5+F.epsilon) == F.mant_dig-2);


       // Numbers that are close
       assert(feqrel!(F)(0x1.Bp+84, 0x1.B8p+84) == 5);
       assert(feqrel!(F)(0x1.8p+10, 0x1.Cp+10) == 2);
       assert(feqrel!(F)(1.5 * (1 - F.epsilon), 1.0L) == 2);
       assert(feqrel!(F)(1.5, 1.0) == 1);
       assert(feqrel!(F)(2 * (1 - F.epsilon), 1.0L) == 1);

       // Factors of 2
       assert(feqrel(F.max, F.infinity) == 0);
       assert(feqrel!(F)(2 * (1 - F.epsilon), 1.0L) == 1);
       assert(feqrel!(F)(1.0, 2.0) == 0);
       assert(feqrel!(F)(4.0, 1.0) == 0);

       // Extreme inequality
       assert(feqrel(F.nan, F.nan) == 0);
       assert(feqrel!(F)(0.0L, -F.nan) == 0);
       assert(feqrel(F.nan, F.infinity) == 0);
       assert(feqrel(F.infinity, -F.infinity) == 0);
       assert(feqrel(F.max, -F.max) == 0);
    }

    assert(feqrel(7.1824L, 7.1824L) == real.mant_dig);
    assert(feqrel(real.min_normal / 8, real.min_normal / 17) == 3);

    testFeqrel!(real)();
    testFeqrel!(double)();
    testFeqrel!(float)();
}

package: // Not public yet
/* Return the value that lies halfway between x and y on the IEEE number line.
 *
 * Formally, the result is the arithmetic mean of the binary significands of x
 * and y, multiplied by the geometric mean of the binary exponents of x and y.
 * x and y must have the same sign, and must not be NaN.
 * Note: this function is useful for ensuring O(log n) behaviour in algorithms
 * involving a 'binary chop'.
 *
 * Special cases:
 * If x and y are within a factor of 2, (ie, feqrel(x, y) > 0), the return value
 * is the arithmetic mean (x + y) / 2.
 * If x and y are even powers of 2, the return value is the geometric mean,
 *   ieeeMean(x, y) = sqrt(x * y).
 *
 */
T ieeeMean(T)(T x, T y)  @trusted pure nothrow
in {
    // both x and y must have the same sign, and must not be NaN.
    assert(signbit(x) == signbit(y));
    assert(x<>=0 && y<>=0);
}
body {
    // Runtime behaviour for contract violation:
    // If signs are opposite, or one is a NaN, return 0.
    if (!((x>=0 && y>=0) || (x<=0 && y<=0))) return 0.0;

    // The implementation is simple: cast x and y to integers,
    // average them (avoiding overflow), and cast the result back to a floating-point number.

    alias floatTraits!(real) F;
    T u;
    static if (T.mant_dig==64) { // real80
        // There's slight additional complexity because they are actually
        // 79-bit reals...
        ushort *ue = cast(ushort *)&u;
        ulong *ul = cast(ulong *)&u;
        ushort *xe = cast(ushort *)&x;
        ulong *xl = cast(ulong *)&x;
        ushort *ye = cast(ushort *)&y;
        ulong *yl = cast(ulong *)&y;
        // Ignore the useless implicit bit. (Bonus: this prevents overflows)
        ulong m = ((*xl) & 0x7FFF_FFFF_FFFF_FFFFL) + ((*yl) & 0x7FFF_FFFF_FFFF_FFFFL);

        // @@@ BUG? @@@
        // Cast shouldn't be here
        ushort e = cast(ushort) ((xe[F.EXPPOS_SHORT] & F.EXPMASK)
                                 + (ye[F.EXPPOS_SHORT] & F.EXPMASK));
        if (m & 0x8000_0000_0000_0000L) {
            ++e;
            m &= 0x7FFF_FFFF_FFFF_FFFFL;
        }
        // Now do a multi-byte right shift
        uint c = e & 1; // carry
        e >>= 1;
    m >>>= 1;
    if (c) m |= 0x4000_0000_0000_0000L; // shift carry into significand
    if (e) *ul = m | 0x8000_0000_0000_0000L; // set implicit bit...
    else *ul = m; // ... unless exponent is 0 (subnormal or zero).
    ue[4]= e | (xe[F.EXPPOS_SHORT]& 0x8000); // restore sign bit
    } else static if(T.mant_dig == 113) { //quadruple
        // This would be trivial if 'ucent' were implemented...
        ulong *ul = cast(ulong *)&u;
        ulong *xl = cast(ulong *)&x;
        ulong *yl = cast(ulong *)&y;
        // Multi-byte add, then multi-byte right shift.
        ulong mh = ((xl[MANTISSA_MSB] & 0x7FFF_FFFF_FFFF_FFFFL)
                    + (yl[MANTISSA_MSB] & 0x7FFF_FFFF_FFFF_FFFFL));
        // Discard the lowest bit (to avoid overflow)
        ulong ml = (xl[MANTISSA_LSB]>>>1) + (yl[MANTISSA_LSB]>>>1);
        // add the lowest bit back in, if necessary.
        if (xl[MANTISSA_LSB] & yl[MANTISSA_LSB] & 1) {
            ++ml;
            if (ml==0) ++mh;
        }
        mh >>>=1;
        ul[MANTISSA_MSB] = mh | (xl[MANTISSA_MSB] & 0x8000_0000_0000_0000);
        ul[MANTISSA_LSB] = ml;
    } else static if (T.mant_dig == double.mant_dig) {
        ulong *ul = cast(ulong *)&u;
        ulong *xl = cast(ulong *)&x;
        ulong *yl = cast(ulong *)&y;
        ulong m = (((*xl) & 0x7FFF_FFFF_FFFF_FFFFL)
                   + ((*yl) & 0x7FFF_FFFF_FFFF_FFFFL)) >>> 1;
                   m |= ((*xl) & 0x8000_0000_0000_0000L);
                   *ul = m;
    } else static if (T.mant_dig == float.mant_dig) {
        uint *ul = cast(uint *)&u;
        uint *xl = cast(uint *)&x;
        uint *yl = cast(uint *)&y;
        uint m = (((*xl) & 0x7FFF_FFFF) + ((*yl) & 0x7FFF_FFFF)) >>> 1;
        m |= ((*xl) & 0x8000_0000);
        *ul = m;
    } else {
        assert(0, "Not implemented");
    }
    return u;
}

unittest {
    assert(ieeeMean(-0.0,-1e-20)<0);
    assert(ieeeMean(0.0,1e-20)>0);

    assert(ieeeMean(1.0L,4.0L)==2L);
    assert(ieeeMean(2.0*1.013,8.0*1.013)==4*1.013);
    assert(ieeeMean(-1.0L,-4.0L)==-2L);
    assert(ieeeMean(-1.0,-4.0)==-2);
    assert(ieeeMean(-1.0f,-4.0f)==-2f);
    assert(ieeeMean(-1.0,-2.0)==-1.5);
    assert(ieeeMean(-1*(1+8*real.epsilon),-2*(1+8*real.epsilon))
                 ==-1.5*(1+5*real.epsilon));
    assert(ieeeMean(0x1p60,0x1p-10)==0x1p25);
    static if (real.mant_dig==64) { // x87, 80-bit reals
      assert(ieeeMean(1.0L,real.infinity)==0x1p8192L);
      assert(ieeeMean(0.0L,real.infinity)==1.5);
    }
    assert(ieeeMean(0.5*real.min_normal*(1-4*real.epsilon),0.5*real.min_normal)
           == 0.5*real.min_normal*(1-2*real.epsilon));
}

public:


/***********************************
 * Evaluate polynomial A(x) = $(SUB a, 0) + $(SUB a, 1)x + $(SUB a, 2)$(POWER x,2)
 *                          + $(SUB a,3)$(POWER x,3); ...
 *
 * Uses Horner's rule A(x) = $(SUB a, 0) + x($(SUB a, 1) + x($(SUB a, 2)
 *                         + x($(SUB a, 3) + ...)))
 * Params:
 *      A =     array of coefficients $(SUB a, 0), $(SUB a, 1), etc.
 */
real poly(real x, const real[] A) @trusted pure nothrow
in
{
    assert(A.length > 0);
}
body
{
    version (D_InlineAsm_X86)
    {
        version (Windows)
        {
        // BUG: This code assumes a frame pointer in EBP.
            asm // assembler by W. Bright
            {
                // EDX = (A.length - 1) * real.sizeof
                mov     ECX,A[EBP]              ; // ECX = A.length
                dec     ECX                     ;
                lea     EDX,[ECX][ECX*8]        ;
                add     EDX,ECX                 ;
                add     EDX,A+4[EBP]            ;
                fld     real ptr [EDX]          ; // ST0 = coeff[ECX]
                jecxz   return_ST               ;
                fld     x[EBP]                  ; // ST0 = x
                fxch    ST(1)                   ; // ST1 = x, ST0 = r
                align   4                       ;
        L2:     fmul    ST,ST(1)                ; // r *= x
                fld     real ptr -10[EDX]       ;
                sub     EDX,10                  ; // deg--
                faddp   ST(1),ST                ;
                dec     ECX                     ;
                jne     L2                      ;
                fxch    ST(1)                   ; // ST1 = r, ST0 = x
                fstp    ST(0)                   ; // dump x
                align   4                       ;
        return_ST:                              ;
                ;
            }
        }
        else version (linux)
        {
            asm // assembler by W. Bright
            {
                // EDX = (A.length - 1) * real.sizeof
                mov     ECX,A[EBP]              ; // ECX = A.length
                dec     ECX                     ;
                lea     EDX,[ECX*8]             ;
                lea     EDX,[EDX][ECX*4]        ;
                add     EDX,A+4[EBP]            ;
                fld     real ptr [EDX]          ; // ST0 = coeff[ECX]
                jecxz   return_ST               ;
                fld     x[EBP]                  ; // ST0 = x
                fxch    ST(1)                   ; // ST1 = x, ST0 = r
                align   4                       ;
        L2:     fmul    ST,ST(1)                ; // r *= x
                fld     real ptr -12[EDX]       ;
                sub     EDX,12                  ; // deg--
                faddp   ST(1),ST                ;
                dec     ECX                     ;
                jne     L2                      ;
                fxch    ST(1)                   ; // ST1 = r, ST0 = x
                fstp    ST(0)                   ; // dump x
                align   4                       ;
        return_ST:                              ;
                ;
            }
        }
        else version (OSX)
        {
            asm // assembler by W. Bright
            {
                // EDX = (A.length - 1) * real.sizeof
                mov     ECX,A[EBP]              ; // ECX = A.length
                dec     ECX                     ;
                lea     EDX,[ECX*8]             ;
                add     EDX,EDX                 ;
                add     EDX,A+4[EBP]            ;
                fld     real ptr [EDX]          ; // ST0 = coeff[ECX]
                jecxz   return_ST               ;
                fld     x[EBP]                  ; // ST0 = x
                fxch    ST(1)                   ; // ST1 = x, ST0 = r
                align   4                       ;
        L2:     fmul    ST,ST(1)                ; // r *= x
                fld     real ptr -16[EDX]       ;
                sub     EDX,16                  ; // deg--
                faddp   ST(1),ST                ;
                dec     ECX                     ;
                jne     L2                      ;
                fxch    ST(1)                   ; // ST1 = r, ST0 = x
                fstp    ST(0)                   ; // dump x
                align   4                       ;
        return_ST:                              ;
                ;
            }
        }
        else version (FreeBSD)
        {
            asm // assembler by W. Bright
            {
                // EDX = (A.length - 1) * real.sizeof
                mov     ECX,A[EBP]              ; // ECX = A.length
                dec     ECX                     ;
                lea     EDX,[ECX*8]             ;
                lea     EDX,[EDX][ECX*4]        ;
                add     EDX,A+4[EBP]            ;
                fld     real ptr [EDX]          ; // ST0 = coeff[ECX]
                jecxz   return_ST               ;
                fld     x[EBP]                  ; // ST0 = x
                fxch    ST(1)                   ; // ST1 = x, ST0 = r
                align   4                       ;
        L2:     fmul    ST,ST(1)                ; // r *= x
                fld     real ptr -12[EDX]       ;
                sub     EDX,12                  ; // deg--
                faddp   ST(1),ST                ;
                dec     ECX                     ;
                jne     L2                      ;
                fxch    ST(1)                   ; // ST1 = r, ST0 = x
                fstp    ST(0)                   ; // dump x
                align   4                       ;
        return_ST:                              ;
                ;
            }
        }
        else
        {
            static assert(0);
        }
    }
    else
    {
        sizediff_t i = A.length - 1;
        real r = A[i];
        while (--i >= 0)
        {
            r *= x;
            r += A[i];
        }
        return r;
    }
}

unittest
{
    debug (math) printf("math.poly.unittest\n");
    real x = 3.1;
    static real pp[] = [56.1, 32.7, 6];

    assert( poly(x, pp) == (56.1L + (32.7L + 6L * x) * x) );
}

/**
   Computes whether $(D lhs) is approximately equal to $(D rhs)
   admitting a maximum relative difference $(D maxRelDiff) and a
   maximum absolute difference $(D maxAbsDiff).

   If the two inputs are ranges, $(D approxEqual) returns true if and
   only if the ranges have the same number of elements and if $(D
   approxEqual) evaluates to $(D true) for each pair of elements.
 */
bool approxEqual(T, U, V)(T lhs, U rhs, V maxRelDiff, V maxAbsDiff = 1e-5)
{
    static if (isInputRange!T)
    {
        static if (isInputRange!U)
        {
            // Two ranges
            for (;; lhs.popFront(), rhs.popFront())
            {
                if (lhs.empty) return rhs.empty;
                if (rhs.empty) return lhs.empty;
                if (!approxEqual(lhs.front, rhs.front, maxRelDiff, maxAbsDiff))
                    return false;
            }
        }
        else
        {
            // lhs is range, rhs is number
            for (; !lhs.empty; lhs.popFront())
            {
                if (!approxEqual(lhs.front, rhs, maxRelDiff, maxAbsDiff))
                    return false;
            }
            return true;
        }
    }
    else
    {
        static if (isInputRange!U)
        {
            // lhs is number, rhs is array
            return approxEqual(rhs, lhs, maxRelDiff, maxAbsDiff);
        }
        else
        {
            // two numbers
            //static assert(is(T : real) && is(U : real));
            if (rhs == 0)
            {
                return fabs(lhs) <= maxAbsDiff;
            }
            static if (is(typeof(lhs.infinity)) && is(typeof(rhs.infinity)))
            {
                if (lhs == lhs.infinity && rhs == rhs.infinity ||
                    lhs == -lhs.infinity && rhs == -rhs.infinity) return true;
            }
            return fabs((lhs - rhs) / rhs) <= maxRelDiff
                || maxAbsDiff != 0 && fabs(lhs - rhs) <= maxAbsDiff;
        }
    }
}

/**
   Returns $(D approxEqual(lhs, rhs, 1e-2, 1e-5)).
 */
bool approxEqual(T, U)(T lhs, U rhs)
{
    return approxEqual(lhs, rhs, 1e-2, 1e-5);
}

unittest
{
    assert(approxEqual(1.0, 1.0099));
    assert(!approxEqual(1.0, 1.011));
    float[] arr1 = [ 1.0, 2.0, 3.0 ];
    double[] arr2 = [ 1.001, 1.999, 3 ];
    assert(approxEqual(arr1, arr2));

    real num = real.infinity;
    assert(num == real.infinity);  // Passes.
    assert(approxEqual(num, real.infinity));  // Fails.
    num = -real.infinity;
    assert(num == -real.infinity);  // Passes.
    assert(approxEqual(num, -real.infinity));  // Fails.
}

// Included for backwards compatibility with Phobos1
alias isNaN isnan;
alias isFinite isfinite;
alias isNormal isnormal;
alias isSubnormal issubnormal;
alias isInfinity isinf;

/* **********************************
 * Building block functions, they
 * translate to a single x87 instruction.
 */

real yl2x(real x, real y)   @safe pure nothrow;       // y * log2(x)
real yl2xp1(real x, real y) @safe pure nothrow;       // y * log2(x + 1)

unittest
{
    version (INLINE_YL2X)
    {
        assert(yl2x(1024, 1) == 10);
        assert(yl2xp1(1023, 1) == 10);
    }
}

unittest
{
    real num = real.infinity;
    assert(num == real.infinity);  // Passes.
    assert(approxEqual(num, real.infinity));  // Fails.
}