/
exponential.d
4313 lines (3938 loc) · 143 KB
/
exponential.d
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// Written in the D programming language.
/**
This is a submodule of $(MREF std, math).
It contains several exponential and logarithm functions.
Copyright: Copyright The D Language Foundation 2000 - 2011.
D implementations of exp, expm1, exp2, log, log10, log1p, and log2
functions are based on the CEPHES math library, which is Copyright
(C) 2001 Stephen L. Moshier $(LT)steve@moshier.net$(GT) and are
incorporated herein by permission of the author. The author reserves
the right to distribute this material elsewhere under different
copying permissions. These modifications are distributed here under
the following terms:
License: $(HTTP www.boost.org/LICENSE_1_0.txt, Boost License 1.0).
Authors: $(HTTP digitalmars.com, Walter Bright), Don Clugston,
Conversion of CEPHES math library to D by Iain Buclaw and David Nadlinger
Source: $(PHOBOSSRC std/math/exponential.d)
Macros:
TABLE_SV = <table border="1" cellpadding="4" cellspacing="0">
<caption>Special Values</caption>
$0</table>
NAN = $(RED NAN)
PLUSMN = ±
INFIN = ∞
PLUSMNINF = ±∞
LT = <
GT = >
*/
module std.math.exponential;
import std.traits : isFloatingPoint, isIntegral, isSigned, isUnsigned, Largest, Unqual;
static import core.math;
static import core.stdc.math;
version (DigitalMars)
{
version (OSX) { } // macOS 13 (M1) has issues emulating instruction
else version = INLINE_YL2X; // x87 has opcodes for these
}
version (D_InlineAsm_X86) version = InlineAsm_X86_Any;
version (D_InlineAsm_X86_64) version = InlineAsm_X86_Any;
version (InlineAsm_X86_Any) version = InlineAsm_X87;
version (InlineAsm_X87)
{
static assert(real.mant_dig == 64);
version (CRuntime_Microsoft) version = InlineAsm_X87_MSVC;
}
version (D_HardFloat)
{
// FloatingPointControl.clearExceptions() depends on version IeeeFlagsSupport
version (IeeeFlagsSupport) version = FloatingPointControlSupport;
}
/**
* Compute the value of x $(SUPERSCRIPT n), where n is an integer
*/
Unqual!F pow(F, G)(F x, G n) @nogc @trusted pure nothrow
if (isFloatingPoint!(F) && isIntegral!(G))
{
import std.traits : Unsigned;
real p = 1.0, v = void;
Unsigned!(Unqual!G) m = n;
if (n < 0)
{
if (n == -1) return 1 / x;
m = cast(typeof(m))(0 - 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;
}
///
@safe pure nothrow @nogc unittest
{
import std.math.operations : feqrel;
assert(pow(2.0, 5) == 32.0);
assert(pow(1.5, 9).feqrel(38.4433) > 16);
assert(pow(real.nan, 2) is real.nan);
assert(pow(real.infinity, 2) == real.infinity);
}
@safe pure nothrow @nogc unittest
{
import std.math.operations : isClose, feqrel;
// 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);
assert(isClose(pow(xd, neg2), cast(double) (1 / (x * x)), 1e-15));
assert(isClose(pow(xf, neg8), cast(float) (1 / ((x * x) * (x * x) * (x * x) * (x * x))), 1e-15));
assert(feqrel(pow(x, neg3), 1 / (x * x * x)) >= real.mant_dig - 1);
}
@safe @nogc nothrow unittest
{
import std.math.operations : isClose;
assert(isClose(pow(2.0L, 10L), 1024, 1e-18));
}
// https://issues.dlang.org/show_bug.cgi?id=21601
@safe @nogc nothrow pure unittest
{
// When reals are large enough the results of pow(b, e) can be
// calculated correctly, if b is of type float or double and e is
// not too large.
static if (real.mant_dig >= 64)
{
// expected result: 3.790e-42
assert(pow(-513645318757045764096.0f, -2) > 0.0);
// expected result: 3.763915357831797e-309
assert(pow(-1.6299717435255677e+154, -2) > 0.0);
}
}
@safe @nogc nothrow unittest
{
import std.math.operations : isClose;
import std.math.traits : isInfinity;
static float f1 = 19100.0f;
static float f2 = 0.000012f;
assert(isClose(pow(f1,9), 3.3829868e+38f));
assert(isInfinity(pow(f1,10)));
assert(pow(f2,9) > 0.0f);
assert(isClose(pow(f2,10), 0.0f, 0.0, float.min_normal));
static double d1 = 21800.0;
static double d2 = 0.000012;
assert(isClose(pow(d1,71), 1.0725339442974e+308));
assert(isInfinity(pow(d1,72)));
assert(pow(d2,65) > 0.0f);
assert(isClose(pow(d2,66), 0.0, 0.0, double.min_normal));
static if (real.mant_dig == 64) // x87
{
static real r1 = 21950.0L;
static real r2 = 0.000011L;
assert(isClose(pow(r1,1136), 7.4066175654969242752260330529e+4931L));
assert(isInfinity(pow(r1,1137)));
assert(pow(r2,998) > 0.0L);
assert(isClose(pow(r2,999), 0.0L, 0.0, real.min_normal));
}
}
@safe @nogc nothrow pure unittest
{
import std.math.operations : isClose;
enum f1 = 19100.0f;
enum f2 = 0.000012f;
static assert(isClose(pow(f1,9), 3.3829868e+38f));
static assert(pow(f1,10) > float.max);
static assert(pow(f2,9) > 0.0f);
static assert(isClose(pow(f2,10), 0.0f, 0.0, float.min_normal));
enum d1 = 21800.0;
enum d2 = 0.000012;
static assert(isClose(pow(d1,71), 1.0725339442974e+308));
static assert(pow(d1,72) > double.max);
static assert(pow(d2,65) > 0.0f);
static assert(isClose(pow(d2,66), 0.0, 0.0, double.min_normal));
static if (real.mant_dig == 64) // x87
{
enum r1 = 21950.0L;
enum r2 = 0.000011L;
static assert(isClose(pow(r1,1136), 7.4066175654969242752260330529e+4931L));
static assert(pow(r1,1137) > real.max);
static assert(pow(r2,998) > 0.0L);
static assert(isClose(pow(r2,999), 0.0L, 0.0, real.min_normal));
}
}
/**
* Compute the power of two integral numbers.
*
* Params:
* x = base
* n = exponent
*
* Returns:
* x raised to the power of n. If n is negative the result is 1 / pow(x, -n),
* which is calculated as integer division with remainder. This may result in
* a division by zero error.
*
* If both x and n are 0, the result is 1.
*
* Throws:
* If x is 0 and n is negative, the result is the same as the result of a
* division by zero.
*/
typeof(Unqual!(F).init * Unqual!(G).init) pow(F, G)(F x, G n) @nogc @trusted pure nothrow
if (isIntegral!(F) && isIntegral!(G))
{
import std.traits : isSigned;
typeof(return) p, v = void;
Unqual!G m = n;
static if (isSigned!(F))
{
if (x == -1) return cast(typeof(return)) (m & 1 ? -1 : 1);
}
static if (isSigned!(G))
{
if (x == 0 && m <= -1) return x / 0;
}
if (x == 1) return 1;
static if (isSigned!(G))
{
if (m < 0) return 0;
}
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;
}
///
@safe pure nothrow @nogc unittest
{
assert(pow(2, 3) == 8);
assert(pow(3, 2) == 9);
assert(pow(2, 10) == 1_024);
assert(pow(2, 20) == 1_048_576);
assert(pow(2, 30) == 1_073_741_824);
assert(pow(0, 0) == 1);
assert(pow(1, -5) == 1);
assert(pow(1, -6) == 1);
assert(pow(-1, -5) == -1);
assert(pow(-1, -6) == 1);
assert(pow(-2, 5) == -32);
assert(pow(-2, -5) == 0);
assert(pow(cast(double) -2, -5) == -0.03125);
}
@safe pure nothrow @nogc 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);
}
// https://issues.dlang.org/show_bug.cgi?id=7006
@safe pure nothrow @nogc unittest
{
assert(pow(5, -1) == 0);
assert(pow(-5, -1) == 0);
assert(pow(5, -2) == 0);
assert(pow(-5, -2) == 0);
assert(pow(-1, int.min) == 1);
assert(pow(-2, int.min) == 0);
assert(pow(4294967290UL,2) == 18446744022169944100UL);
assert(pow(0,uint.max) == 0);
}
/**Computes integer to floating point powers.*/
real pow(I, F)(I x, F y) @nogc @trusted pure nothrow
if (isIntegral!I && isFloatingPoint!F)
{
return pow(cast(real) x, cast(Unqual!F) y);
}
///
@safe pure nothrow @nogc unittest
{
assert(pow(2, 5.0) == 32.0);
assert(pow(7, 3.0) == 343.0);
assert(pow(2, real.nan) is real.nan);
assert(pow(2, real.infinity) == real.infinity);
}
/**
* Calculates x$(SUPERSCRIPT 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) @nogc @trusted pure nothrow
if (isFloatingPoint!(F) && isFloatingPoint!(G))
{
import core.math : fabs, sqrt;
import std.math.traits : isInfinity, isNaN, signbit;
alias Float = typeof(return);
static real impl(real x, real y) @nogc pure nothrow
{
// Special cases.
if (isNaN(y))
return y;
if (isNaN(x) && y != 0.0)
return x;
// Even if x is NaN.
if (y == 0.0)
return 1.0;
if (y == 1.0)
return x;
if (isInfinity(y))
{
if (isInfinity(x))
{
if (!signbit(y) && !signbit(x))
return F.infinity;
else
return F.nan;
}
else if (fabs(x) > 1)
{
if (signbit(y))
return +0.0;
else
return F.infinity;
}
else if (fabs(x) == 1)
{
return F.nan;
}
else // < 1
{
if (signbit(y))
return F.infinity;
else
return +0.0;
}
}
if (isInfinity(x))
{
if (signbit(x))
{
long i = cast(long) y;
if (y > 0.0)
{
if (i == y && i & 1)
return -F.infinity;
else if (i == y)
return F.infinity;
else
return -F.nan;
}
else if (y < 0.0)
{
if (i == y && i & 1)
return -0.0;
else if (i == y)
return +0.0;
else
return F.nan;
}
}
else
{
if (y > 0.0)
return F.infinity;
else if (y < 0.0)
return +0.0;
}
}
if (x == 0.0)
{
if (signbit(x))
{
long i = cast(long) y;
if (y > 0.0)
{
if (i == y && i & 1)
return -0.0;
else
return +0.0;
}
else if (y < 0.0)
{
if (i == y && i & 1)
return -F.infinity;
else
return F.infinity;
}
}
else
{
if (y > 0.0)
return +0.0;
else if (y < 0.0)
return F.infinity;
}
}
if (x == 1.0)
return 1.0;
if (y >= F.max)
{
if ((x > 0.0 && x < 1.0) || (x > -1.0 && x < 0.0))
return 0.0;
if (x > 1.0 || x < -1.0)
return F.infinity;
}
if (y <= -F.max)
{
if ((x > 0.0 && x < 1.0) || (x > -1.0 && x < 0.0))
return F.infinity;
if (x > 1.0 || x < -1.0)
return 0.0;
}
if (x >= F.max)
{
if (y > 0.0)
return F.infinity;
else
return 0.0;
}
if (x <= -F.max)
{
long i = cast(long) y;
if (y > 0.0)
{
if (i == y && i & 1)
return -F.infinity;
else
return F.infinity;
}
else if (y < 0.0)
{
if (i == y && i & 1)
return -0.0;
else
return +0.0;
}
}
// Integer power of x.
long iy = cast(long) y;
if (iy == y && fabs(y) < 32_768.0)
return pow(x, iy);
real sign = 1.0;
if (x < 0)
{
// Result is real only if y is an integer
// Check for a non-zero fractional part
enum maxOdd = pow(2.0L, real.mant_dig) - 1.0L;
static if (maxOdd > ulong.max)
{
// Generic method, for any FP type
import std.math.rounding : floor;
if (floor(y) != y)
return sqrt(x); // Complex result -- create a NaN
const hy = 0.5 * y;
if (floor(hy) != hy)
sign = -1.0;
}
else
{
// Much faster, if ulong has enough precision
const absY = fabs(y);
if (absY <= maxOdd)
{
const uy = cast(ulong) absY;
if (uy != absY)
return sqrt(x); // Complex result -- create a NaN
if (uy & 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( core.math.yl2x(x, y) );
}
else
{
// 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).
Float w = exp2(y * log2(x));
return sign * w;
}
}
return impl(x, y);
}
///
@safe pure nothrow @nogc unittest
{
import std.math.operations : isClose;
assert(isClose(pow(2.0, 3.0), 8.0));
assert(isClose(pow(1.5, 10.0), 57.6650390625));
// square root of 9
assert(isClose(pow(9.0, 0.5), 3.0));
// 10th root of 1024
assert(isClose(pow(1024.0, 0.1), 2.0));
assert(isClose(pow(-4.0, 3.0), -64.0));
// reciprocal of 4 ^^ 2
assert(isClose(pow(4.0, -2.0), 0.0625));
// reciprocal of (-2) ^^ 3
assert(isClose(pow(-2.0, -3.0), -0.125));
assert(isClose(pow(-2.5, 3.0), -15.625));
// reciprocal of 2.5 ^^ 3
assert(isClose(pow(2.5, -3.0), 0.064));
// reciprocal of (-2.5) ^^ 3
assert(isClose(pow(-2.5, -3.0), -0.064));
// reciprocal of square root of 4
assert(isClose(pow(4.0, -0.5), 0.5));
// per definition
assert(isClose(pow(0.0, 0.0), 1.0));
}
///
@safe pure nothrow @nogc unittest
{
import std.math.operations : isClose;
// the result is a complex number
// which cannot be represented as floating point number
import std.math.traits : isNaN;
assert(isNaN(pow(-2.5, -1.5)));
// use the ^^-operator of std.complex instead
import std.complex : complex;
auto c1 = complex(-2.5, 0.0);
auto c2 = complex(-1.5, 0.0);
auto result = c1 ^^ c2;
// exact result apparently depends on `real` precision => increased tolerance
assert(isClose(result.re, -4.64705438e-17, 2e-4));
assert(isClose(result.im, 2.52982e-1, 2e-4));
}
@safe pure nothrow @nogc unittest
{
import std.math.traits : isNaN;
assert(pow(1.5, real.infinity) == real.infinity);
assert(pow(0.5, real.infinity) == 0.0);
assert(pow(1.5, -real.infinity) == 0.0);
assert(pow(0.5, -real.infinity) == real.infinity);
assert(pow(real.infinity, 1.0) == real.infinity);
assert(pow(real.infinity, -1.0) == 0.0);
assert(pow(real.infinity, real.infinity) == real.infinity);
assert(pow(-real.infinity, 1.0) == -real.infinity);
assert(pow(-real.infinity, 2.0) == real.infinity);
assert(pow(-real.infinity, -1.0) == -0.0);
assert(pow(-real.infinity, -2.0) == 0.0);
assert(isNaN(pow(1.0, real.infinity)));
assert(pow(0.0, -1.0) == real.infinity);
assert(pow(real.nan, 0.0) == 1.0);
assert(isNaN(pow(real.nan, 3.0)));
assert(isNaN(pow(3.0, real.nan)));
}
@safe @nogc nothrow unittest
{
import std.math.operations : isClose;
assert(isClose(pow(2.0L, 10.0L), 1024, 1e-18));
}
@safe pure nothrow @nogc unittest
{
import std.math.operations : isClose;
import std.math.traits : isIdentical, isNaN;
import std.math.constants : PI;
// 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));
// https://issues.dlang.org/show_bug.cgi?id=14786 fixed
immutable real maxOdd = pow(2.0L, real.mant_dig) - 1.0L;
assert(pow(-1.0L, maxOdd) == -1.0L);
assert(pow(-1.0L, -maxOdd) == -1.0L);
assert(pow(-1.0L, maxOdd + 1.0L) == 1.0L);
assert(pow(-1.0L, -maxOdd + 1.0L) == 1.0L);
assert(pow(-1.0L, maxOdd - 1.0L) == 1.0L);
assert(pow(-1.0L, -maxOdd - 1.0L) == 1.0L);
// Now, actual numbers.
assert(isClose(pow(two, three), 8.0));
assert(isClose(pow(two, -2.5), 0.1767766953));
// Test integer to float power.
immutable uint twoI = 2;
assert(isClose(pow(twoI, three), 8.0));
}
// https://issues.dlang.org/show_bug.cgi?id=20508
@safe pure nothrow @nogc unittest
{
import std.math.traits : isNaN;
assert(isNaN(pow(-double.infinity, 0.5)));
assert(isNaN(pow(-real.infinity, real.infinity)));
assert(isNaN(pow(-real.infinity, -real.infinity)));
assert(isNaN(pow(-real.infinity, 1.234)));
assert(isNaN(pow(-real.infinity, -0.751)));
assert(pow(-real.infinity, 0.0) == 1.0);
}
/** Computes the value of a positive integer `x`, raised to the power `n`, modulo `m`.
*
* Params:
* x = base
* n = exponent
* m = modulus
*
* Returns:
* `x` to the power `n`, modulo `m`.
* The return type is the largest of `x`'s and `m`'s type.
*
* The function requires that all values have unsigned types.
*/
Unqual!(Largest!(F, H)) powmod(F, G, H)(F x, G n, H m)
if (isUnsigned!F && isUnsigned!G && isUnsigned!H)
{
import std.meta : AliasSeq;
alias T = Unqual!(Largest!(F, H));
static if (T.sizeof <= 4)
{
alias DoubleT = AliasSeq!(void, ushort, uint, void, ulong)[T.sizeof];
}
static T mulmod(T a, T b, T c)
{
static if (T.sizeof == 8)
{
static T addmod(T a, T b, T c)
{
b = c - b;
if (a >= b)
return a - b;
else
return c - b + a;
}
T result = 0, tmp;
b %= c;
while (a > 0)
{
if (a & 1)
result = addmod(result, b, c);
a >>= 1;
b = addmod(b, b, c);
}
return result;
}
else
{
DoubleT result = cast(DoubleT) (cast(DoubleT) a * cast(DoubleT) b);
return result % c;
}
}
T base = x, result = 1, modulus = m;
Unqual!G exponent = n;
while (exponent > 0)
{
if (exponent & 1)
result = mulmod(result, base, modulus);
base = mulmod(base, base, modulus);
exponent >>= 1;
}
return result;
}
///
@safe pure nothrow @nogc unittest
{
assert(powmod(1U, 10U, 3U) == 1);
assert(powmod(3U, 2U, 6U) == 3);
assert(powmod(5U, 5U, 15U) == 5);
assert(powmod(2U, 3U, 5U) == 3);
assert(powmod(2U, 4U, 5U) == 1);
assert(powmod(2U, 5U, 5U) == 2);
}
@safe pure nothrow @nogc unittest
{
ulong a = 18446744073709551615u, b = 20u, c = 18446744073709551610u;
assert(powmod(a, b, c) == 95367431640625u);
a = 100; b = 7919; c = 18446744073709551557u;
assert(powmod(a, b, c) == 18223853583554725198u);
a = 117; b = 7919; c = 18446744073709551557u;
assert(powmod(a, b, c) == 11493139548346411394u);
a = 134; b = 7919; c = 18446744073709551557u;
assert(powmod(a, b, c) == 10979163786734356774u);
a = 151; b = 7919; c = 18446744073709551557u;
assert(powmod(a, b, c) == 7023018419737782840u);
a = 168; b = 7919; c = 18446744073709551557u;
assert(powmod(a, b, c) == 58082701842386811u);
a = 185; b = 7919; c = 18446744073709551557u;
assert(powmod(a, b, c) == 17423478386299876798u);
a = 202; b = 7919; c = 18446744073709551557u;
assert(powmod(a, b, c) == 5522733478579799075u);
a = 219; b = 7919; c = 18446744073709551557u;
assert(powmod(a, b, c) == 15230218982491623487u);
a = 236; b = 7919; c = 18446744073709551557u;
assert(powmod(a, b, c) == 5198328724976436000u);
a = 0; b = 7919; c = 18446744073709551557u;
assert(powmod(a, b, c) == 0);
a = 123; b = 0; c = 18446744073709551557u;
assert(powmod(a, b, c) == 1);
immutable ulong a1 = 253, b1 = 7919, c1 = 18446744073709551557u;
assert(powmod(a1, b1, c1) == 3883707345459248860u);
uint x = 100 ,y = 7919, z = 1844674407u;
assert(powmod(x, y, z) == 1613100340u);
x = 134; y = 7919; z = 1844674407u;
assert(powmod(x, y, z) == 734956622u);
x = 151; y = 7919; z = 1844674407u;
assert(powmod(x, y, z) == 1738696945u);
x = 168; y = 7919; z = 1844674407u;
assert(powmod(x, y, z) == 1247580927u);
x = 185; y = 7919; z = 1844674407u;
assert(powmod(x, y, z) == 1293855176u);
x = 202; y = 7919; z = 1844674407u;
assert(powmod(x, y, z) == 1566963682u);
x = 219; y = 7919; z = 1844674407u;
assert(powmod(x, y, z) == 181227807u);
x = 236; y = 7919; z = 1844674407u;
assert(powmod(x, y, z) == 217988321u);
x = 253; y = 7919; z = 1844674407u;
assert(powmod(x, y, z) == 1588843243u);
x = 0; y = 7919; z = 184467u;
assert(powmod(x, y, z) == 0);
x = 123; y = 0; z = 1844674u;
assert(powmod(x, y, z) == 1);
immutable ubyte x1 = 117;
immutable uint y1 = 7919;
immutable uint z1 = 1844674407u;
auto res = powmod(x1, y1, z1);
assert(is(typeof(res) == uint));
assert(res == 9479781u);
immutable ushort x2 = 123;
immutable uint y2 = 203;
immutable ubyte z2 = 113;
auto res2 = powmod(x2, y2, z2);
assert(is(typeof(res2) == ushort));
assert(res2 == 42u);
}
/**
* Calculates e$(SUPERSCRIPT x).
*
* $(TABLE_SV
* $(TR $(TH x) $(TH e$(SUPERSCRIPT x)) )
* $(TR $(TD +$(INFIN)) $(TD +$(INFIN)) )
* $(TR $(TD -$(INFIN)) $(TD +0.0) )
* $(TR $(TD $(NAN)) $(TD $(NAN)) )
* )
*/
pragma(inline, true)
real exp(real x) @trusted pure nothrow @nogc // TODO: @safe
{
version (InlineAsm_X87)
{
import std.math.constants : LOG2E;
// e^^x = 2^^(LOG2E*x)
// (This is valid because the overflow & underflow limits for exp
// and exp2 are so similar).
if (!__ctfe)
return exp2Asm(LOG2E*x);
}
return expImpl(x);
}
/// ditto
pragma(inline, true)
double exp(double x) @safe pure nothrow @nogc { return __ctfe ? cast(double) exp(cast(real) x) : expImpl(x); }
/// ditto
pragma(inline, true)
float exp(float x) @safe pure nothrow @nogc { return __ctfe ? cast(float) exp(cast(real) x) : expImpl(x); }
///
@safe unittest
{
import std.math.operations : feqrel;
import std.math.constants : E;
assert(exp(0.0) == 1.0);
assert(exp(3.0).feqrel(E * E * E) > 16);