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#if defined(AX_CONFIG_FP_SIMD)
#include <float.h>
#include <math.h>
#include <stddef.h>
#include <stdint.h>
#include <stdio.h>
#include "libm.h"
#define OFF 0x3fe6955500000000
#define POW_LOG_TABLE_BITS 7
#define POW_LOG_POLY_ORDER 8
#define N (1 << POW_LOG_TABLE_BITS)
struct pow_log_data {
double ln2hi;
double ln2lo;
double poly[POW_LOG_POLY_ORDER - 1]; /* First coefficient is 1. */
/* Note: the pad field is unused, but allows slightly faster indexing. */
struct {
double invc, pad, logc, logctail;
} tab[1 << POW_LOG_TABLE_BITS];
};
const struct
pow_log_data
__pow_log_data =
{
.ln2hi = 0x1.62e42fefa3800p-1,
.ln2lo = 0x1.ef35793c76730p-45,
.poly =
{
-0x1p-1,
0x1.555555555556p-2 * -2,
-0x1.0000000000006p-2 * -2,
0x1.999999959554ep-3 * 4,
-0x1.555555529a47ap-3 * 4,
0x1.2495b9b4845e9p-3 * -8,
-0x1.0002b8b263fc3p-3 * -8,
},
.tab =
{
#define A(a, b, c) {a, 0, b, c},
A(0x1.6a00000000000p+0, -0x1.62c82f2b9c800p-2, 0x1.ab42428375680p-48)
A(0x1.6800000000000p+0, -0x1.5d1bdbf580800p-2, -0x1.ca508d8e0f720p-46)
A(0x1.6600000000000p+0, -0x1.5767717455800p-2,
-0x1.362a4d5b6506dp-45)
A(0x1.6400000000000p+0, -0x1.51aad872df800p-2,
-0x1.684e49eb067d5p-49) A(0x1.6200000000000p+0,
-0x1.4be5f95777800p-2,
-0x1.41b6993293ee0p-47) A(0x1.6000000000000p+0, -0x1.4618bc21c6000p-2, 0x1.3d82f484c84ccp-46) A(0x1.5e00000000000p+0, -0x1.404308686a800p-2, 0x1.c42f3ed820b3ap-50) A(0x1.5c00000000000p+0, -0x1.3a64c55694800p-2, 0x1.0b1c686519460p-45) A(0x1.5a00000000000p+0, -0x1.347dd9a988000p-2, 0x1.5594dd4c58092p-45) A(0x1.5800000000000p+0, -0x1.2e8e2bae12000p-2, 0x1.67b1e99b72bd8p-45) A(0x1.5600000000000p+0,
-0x1.2895a13de8800p-2, 0x1.5ca14b6cfb03fp-46) A(0x1.5600000000000p+0,
-0x1.2895a13de8800p-2, 0x1.5ca14b6cfb03fp-46) A(0x1.5400000000000p+0,
-0x1.22941fbcf7800p-2,
-0x1.65a242853da76p-46) A(0x1.5200000000000p+0,
-0x1.1c898c1699800p-2,
-0x1.fafbc68e75404p-46) A(0x1.5000000000000p+0,
-0x1.1675cababa800p-2, 0x1.f1fc63382a8f0p-46) A(0x1.4e00000000000p+0,
-0x1.1058bf9ae4800p-2,
-0x1.6a8c4fd055a66p-45) A(0x1.4c00000000000p+0,
-0x1.0a324e2739000p-2, -0x1.c6bee7ef4030ep-47) A(0x1.4a00000000000p+0,
-0x1.0402594b4d000p-2, -0x1.036b89ef42d7fp-48) A(0x1.4a00000000000p+0,
-0x1.0402594b4d000p-2, -0x1.036b89ef42d7fp-48) A(0x1.4800000000000p+0,
-0x1.fb9186d5e4000p-3,
0x1.d572aab993c87p-47) A(0x1.4600000000000p+0,
-0x1.ef0adcbdc6000p-3,
0x1.b26b79c86af24p-45) A(0x1.4400000000000p+0,
-0x1.e27076e2af000p-3, -0x1.72f4f543fff10p-46) A(0x1.4200000000000p+0,
-0x1.d5c216b4fc000p-3, 0x1.1ba91bbca681bp-45) A(0x1.4000000000000p+0, -0x1.c8ff7c79aa000p-3, 0x1.7794f689f8434p-45) A(0x1.4000000000000p+0, -0x1.c8ff7c79aa000p-3, 0x1.7794f689f8434p-45) A(0x1.3e00000000000p+0,
-0x1.bc286742d9000p-3, 0x1.94eb0318bb78fp-46) A(0x1.3c00000000000p+0,
-0x1.af3c94e80c000p-3, 0x1.a4e633fcd9066p-52) A(0x1.3a00000000000p+0,
-0x1.a23bc1fe2b000p-3,
-0x1.58c64dc46c1eap-45) A(0x1.3a00000000000p+0,
-0x1.a23bc1fe2b000p-3, -0x1.58c64dc46c1eap-45) A(0x1.3800000000000p+0,
-0x1.9525a9cf45000p-3, -0x1.ad1d904c1d4e3p-45) A(0x1.3600000000000p+0,
-0x1.87fa06520d000p-3, 0x1.bbdbf7fdbfa09p-45) A(0x1.3400000000000p+0,
-0x1.7ab890210e000p-3, 0x1.bdb9072534a58p-45) A(0x1.3400000000000p+0,
-0x1.7ab890210e000p-3, 0x1.bdb9072534a58p-45) A(0x1.3200000000000p+0, -0x1.6d60fe719d000p-3, -0x1.0e46aa3b2e266p-46) A(0x1.3000000000000p+0,
-0x1.5ff3070a79000p-3, -0x1.e9e439f105039p-46) A(0x1.3000000000000p+0,
-0x1.5ff3070a79000p-3, -0x1.e9e439f105039p-46) A(0x1.2e00000000000p+0,
-0x1.526e5e3a1b000p-3, -0x1.0de8b90075b8fp-45) A(0x1.2c00000000000p+0,
-0x1.44d2b6ccb8000p-3, 0x1.70cc16135783cp-46) A(0x1.2c00000000000p+0, -0x1.44d2b6ccb8000p-3, 0x1.70cc16135783cp-46) A(0x1.2a00000000000p+0,
-0x1.371fc201e9000p-3, 0x1.178864d27543ap-48) A(0x1.2800000000000p+0,
-0x1.29552f81ff000p-3, -0x1.48d301771c408p-45) A(0x1.2600000000000p+0, -0x1.1b72ad52f6000p-3, -0x1.e80a41811a396p-45) A(0x1.2600000000000p+0, -0x1.1b72ad52f6000p-3, -0x1.e80a41811a396p-45) A(0x1.2400000000000p+0,
-0x1.0d77e7cd09000p-3,
0x1.a699688e85bf4p-47) A(0x1.2400000000000p+0, -0x1.0d77e7cd09000p-3, 0x1.a699688e85bf4p-47) A(0x1.2200000000000p+0, -0x1.fec9131dbe000p-4, -0x1.575545ca333f2p-45) A(0x1.2000000000000p+0,
-0x1.e27076e2b0000p-4, 0x1.a342c2af0003cp-45) A(0x1.2000000000000p+0, -0x1.e27076e2b0000p-4, 0x1.a342c2af0003cp-45) A(0x1.1e00000000000p+0, -0x1.c5e548f5bc000p-4, -0x1.d0c57585fbe06p-46) A(0x1.1c00000000000p+0,
-0x1.a926d3a4ae000p-4, 0x1.53935e85baac8p-45) A(0x1.1c00000000000p+0,
-0x1.a926d3a4ae000p-4, 0x1.53935e85baac8p-45) A(0x1.1a00000000000p+0, -0x1.8c345d631a000p-4, 0x1.37c294d2f5668p-46) A(0x1.1a00000000000p+0,
-0x1.8c345d631a000p-4,
0x1.37c294d2f5668p-46) A(0x1.1800000000000p+0, -0x1.6f0d28ae56000p-4,
-0x1.69737c93373dap-45) A(0x1.1600000000000p+0,
-0x1.51b073f062000p-4, 0x1.f025b61c65e57p-46) A(0x1.1600000000000p+0,
-0x1.51b073f062000p-4, 0x1.f025b61c65e57p-46) A(0x1.1400000000000p+0, -0x1.341d7961be000p-4, 0x1.c5edaccf913dfp-45) A(0x1.1400000000000p+0,
-0x1.341d7961be000p-4, 0x1.c5edaccf913dfp-45) A(0x1.1200000000000p+0, -0x1.16536eea38000p-4, 0x1.47c5e768fa309p-46) A(0x1.1000000000000p+0,
-0x1.f0a30c0118000p-5, 0x1.d599e83368e91p-45) A(0x1.1000000000000p+0, -0x1.f0a30c0118000p-5,
0x1.d599e83368e91p-45) A(0x1.0e00000000000p+0,
-0x1.b42dd71198000p-5, 0x1.c827ae5d6704cp-46) A(0x1.0e00000000000p+0,
-0x1.b42dd71198000p-5, 0x1.c827ae5d6704cp-46) A(0x1.0c00000000000p+0,
-0x1.77458f632c000p-5, -0x1.cfc4634f2a1eep-45) A(0x1.0c00000000000p+0, -0x1.77458f632c000p-5, -0x1.cfc4634f2a1eep-45) A(0x1.0a00000000000p+0, -0x1.39e87b9fec000p-5, 0x1.502b7f526feaap-48) A(0x1.0a00000000000p+0,
-0x1.39e87b9fec000p-5, 0x1.502b7f526feaap-48) A(0x1.0800000000000p+0, -0x1.f829b0e780000p-6, -0x1.980267c7e09e4p-45) A(0x1.0800000000000p+0, -0x1.f829b0e780000p-6, -0x1.980267c7e09e4p-45) A(0x1.0600000000000p+0,
-0x1.7b91b07d58000p-6, -0x1.88d5493faa639p-45) A(0x1.0400000000000p+0,
-0x1.fc0a8b0fc0000p-7, -0x1.f1e7cf6d3a69cp-50) A(0x1.0400000000000p+0, -0x1.fc0a8b0fc0000p-7, -0x1.f1e7cf6d3a69cp-50) A(0x1.0200000000000p+0, -0x1.fe02a6b100000p-8, -0x1.9e23f0dda40e4p-46) A(0x1.0200000000000p+0,
-0x1.fe02a6b100000p-8,
-0x1.9e23f0dda40e4p-46) A(0x1.0000000000000p+0,
0x0.0000000000000p+0, 0x0.0000000000000p+0) A(0x1.0000000000000p+0,
0x0.0000000000000p+0,
0x0.0000000000000p+0) A(0x1.fc00000000000p-1,
0x1.0101575890000p-7, -0x1.0c76b999d2be8p-46) A(0x1.f800000000000p-1,
0x1.0205658938000p-6, -0x1.3dc5b06e2f7d2p-45) A(0x1.f400000000000p-1,
0x1.8492528c90000p-6,
-0x1.aa0ba325a0c34p-45) A(0x1.f000000000000p-1,
0x1.0415d89e74000p-5,
0x1.111c05cf1d753p-47) A(0x1.ec00000000000p-1, 0x1.466aed42e0000p-5, -0x1.c167375bdfd28p-45) A(0x1.e800000000000p-1,
0x1.894aa149fc000p-5,
-0x1.97995d05a267dp-46) A(0x1.e400000000000p-1,
0x1.ccb73cdddc000p-5, -0x1.a68f247d82807p-46) A(0x1.e200000000000p-1,
0x1.eea31c006c000p-5,
-0x1.e113e4fc93b7bp-47) A(0x1.de00000000000p-1,
0x1.1973bd1466000p-4,
-0x1.5325d560d9e9bp-45) A(0x1.da00000000000p-1, 0x1.3bdf5a7d1e000p-4, 0x1.cc85ea5db4ed7p-45) A(0x1.d600000000000p-1, 0x1.5e95a4d97a000p-4, -0x1.c69063c5d1d1ep-45) A(0x1.d400000000000p-1, 0x1.700d30aeac000p-4, 0x1.c1e8da99ded32p-49) A(0x1.d000000000000p-1,
0x1.9335e5d594000p-4,
0x1.3115c3abd47dap-45) A(0x1.cc00000000000p-1,
0x1.b6ac88dad6000p-4,
-0x1.390802bf768e5p-46) A(0x1.ca00000000000p-1,
0x1.c885801bc4000p-4,
0x1.646d1c65aacd3p-45) A(0x1.c600000000000p-1,
0x1.ec739830a2000p-4,
-0x1.dc068afe645e0p-45) A(0x1.c400000000000p-1,
0x1.fe89139dbe000p-4,
-0x1.534d64fa10afdp-45) A(0x1.c000000000000p-1, 0x1.1178e8227e000p-3, 0x1.1ef78ce2d07f2p-45) A(0x1.be00000000000p-1, 0x1.1aa2b7e23f000p-3, 0x1.ca78e44389934p-45) A(0x1.ba00000000000p-1, 0x1.2d1610c868000p-3, 0x1.39d6ccb81b4a1p-47) A(0x1.b800000000000p-1, 0x1.365fcb0159000p-3, 0x1.62fa8234b7289p-51) A(0x1.b400000000000p-1, 0x1.4913d8333b000p-3, 0x1.5837954fdb678p-45) A(0x1.b200000000000p-1, 0x1.527e5e4a1b000p-3,
0x1.633e8e5697dc7p-45) A(0x1.ae00000000000p-1,
0x1.6574ebe8c1000p-3,
0x1.9cf8b2c3c2e78p-46) A(0x1.ac00000000000p-1,
0x1.6f0128b757000p-3, -0x1.5118de59c21e1p-45) A(0x1.aa00000000000p-1, 0x1.7898d85445000p-3, -0x1.c661070914305p-46) A(0x1.a600000000000p-1,
0x1.8beafeb390000p-3, -0x1.73d54aae92cd1p-47) A(0x1.a400000000000p-1, 0x1.95a5adcf70000p-3, 0x1.7f22858a0ff6fp-47) A(0x1.a000000000000p-1, 0x1.a93ed3c8ae000p-3, -0x1.8724350562169p-45) A(0x1.9e00000000000p-1, 0x1.b31d8575bd000p-3, -0x1.c358d4eace1aap-47) A(0x1.9c00000000000p-1, 0x1.bd087383be000p-3, -0x1.d4bc4595412b6p-45) A(0x1.9a00000000000p-1, 0x1.c6ffbc6f01000p-3, -0x1.1ec72c5962bd2p-48) A(0x1.9600000000000p-1, 0x1.db13db0d49000p-3, -0x1.aff2af715b035p-45) A(0x1.9400000000000p-1,
0x1.e530effe71000p-3,
0x1.212276041f430p-51) A(0x1.9200000000000p-1, 0x1.ef5ade4dd0000p-3, -0x1.a211565bb8e11p-51) A(0x1.9000000000000p-1, 0x1.f991c6cb3b000p-3, 0x1.bcbecca0cdf30p-46) A(0x1.8c00000000000p-1, 0x1.07138604d5800p-2, 0x1.89cdb16ed4e91p-48) A(0x1.8a00000000000p-1,
0x1.0c42d67616000p-2,
0x1.7188b163ceae9p-45) A(0x1.8800000000000p-1, 0x1.1178e8227e800p-2, -0x1.c210e63a5f01cp-45) A(0x1.8600000000000p-1,
0x1.16b5ccbacf800p-2,
0x1.b9acdf7a51681p-45) A(0x1.8400000000000p-1,
0x1.1bf99635a6800p-2,
0x1.ca6ed5147bdb7p-45) A(0x1.8200000000000p-1,
0x1.214456d0eb800p-2,
0x1.a87deba46baeap-47) A(0x1.7e00000000000p-1,
0x1.2bef07cdc9000p-2, 0x1.a9cfa4a5004f4p-45) A(0x1.7c00000000000p-1, 0x1.314f1e1d36000p-2, -0x1.8e27ad3213cb8p-45) A(0x1.7a00000000000p-1, 0x1.36b6776be1000p-2, 0x1.16ecdb0f177c8p-46) A(0x1.7800000000000p-1,
0x1.3c25277333000p-2,
0x1.83b54b606bd5cp-46) A(0x1.7600000000000p-1,
0x1.419b423d5e800p-2,
0x1.8e436ec90e09dp-47) A(0x1.7400000000000p-1, 0x1.4718dc271c800p-2, -0x1.f27ce0967d675p-45) A(0x1.7200000000000p-1, 0x1.4c9e09e173000p-2, -0x1.e20891b0ad8a4p-45) A(0x1.7000000000000p-1,
0x1.522ae0738a000p-2,
0x1.ebe708164c759p-45) A(0x1.6e00000000000p-1, 0x1.57bf753c8d000p-2, 0x1.fadedee5d40efp-46) A(0x1.6c00000000000p-1,
0x1.5d5bddf596000p-2,
-0x1.a0b2a08a465dcp-47)},
};
#define T __pow_log_data.tab
#undef A
#define A __pow_log_data.poly
#define Ln2hi __pow_log_data.ln2hi
#define Ln2lo __pow_log_data.ln2lo
/* Top 12 bits of a double (sign and exponent bits). */
static inline uint32_t top12(double x)
{
return asuint64(x) >> 52;
}
/* Compute y+TAIL = log(x) where the rounded result is y and TAIL has about
additional 15 bits precision. IX is the bit representation of x, but
normalized in the subnormal range using the sign bit for the exponent. */
static inline double_t log_inline(uint64_t ix, double_t *tail)
{
/* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
double_t z, r, y, invc, logc, logctail, kd, hi, t1, t2, lo, lo1, lo2, p;
uint64_t iz, tmp;
int k, i;
/* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
The range is split into N subintervals.
The ith subinterval contains z and c is near its center. */
tmp = ix - OFF;
i = (tmp >> (52 - POW_LOG_TABLE_BITS)) % N;
k = (int64_t)tmp >> 52; /* arithmetic shift */
iz = ix - (tmp & 0xfffULL << 52);
z = asdouble(iz);
kd = (double_t)k;
/* log(x) = k*Ln2 + log(c) + log1p(z/c-1). */
invc = T[i].invc;
logc = T[i].logc;
logctail = T[i].logctail;
/* Note: 1/c is j/N or j/N/2 where j is an integer in [N,2N) and
|z/c - 1| < 1/N, so r = z/c - 1 is exactly representible. */
#if __FP_FAST_FMA
r = __builtin_fma(z, invc, -1.0);
#else
/* Split z such that rhi, rlo and rhi*rhi are exact and |rlo| <= |r|. */
double_t zhi = asdouble((iz + (1ULL << 31)) & (-1ULL << 32));
double_t zlo = z - zhi;
double_t rhi = zhi * invc - 1.0;
double_t rlo = zlo * invc;
r = rhi + rlo;
#endif
/* k*Ln2 + log(c) + r. */
t1 = kd * Ln2hi + logc;
t2 = t1 + r;
lo1 = kd * Ln2lo + logctail;
lo2 = t1 - t2 + r;
/* Evaluation is optimized assuming superscalar pipelined execution. */
double_t ar, ar2, ar3, lo3, lo4;
ar = A[0] * r; /* A[0] = -0.5. */
ar2 = r * ar;
ar3 = r * ar2;
/* k*Ln2 + log(c) + r + A[0]*r*r. */
#if __FP_FAST_FMA
hi = t2 + ar2;
lo3 = __builtin_fma(ar, r, -ar2);
lo4 = t2 - hi + ar2;
#else
double_t arhi = A[0] * rhi;
double_t arhi2 = rhi * arhi;
hi = t2 + arhi2;
lo3 = rlo * (ar + arhi);
lo4 = t2 - hi + arhi2;
#endif
/* p = log1p(r) - r - A[0]*r*r. */
p = (ar3 * (A[1] + r * A[2] + ar2 * (A[3] + r * A[4] + ar2 * (A[5] + r * A[6]))));
lo = lo1 + lo2 + lo3 + lo4 + p;
y = hi + lo;
*tail = hi - y + lo;
return y;
}
#undef N
#undef T
#define EXP_TABLE_BITS 7
#define EXP_POLY_ORDER 5
#define EXP_USE_TOINT_NARROW 0
#define EXP2_POLY_ORDER 5
struct exp_data {
double invln2N;
double shift;
double negln2hiN;
double negln2loN;
double poly[4]; /* Last four coefficients. */
double exp2_shift;
double exp2_poly[EXP2_POLY_ORDER];
uint64_t tab[2 * (1 << EXP_TABLE_BITS)];
};
#define N (1 << EXP_TABLE_BITS)
const struct exp_data __exp_data = {
// N/ln2
.invln2N = 0x1.71547652b82fep0 * N,
// -ln2/N
.negln2hiN = -0x1.62e42fefa0000p-8,
.negln2loN = -0x1.cf79abc9e3b3ap-47,
// Used for rounding when !TOINT_INTRINSICS
#if EXP_USE_TOINT_NARROW
.shift = 0x1800000000.8p0,
#else
.shift = 0x1.8p52,
#endif
// exp polynomial coefficients.
.poly =
{
// abs error: 1.555*2^-66
// ulp error: 0.509 (0.511 without fma)
// if |x| < ln2/256+eps
// abs error if |x| < ln2/256+0x1p-15: 1.09*2^-65
// abs error if |x| < ln2/128: 1.7145*2^-56
0x1.ffffffffffdbdp-2,
0x1.555555555543cp-3,
0x1.55555cf172b91p-5,
0x1.1111167a4d017p-7,
},
.exp2_shift = 0x1.8p52 / N,
// exp2 polynomial coefficients.
.exp2_poly =
{
// abs error: 1.2195*2^-65
// ulp error: 0.507 (0.511 without fma)
// if |x| < 1/256
// abs error if |x| < 1/128: 1.9941*2^-56
0x1.62e42fefa39efp-1,
0x1.ebfbdff82c424p-3,
0x1.c6b08d70cf4b5p-5,
0x1.3b2abd24650ccp-7,
0x1.5d7e09b4e3a84p-10,
},
// 2^(k/N) ~= H[k]*(1 + T[k]) for int k in [0,N)
// tab[2*k] = asuint64(T[k])
// tab[2*k+1] = asuint64(H[k]) - (k << 52)/N
.tab =
{
0x0,
0x3ff0000000000000,
0x3c9b3b4f1a88bf6e,
0x3feff63da9fb3335,
0xbc7160139cd8dc5d,
0x3fefec9a3e778061,
0xbc905e7a108766d1,
0x3fefe315e86e7f85,
0x3c8cd2523567f613,
0x3fefd9b0d3158574,
0xbc8bce8023f98efa,
0x3fefd06b29ddf6de,
0x3c60f74e61e6c861,
0x3fefc74518759bc8,
0x3c90a3e45b33d399,
0x3fefbe3ecac6f383,
0x3c979aa65d837b6d,
0x3fefb5586cf9890f,
0x3c8eb51a92fdeffc,
0x3fefac922b7247f7,
0x3c3ebe3d702f9cd1,
0x3fefa3ec32d3d1a2,
0xbc6a033489906e0b,
0x3fef9b66affed31b,
0xbc9556522a2fbd0e,
0x3fef9301d0125b51,
0xbc5080ef8c4eea55,
0x3fef8abdc06c31cc,
0xbc91c923b9d5f416,
0x3fef829aaea92de0,
0x3c80d3e3e95c55af,
0x3fef7a98c8a58e51,
0xbc801b15eaa59348,
0x3fef72b83c7d517b,
0xbc8f1ff055de323d,
0x3fef6af9388c8dea,
0x3c8b898c3f1353bf,
0x3fef635beb6fcb75,
0xbc96d99c7611eb26,
0x3fef5be084045cd4,
0x3c9aecf73e3a2f60,
0x3fef54873168b9aa,
0xbc8fe782cb86389d,
0x3fef4d5022fcd91d,
0x3c8a6f4144a6c38d,
0x3fef463b88628cd6,
0x3c807a05b0e4047d,
0x3fef3f49917ddc96,
0x3c968efde3a8a894,
0x3fef387a6e756238,
0x3c875e18f274487d,
0x3fef31ce4fb2a63f,
0x3c80472b981fe7f2,
0x3fef2b4565e27cdd,
0xbc96b87b3f71085e,
0x3fef24dfe1f56381,
0x3c82f7e16d09ab31,
0x3fef1e9df51fdee1,
0xbc3d219b1a6fbffa,
0x3fef187fd0dad990,
0x3c8b3782720c0ab4,
0x3fef1285a6e4030b,
0x3c6e149289cecb8f,
0x3fef0cafa93e2f56,
0x3c834d754db0abb6,
0x3fef06fe0a31b715,
0x3c864201e2ac744c,
0x3fef0170fc4cd831,
0x3c8fdd395dd3f84a,
0x3feefc08b26416ff,
0xbc86a3803b8e5b04,
0x3feef6c55f929ff1,
0xbc924aedcc4b5068,
0x3feef1a7373aa9cb,
0xbc9907f81b512d8e,
0x3feeecae6d05d866,
0xbc71d1e83e9436d2,
0x3feee7db34e59ff7,
0xbc991919b3ce1b15,
0x3feee32dc313a8e5,
0x3c859f48a72a4c6d,
0x3feedea64c123422,
0xbc9312607a28698a,
0x3feeda4504ac801c,
0xbc58a78f4817895b,
0x3feed60a21f72e2a,
0xbc7c2c9b67499a1b,
0x3feed1f5d950a897,
0x3c4363ed60c2ac11,
0x3feece086061892d,
0x3c9666093b0664ef,
0x3feeca41ed1d0057,
0x3c6ecce1daa10379,
0x3feec6a2b5c13cd0,
0x3c93ff8e3f0f1230,
0x3feec32af0d7d3de,
0x3c7690cebb7aafb0,
0x3feebfdad5362a27,
0x3c931dbdeb54e077,
0x3feebcb299fddd0d,
0xbc8f94340071a38e,
0x3feeb9b2769d2ca7,
0xbc87deccdc93a349,
0x3feeb6daa2cf6642,
0xbc78dec6bd0f385f,
0x3feeb42b569d4f82,
0xbc861246ec7b5cf6,
0x3feeb1a4ca5d920f,
0x3c93350518fdd78e,
0x3feeaf4736b527da,
0x3c7b98b72f8a9b05,
0x3feead12d497c7fd,
0x3c9063e1e21c5409,
0x3feeab07dd485429,
0x3c34c7855019c6ea,
0x3feea9268a5946b7,
0x3c9432e62b64c035,
0x3feea76f15ad2148,
0xbc8ce44a6199769f,
0x3feea5e1b976dc09,
0xbc8c33c53bef4da8,
0x3feea47eb03a5585,
0xbc845378892be9ae,
0x3feea34634ccc320,
0xbc93cedd78565858,
0x3feea23882552225,
0x3c5710aa807e1964,
0x3feea155d44ca973,
0xbc93b3efbf5e2228,
0x3feea09e667f3bcd,
0xbc6a12ad8734b982,
0x3feea012750bdabf,
0xbc6367efb86da9ee,
0x3fee9fb23c651a2f,
0xbc80dc3d54e08851,
0x3fee9f7df9519484,
0xbc781f647e5a3ecf,
0x3fee9f75e8ec5f74,
0xbc86ee4ac08b7db0,
0x3fee9f9a48a58174,
0xbc8619321e55e68a,
0x3fee9feb564267c9,
0x3c909ccb5e09d4d3,
0x3feea0694fde5d3f,
0xbc7b32dcb94da51d,
0x3feea11473eb0187,
0x3c94ecfd5467c06b,
0x3feea1ed0130c132,
0x3c65ebe1abd66c55,
0x3feea2f336cf4e62,
0xbc88a1c52fb3cf42,
0x3feea427543e1a12,
0xbc9369b6f13b3734,
0x3feea589994cce13,
0xbc805e843a19ff1e,
0x3feea71a4623c7ad,
0xbc94d450d872576e,
0x3feea8d99b4492ed,
0x3c90ad675b0e8a00,
0x3feeaac7d98a6699,
0x3c8db72fc1f0eab4,
0x3feeace5422aa0db,
0xbc65b6609cc5e7ff,
0x3feeaf3216b5448c,
0x3c7bf68359f35f44,
0x3feeb1ae99157736,
0xbc93091fa71e3d83,
0x3feeb45b0b91ffc6,
0xbc5da9b88b6c1e29,
0x3feeb737b0cdc5e5,
0xbc6c23f97c90b959,
0x3feeba44cbc8520f,
0xbc92434322f4f9aa,
0x3feebd829fde4e50,
0xbc85ca6cd7668e4b,
0x3feec0f170ca07ba,
0x3c71affc2b91ce27,
0x3feec49182a3f090,
0x3c6dd235e10a73bb,
0x3feec86319e32323,
0xbc87c50422622263,
0x3feecc667b5de565,
0x3c8b1c86e3e231d5,
0x3feed09bec4a2d33,
0xbc91bbd1d3bcbb15,
0x3feed503b23e255d,
0x3c90cc319cee31d2,
0x3feed99e1330b358,
0x3c8469846e735ab3,
0x3feede6b5579fdbf,
0xbc82dfcd978e9db4,
0x3feee36bbfd3f37a,
0x3c8c1a7792cb3387,
0x3feee89f995ad3ad,
0xbc907b8f4ad1d9fa,
0x3feeee07298db666,
0xbc55c3d956dcaeba,
0x3feef3a2b84f15fb,
0xbc90a40e3da6f640,
0x3feef9728de5593a,
0xbc68d6f438ad9334,
0x3feeff76f2fb5e47,
0xbc91eee26b588a35,
0x3fef05b030a1064a,
0x3c74ffd70a5fddcd,
0x3fef0c1e904bc1d2,
0xbc91bdfbfa9298ac,
0x3fef12c25bd71e09,
0x3c736eae30af0cb3,
0x3fef199bdd85529c,
0x3c8ee3325c9ffd94,
0x3fef20ab5fffd07a,
0x3c84e08fd10959ac,
0x3fef27f12e57d14b,
0x3c63cdaf384e1a67,
0x3fef2f6d9406e7b5,
0x3c676b2c6c921968,
0x3fef3720dcef9069,
0xbc808a1883ccb5d2,
0x3fef3f0b555dc3fa,
0xbc8fad5d3ffffa6f,
0x3fef472d4a07897c,
0xbc900dae3875a949,
0x3fef4f87080d89f2,
0x3c74a385a63d07a7,
0x3fef5818dcfba487,
0xbc82919e2040220f,
0x3fef60e316c98398,
0x3c8e5a50d5c192ac,
0x3fef69e603db3285,
0x3c843a59ac016b4b,
0x3fef7321f301b460,
0xbc82d52107b43e1f,
0x3fef7c97337b9b5f,
0xbc892ab93b470dc9,
0x3fef864614f5a129,
0x3c74b604603a88d3,
0x3fef902ee78b3ff6,
0x3c83c5ec519d7271,
0x3fef9a51fbc74c83,
0xbc8ff7128fd391f0,
0x3fefa4afa2a490da,
0xbc8dae98e223747d,
0x3fefaf482d8e67f1,
0x3c8ec3bc41aa2008,
0x3fefba1bee615a27,
0x3c842b94c3a9eb32,
0x3fefc52b376bba97,
0x3c8a64a931d185ee,
0x3fefd0765b6e4540,
0xbc8e37bae43be3ed,
0x3fefdbfdad9cbe14,
0x3c77893b4d91cd9d,
0x3fefe7c1819e90d8,
0x3c5305c14160cc89,
0x3feff3c22b8f71f1,
},
};
#define InvLn2N __exp_data.invln2N
#define NegLn2hiN __exp_data.negln2hiN
#define NegLn2loN __exp_data.negln2loN
#define Shift __exp_data.shift
#define T __exp_data.tab
#define C2 __exp_data.poly[5 - EXP_POLY_ORDER]
#define C3 __exp_data.poly[6 - EXP_POLY_ORDER]
#define C4 __exp_data.poly[7 - EXP_POLY_ORDER]
#define C5 __exp_data.poly[8 - EXP_POLY_ORDER]
#define C6 __exp_data.poly[9 - EXP_POLY_ORDER]
static inline double specialcase(double_t tmp, uint64_t sbits, uint64_t ki)
{
double_t scale, y;
if ((ki & 0x80000000) == 0) {
/* k > 0, the exponent of scale might have overflowed by <= 460. */
sbits -= 1009ull << 52;
scale = asdouble(sbits);
y = 0x1p1009 * (scale + scale * tmp);
return eval_as_double(y);
}
/* k < 0, need special care in the subnormal range. */
sbits += 1022ull << 52;
/* Note: sbits is signed scale. */
scale = asdouble(sbits);
y = scale + scale * tmp;
if (fabs(y) < 1.0) {
/* Round y to the right precision before scaling it into the subnormal
range to avoid double rounding that can cause 0.5+E/2 ulp error where
E is the worst-case ulp error outside the subnormal range. So this
is only useful if the goal is better than 1 ulp worst-case error. */
double_t hi, lo, one = 1.0;
if (y < 0.0)
one = -1.0;
lo = scale - y + scale * tmp;
hi = one + y;
lo = one - hi + y + lo;
y = eval_as_double(hi + lo) - one;
/* Fix the sign of 0. */
if (y == 0.0)
y = asdouble(sbits & 0x8000000000000000);
/* The underflow exception needs to be signaled explicitly. */
fp_force_eval(fp_barrier(0x1p-1022) * 0x1p-1022);
}
y = 0x1p-1022 * y;
return eval_as_double(y);
}
#define SIGN_BIAS (0x800 << EXP_TABLE_BITS)
double __math_xflow(uint32_t sign, double y)
{
return eval_as_double(fp_barrier(sign ? -y : y) * y);
}
double __math_uflow(uint32_t sign)
{
return __math_xflow(sign, 0x1p-767);
}
double __math_oflow(uint32_t sign)
{
return __math_xflow(sign, 0x1p769);
}
/* Computes sign*exp(x+xtail) where |xtail| < 2^-8/N and |xtail| <= |x|.
The sign_bias argument is SIGN_BIAS or 0 and sets the sign to -1 or 1. */
static inline double exp_inline(double_t x, double_t xtail, uint32_t sign_bias)
{
uint32_t abstop;
uint64_t ki, idx, top, sbits;
/* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
double_t kd, z, r, r2, scale, tail, tmp;
abstop = top12(x) & 0x7ff;
if (predict_false(abstop - top12(0x1p-54) >= top12(512.0) - top12(0x1p-54))) {
if (abstop - top12(0x1p-54) >= 0x80000000) {
/* Avoid spurious underflow for tiny x. */
/* Note: 0 is common input. */
double_t one = WANT_ROUNDING ? 1.0 + x : 1.0;
return sign_bias ? -one : one;
}
if (abstop >= top12(1024.0)) {
/* Note: inf and nan are already handled. */
if (asuint64(x) >> 63)
return __math_uflow(sign_bias);
else
return __math_oflow(sign_bias);
}
/* Large x is special cased below. */
abstop = 0;
}
/* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)]. */
/* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N]. */
z = InvLn2N * x;
#if TOINT_INTRINSICS
kd = roundtoint(z);
ki = converttoint(z);
#elif EXP_USE_TOINT_NARROW
/* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes. */
kd = eval_as_double(z + Shift);
ki = asuint64(kd) >> 16;
kd = (double_t)(int32_t)ki;
#else
/* z - kd is in [-1, 1] in non-nearest rounding modes. */
kd = eval_as_double(z + Shift);
ki = asuint64(kd);
kd -= Shift;
#endif
r = x + kd * NegLn2hiN + kd * NegLn2loN;
/* The code assumes 2^-200 < |xtail| < 2^-8/N. */
r += xtail;
/* 2^(k/N) ~= scale * (1 + tail). */
idx = 2 * (ki % N);
top = (ki + sign_bias) << (52 - EXP_TABLE_BITS);
tail = asdouble(T[idx]);
/* This is only a valid scale when -1023*N < k < 1024*N. */
sbits = T[idx + 1] + top;
/* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1). */
/* Evaluation is optimized assuming superscalar pipelined execution. */
r2 = r * r;
/* Without fma the worst case error is 0.25/N ulp larger. */
/* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp. */
tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
if (predict_false(abstop == 0))
return specialcase(tmp, sbits, ki);
scale = asdouble(sbits);
/* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there
is no spurious underflow here even without fma. */
return eval_as_double(scale + scale * tmp);
}
/* Returns 0 if not int, 1 if odd int, 2 if even int. The argument is
the bit representation of a non-zero finite floating-point value. */
static inline int checkint(uint64_t iy)
{
int e = iy >> 52 & 0x7ff;
if (e < 0x3ff)
return 0;
if (e > 0x3ff + 52)
return 2;
if (iy & ((1ULL << (0x3ff + 52 - e)) - 1))
return 0;
if (iy & (1ULL << (0x3ff + 52 - e)))
return 1;
return 2;
}
#if 100 * __GNUC__ + __GNUC_MINOR__ >= 303
#define NAN __builtin_nanf("")
#define INFINITY __builtin_inff()
#else
#define NAN (0.0f / 0.0f)
#define INFINITY 1e5000f
#endif
static inline int zeroinfnan(uint64_t i)
{
return 2 * i - 1 >= 2 * asuint64(INFINITY) - 1;
}
#if WANT_SNAN
#error SNaN is unsupported
#else
#define issignalingf_inline(x) 0
#define issignaling_inline(x) 0
#endif
double pow(double x, double y)
{
uint32_t sign_bias = 0;
uint64_t ix, iy;
uint32_t topx, topy;
ix = asuint64(x);
iy = asuint64(y);
topx = top12(x);
topy = top12(y);
if (predict_false(topx - 0x001 >= 0x7ff - 0x001 || (topy & 0x7ff) - 0x3be >= 0x43e - 0x3be)) {
/* Note: if |y| > 1075 * ln2 * 2^53 ~= 0x1.749p62 then pow(x,y) = inf/0
and if |y| < 2^-54 / 1075 ~= 0x1.e7b6p-65 then pow(x,y) = +-1. */
/* Special cases: (x < 0x1p-126 or inf or nan) or
(|y| < 0x1p-65 or |y| >= 0x1p63 or nan). */
if (predict_false(zeroinfnan(iy))) {
if (2 * iy == 0)
return issignaling_inline(x) ? x + y : 1.0;
if (ix == asuint64(1.0))
return issignaling_inline(y) ? x + y : 1.0;
if (2 * ix > 2 * asuint64(INFINITY) || 2 * iy > 2 * asuint64(INFINITY))
return x + y;
if (2 * ix == 2 * asuint64(1.0))
return 1.0;
if ((2 * ix < 2 * asuint64(1.0)) == !(iy >> 63))
return 0.0; /* |x|<1 && y==inf or |x|>1 && y==-inf. */
return y * y;
}
if (predict_false(zeroinfnan(ix))) {
double_t x2 = x * x;
if (ix >> 63 && checkint(iy) == 1)
x2 = -x2;
/* Without the barrier some versions of clang hoist the 1/x2 and
thus division by zero exception can be signaled spuriously. */
return iy >> 63 ? fp_barrier(1 / x2) : x2;
}
/* Here x and y are non-zero finite. */
if (ix >> 63) {
/* Finite x < 0. */
int yint = checkint(iy);
if (yint == 0)
return __math_invalid(x);
if (yint == 1)
sign_bias = SIGN_BIAS;
ix &= 0x7fffffffffffffff;
topx &= 0x7ff;
}
if ((topy & 0x7ff) - 0x3be >= 0x43e - 0x3be) {
/* Note: sign_bias == 0 here because y is not odd. */
if (ix == asuint64(1.0))
return 1.0;
if ((topy & 0x7ff) < 0x3be) {
/* |y| < 2^-65, x^y ~= 1 + y*log(x). */
if (WANT_ROUNDING)
return ix > asuint64(1.0) ? 1.0 + y : 1.0 - y;
else
return 1.0;
}
return (ix > asuint64(1.0)) == (topy < 0x800) ? __math_oflow(0) : __math_uflow(0);
}
if (topx == 0) {
/* Normalize subnormal x so exponent becomes negative. */
ix = asuint64(x * 0x1p52);
ix &= 0x7fffffffffffffff;
ix -= 52ULL << 52;
}
}
double_t lo;
double_t hi = log_inline(ix, &lo);
double_t ehi, elo;
#if __FP_FAST_FMA
ehi = y * hi;
elo = y * lo + __builtin_fma(y, hi, -ehi);
#else
double_t yhi = asdouble(iy & -1ULL << 27);
double_t ylo = y - yhi;
double_t lhi = asdouble(asuint64(hi) & -1ULL << 27);
double_t llo = hi - lhi + lo;
ehi = yhi * lhi;
elo = ylo * lhi + y * llo; /* |elo| < |ehi| * 2^-25. */
#endif
return exp_inline(ehi, elo, sign_bias);
}
#endif