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/* This file is part of an implementation of the "grisu3" double to string
conversion algorithm described in the research paper
"Printing Floating-Point Numbers Quickly And Accurately with Integers"
by Florian Loitsch, available at
http://www.cs.tufts.edu/~nr/cs257/archive/florian-loitsch/printf.pdf */
#include <stdint.h> // uint64_t etc.
#include <assert.h> // assert
#include <math.h> // ceil
#include <stdio.h> // sprintf
#ifdef _MSC_VER
#pragma warning(disable : 4204) // nonstandard extension used : non-constant aggregate initializer
#endif
#define D64_SIGN 0x8000000000000000ULL
#define D64_EXP_MASK 0x7FF0000000000000ULL
#define D64_FRACT_MASK 0x000FFFFFFFFFFFFFULL
#define D64_IMPLICIT_ONE 0x0010000000000000ULL
#define D64_EXP_POS 52
#define D64_EXP_BIAS 1075
#define DIYFP_FRACT_SIZE 64
#define D_1_LOG2_10 0.30102999566398114 // 1 / lg(10)
#define MIN_TARGET_EXP -60
#define MASK32 0xFFFFFFFFULL
#define CAST_U64(d) (*(uint64_t*)&d)
#define MIN(x,y) ((x) <= (y) ? (x) : (y))
#define MAX(x,y) ((x) >= (y) ? (x) : (y))
#define MIN_CACHED_EXP -348
#define CACHED_EXP_STEP 8
typedef struct diy_fp
{
uint64_t f;
int e;
} diy_fp;
typedef struct power
{
uint64_t fract;
int16_t b_exp, d_exp;
} power;
static const power pow_cache[] =
{
{ 0xfa8fd5a0081c0288ULL, -1220, -348 },
{ 0xbaaee17fa23ebf76ULL, -1193, -340 },
{ 0x8b16fb203055ac76ULL, -1166, -332 },
{ 0xcf42894a5dce35eaULL, -1140, -324 },
{ 0x9a6bb0aa55653b2dULL, -1113, -316 },
{ 0xe61acf033d1a45dfULL, -1087, -308 },
{ 0xab70fe17c79ac6caULL, -1060, -300 },
{ 0xff77b1fcbebcdc4fULL, -1034, -292 },
{ 0xbe5691ef416bd60cULL, -1007, -284 },
{ 0x8dd01fad907ffc3cULL, -980, -276 },
{ 0xd3515c2831559a83ULL, -954, -268 },
{ 0x9d71ac8fada6c9b5ULL, -927, -260 },
{ 0xea9c227723ee8bcbULL, -901, -252 },
{ 0xaecc49914078536dULL, -874, -244 },
{ 0x823c12795db6ce57ULL, -847, -236 },
{ 0xc21094364dfb5637ULL, -821, -228 },
{ 0x9096ea6f3848984fULL, -794, -220 },
{ 0xd77485cb25823ac7ULL, -768, -212 },
{ 0xa086cfcd97bf97f4ULL, -741, -204 },
{ 0xef340a98172aace5ULL, -715, -196 },
{ 0xb23867fb2a35b28eULL, -688, -188 },
{ 0x84c8d4dfd2c63f3bULL, -661, -180 },
{ 0xc5dd44271ad3cdbaULL, -635, -172 },
{ 0x936b9fcebb25c996ULL, -608, -164 },
{ 0xdbac6c247d62a584ULL, -582, -156 },
{ 0xa3ab66580d5fdaf6ULL, -555, -148 },
{ 0xf3e2f893dec3f126ULL, -529, -140 },
{ 0xb5b5ada8aaff80b8ULL, -502, -132 },
{ 0x87625f056c7c4a8bULL, -475, -124 },
{ 0xc9bcff6034c13053ULL, -449, -116 },
{ 0x964e858c91ba2655ULL, -422, -108 },
{ 0xdff9772470297ebdULL, -396, -100 },
{ 0xa6dfbd9fb8e5b88fULL, -369, -92 },
{ 0xf8a95fcf88747d94ULL, -343, -84 },
{ 0xb94470938fa89bcfULL, -316, -76 },
{ 0x8a08f0f8bf0f156bULL, -289, -68 },
{ 0xcdb02555653131b6ULL, -263, -60 },
{ 0x993fe2c6d07b7facULL, -236, -52 },
{ 0xe45c10c42a2b3b06ULL, -210, -44 },
{ 0xaa242499697392d3ULL, -183, -36 },
{ 0xfd87b5f28300ca0eULL, -157, -28 },
{ 0xbce5086492111aebULL, -130, -20 },
{ 0x8cbccc096f5088ccULL, -103, -12 },
{ 0xd1b71758e219652cULL, -77, -4 },
{ 0x9c40000000000000ULL, -50, 4 },
{ 0xe8d4a51000000000ULL, -24, 12 },
{ 0xad78ebc5ac620000ULL, 3, 20 },
{ 0x813f3978f8940984ULL, 30, 28 },
{ 0xc097ce7bc90715b3ULL, 56, 36 },
{ 0x8f7e32ce7bea5c70ULL, 83, 44 },
{ 0xd5d238a4abe98068ULL, 109, 52 },
{ 0x9f4f2726179a2245ULL, 136, 60 },
{ 0xed63a231d4c4fb27ULL, 162, 68 },
{ 0xb0de65388cc8ada8ULL, 189, 76 },
{ 0x83c7088e1aab65dbULL, 216, 84 },
{ 0xc45d1df942711d9aULL, 242, 92 },
{ 0x924d692ca61be758ULL, 269, 100 },
{ 0xda01ee641a708deaULL, 295, 108 },
{ 0xa26da3999aef774aULL, 322, 116 },
{ 0xf209787bb47d6b85ULL, 348, 124 },
{ 0xb454e4a179dd1877ULL, 375, 132 },
{ 0x865b86925b9bc5c2ULL, 402, 140 },
{ 0xc83553c5c8965d3dULL, 428, 148 },
{ 0x952ab45cfa97a0b3ULL, 455, 156 },
{ 0xde469fbd99a05fe3ULL, 481, 164 },
{ 0xa59bc234db398c25ULL, 508, 172 },
{ 0xf6c69a72a3989f5cULL, 534, 180 },
{ 0xb7dcbf5354e9beceULL, 561, 188 },
{ 0x88fcf317f22241e2ULL, 588, 196 },
{ 0xcc20ce9bd35c78a5ULL, 614, 204 },
{ 0x98165af37b2153dfULL, 641, 212 },
{ 0xe2a0b5dc971f303aULL, 667, 220 },
{ 0xa8d9d1535ce3b396ULL, 694, 228 },
{ 0xfb9b7cd9a4a7443cULL, 720, 236 },
{ 0xbb764c4ca7a44410ULL, 747, 244 },
{ 0x8bab8eefb6409c1aULL, 774, 252 },
{ 0xd01fef10a657842cULL, 800, 260 },
{ 0x9b10a4e5e9913129ULL, 827, 268 },
{ 0xe7109bfba19c0c9dULL, 853, 276 },
{ 0xac2820d9623bf429ULL, 880, 284 },
{ 0x80444b5e7aa7cf85ULL, 907, 292 },
{ 0xbf21e44003acdd2dULL, 933, 300 },
{ 0x8e679c2f5e44ff8fULL, 960, 308 },
{ 0xd433179d9c8cb841ULL, 986, 316 },
{ 0x9e19db92b4e31ba9ULL, 1013, 324 },
{ 0xeb96bf6ebadf77d9ULL, 1039, 332 },
{ 0xaf87023b9bf0ee6bULL, 1066, 340 }
};
static int cached_pow(int exp, diy_fp *p)
{
int k = (int)ceil((exp+DIYFP_FRACT_SIZE-1) * D_1_LOG2_10);
int i = (k-MIN_CACHED_EXP-1) / CACHED_EXP_STEP + 1;
p->f = pow_cache[i].fract;
p->e = pow_cache[i].b_exp;
return pow_cache[i].d_exp;
}
static diy_fp minus(diy_fp x, diy_fp y)
{
diy_fp d; d.f = x.f - y.f; d.e = x.e;
assert(x.e == y.e && x.f >= y.f);
return d;
}
static diy_fp multiply(diy_fp x, diy_fp y)
{
uint64_t a, b, c, d, ac, bc, ad, bd, tmp;
diy_fp r;
a = x.f >> 32; b = x.f & MASK32;
c = y.f >> 32; d = y.f & MASK32;
ac = a*c; bc = b*c;
ad = a*d; bd = b*d;
tmp = (bd >> 32) + (ad & MASK32) + (bc & MASK32);
tmp += 1U << 31; // round
r.f = ac + (ad >> 32) + (bc >> 32) + (tmp >> 32);
r.e = x.e + y.e + 64;
return r;
}
static diy_fp normalize_diy_fp(diy_fp n)
{
assert(n.f != 0);
while(!(n.f & 0xFFC0000000000000ULL)) { n.f <<= 10; n.e -= 10; }
while(!(n.f & D64_SIGN)) { n.f <<= 1; --n.e; }
return n;
}
static diy_fp double2diy_fp(double d)
{
diy_fp fp;
uint64_t u64 = CAST_U64(d);
if (!(u64 & D64_EXP_MASK)) { fp.f = u64 & D64_FRACT_MASK; fp.e = 1 - D64_EXP_BIAS; }
else { fp.f = (u64 & D64_FRACT_MASK) + D64_IMPLICIT_ONE; fp.e = (int)((u64 & D64_EXP_MASK) >> D64_EXP_POS) - D64_EXP_BIAS; }
return fp;
}
// pow10_cache[i] = 10^(i-1)
static const unsigned int pow10_cache[] = { 0, 1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000 };
static int largest_pow10(uint32_t n, int n_bits, uint32_t *power)
{
int guess = ((n_bits + 1) * 1233 >> 12) + 1/*skip first entry*/;
if (n < pow10_cache[guess]) --guess; // We don't have any guarantees that 2^n_bits <= n.
*power = pow10_cache[guess];
return guess;
}
static int round_weed(char *buffer, int len, uint64_t wp_W, uint64_t delta, uint64_t rest, uint64_t ten_kappa, uint64_t ulp)
{
uint64_t wp_Wup = wp_W - ulp;
uint64_t wp_Wdown = wp_W + ulp;
while(rest < wp_Wup && delta - rest >= ten_kappa
&& (rest + ten_kappa < wp_Wup || wp_Wup - rest >= rest + ten_kappa - wp_Wup))
{
--buffer[len-1];
rest += ten_kappa;
}
if (rest < wp_Wdown && delta - rest >= ten_kappa
&& (rest + ten_kappa < wp_Wdown || wp_Wdown - rest > rest + ten_kappa - wp_Wdown))
return 0;
return 2*ulp <= rest && rest <= delta - 4*ulp;
}
static int digit_gen(diy_fp low, diy_fp w, diy_fp high, char *buffer, int *length, int *kappa)
{
uint64_t unit = 1;
diy_fp too_low = { low.f - unit, low.e };
diy_fp too_high = { high.f + unit, high.e };
diy_fp unsafe_interval = minus(too_high, too_low);
diy_fp one = { 1ULL << -w.e, w.e };
uint32_t p1 = (uint32_t)(too_high.f >> -one.e);
uint64_t p2 = too_high.f & (one.f - 1);
uint32_t div;
*kappa = largest_pow10(p1, DIYFP_FRACT_SIZE + one.e, &div);
*length = 0;
while(*kappa > 0)
{
uint64_t rest;
int digit = p1 / div;
buffer[*length] = (char)('0' + digit);
++*length;
p1 %= div;
--*kappa;
rest = ((uint64_t)p1 << -one.e) + p2;
if (rest < unsafe_interval.f) return round_weed(buffer, *length, minus(too_high, w).f, unsafe_interval.f, rest, (uint64_t)div << -one.e, unit);
div /= 10;
}
for(;;)
{
int digit;
p2 *= 10;
unit *= 10;
unsafe_interval.f *= 10;
// Integer division by one.
digit = (int)(p2 >> -one.e);
buffer[*length] = (char)('0' + digit);
++*length;
p2 &= one.f - 1; // Modulo by one.
--*kappa;
if (p2 < unsafe_interval.f) return round_weed(buffer, *length, minus(too_high, w).f * unit, unsafe_interval.f, p2, one.f, unit);
}
}
static int grisu3(double v, char *buffer, int *length, int *d_exp)
{
int mk, kappa, success;
diy_fp dfp = double2diy_fp(v);
diy_fp w = normalize_diy_fp(dfp);
// normalize boundaries
diy_fp t = { (dfp.f << 1) + 1, dfp.e - 1 };
diy_fp b_plus = normalize_diy_fp(t);
diy_fp b_minus;
diy_fp c_mk; // Cached power of ten: 10^-k
uint64_t u64 = CAST_U64(v);
assert(v > 0 && v <= 1.7976931348623157e308); // Grisu only handles strictly positive finite numbers.
if (!(u64 & D64_FRACT_MASK) && (u64 & D64_EXP_MASK) != 0) { b_minus.f = (dfp.f << 2) - 1; b_minus.e = dfp.e - 2;} // lower boundary is closer?
else { b_minus.f = (dfp.f << 1) - 1; b_minus.e = dfp.e - 1; }
b_minus.f = b_minus.f << (b_minus.e - b_plus.e);
b_minus.e = b_plus.e;
mk = cached_pow(MIN_TARGET_EXP - DIYFP_FRACT_SIZE - w.e, &c_mk);
w = multiply(w, c_mk);
b_minus = multiply(b_minus, c_mk);
b_plus = multiply(b_plus, c_mk);
success = digit_gen(b_minus, w, b_plus, buffer, length, &kappa);
*d_exp = kappa - mk;
return success;
}
static int i_to_str(int val, char *str)
{
int len, i;
char *s;
char *begin = str;
if (val < 0) { *str++ = '-'; val = -val; }
s = str;
for(;;)
{
int ni = val / 10;
int digit = val - ni*10;
*s++ = (char)('0' + digit);
if (ni == 0)
break;
val = ni;
}
*s = '\0';
len = (int)(s - str);
for(i = 0; i < len/2; ++i)
{
char ch = str[i];
str[i] = str[len-1-i];
str[len-1-i] = ch;
}
return (int)(s - begin);
}
int dtoa_grisu3(double v, char *dst)
{
int d_exp, len, success, decimals, i;
uint64_t u64 = CAST_U64(v);
char *s2 = dst;
assert(dst);
// Prehandle NaNs
if ((u64 << 1) > 0xFFE0000000000000ULL) return sprintf(dst, "NaN(%08X%08X)", (uint32_t)(u64 >> 32), (uint32_t)u64);
// Prehandle negative values.
if ((u64 & D64_SIGN) != 0) { *s2++ = '-'; v = -v; u64 ^= D64_SIGN; }
// Prehandle zero.
if (!u64) { *s2++ = '0'; *s2 = '\0'; return (int)(s2 - dst); }
// Prehandle infinity.
if (u64 == D64_EXP_MASK) { *s2++ = 'i'; *s2++ = 'n'; *s2++ = 'f'; *s2 = '\0'; return (int)(s2 - dst); }
success = grisu3(v, s2, &len, &d_exp);
// If grisu3 was not able to convert the number to a string, then use old sprintf (suboptimal).
if (!success) return sprintf(s2, "%.17g", v) + (int)(s2 - dst);
// We now have an integer string of form "151324135" and a base-10 exponent for that number.
// Next, decide the best presentation for that string by whether to use a decimal point, or the scientific exponent notation 'e'.
// We don't pick the absolute shortest representation, but pick a balance between readability and shortness, e.g.
// 1.545056189557677e-308 could be represented in a shorter form
// 1545056189557677e-323 but that would be somewhat unreadable.
decimals = MIN(-d_exp, MAX(1, len-1));
if (d_exp < 0 && len > 1) // Add decimal point?
{
for(i = 0; i < decimals; ++i) s2[len-i] = s2[len-i-1];
s2[len++ - decimals] = '.';
d_exp += decimals;
// Need scientific notation as well?
if (d_exp != 0) { s2[len++] = 'e'; len += i_to_str(d_exp, s2+len); }
}
else if (d_exp < 0 && d_exp >= -3) // Add decimal point for numbers of form 0.000x where it's shorter?
{
for(i = 0; i < len; ++i) s2[len-d_exp-1-i] = s2[len-i-1];
s2[0] = '.';
for(i = 1; i < -d_exp; ++i) s2[i] = '0';
len += -d_exp;
}
// Add scientific notation?
else if (d_exp < 0 || d_exp > 2) { s2[len++] = 'e'; len += i_to_str(d_exp, s2+len); }
// Add zeroes instead of scientific notation?
else if (d_exp > 0) { while(d_exp-- > 0) s2[len++] = '0'; }
s2[len] = '\0'; // grisu3 doesn't null terminate, so ensure termination.
return (int)(s2+len-dst);
}