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ZIMMERMANN Paul authoredZIMMERMANN Paul authored
acos.c 16.78 KiB
/* Correctly-rounded arc cosine of binary64 value.
Copyright (c) 2024-2025 Alexei Sibidanov.
This file is part of the CORE-MATH project
(https://core-math.gitlabpages.inria.fr/).
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.
*/
#include <stdint.h>
#include <errno.h>
// Warning: clang also defines __GNUC__
#if defined(__GNUC__) && !defined(__clang__)
#pragma GCC diagnostic ignored "-Wunknown-pragmas"
#endif
#pragma STDC FENV_ACCESS ON
/* __builtin_roundeven was introduced in gcc 10:
https://gcc.gnu.org/gcc-10/changes.html,
and in clang 17 */
#if ((defined(__GNUC__) && __GNUC__ >= 10) || (defined(__clang__) && __clang_major__ >= 17)) && (defined(__aarch64__) || defined(__x86_64__) || defined(__i386__))
# define roundeven_finite(x) __builtin_roundeven (x)
#else
/* round x to nearest integer, breaking ties to even */
static double
roundeven_finite (double x)
{
double ix;
# if (defined(__GNUC__) || defined(__clang__)) && (defined(__AVX__) || defined(__SSE4_1__) || (__ARM_ARCH >= 8))
# if defined __AVX__
__asm__("vroundsd $0x8,%1,%1,%0":"=x"(ix):"x"(x));
# elif __ARM_ARCH >= 8
__asm__ ("frintn %d0, %d1":"=w"(ix):"w"(x));
# else /* __SSE4_1__ */
__asm__("roundsd $0x8,%1,%0":"=x"(ix):"x"(x));
# endif
# else
ix = __builtin_round (x); /* nearest, away from 0 */
if (__builtin_fabs (ix - x) == 0.5)
{
/* if ix is odd, we should return ix-1 if x>0, and ix+1 if x<0 */
union { double f; uint64_t n; } u, v;
u.f = ix;
v.f = ix - __builtin_copysign (1.0, x);
if (__builtin_ctz (v.n) > __builtin_ctz (u.n))
ix = v.f;
}
# endif
return ix;
}
#endif
typedef uint64_t u64;
typedef int64_t i64;
typedef unsigned short ushort;
typedef union {double f; uint64_t u;} b64u64_u;
static inline double fasttwosum(double x, double y, double *e){
double s = x + y, z = s - x;
*e = y - z;
return s;
}
static inline double twosum(double xh, double ch, double *l){
double s = xh + ch, d = s - xh;
*l = (ch - d) + (xh + (d - s));
return s;
}
static inline double fastsum(double xh, double xl, double yh, double yl, double *e){
double sl, sh = fasttwosum(xh, yh, &sl);
*e = (xl + yl) + sl;
return sh;
}
static inline double fasttwosub(double x, double y, double *e){
double s = x - y;
*e = (x - s) - y;
return s;
}
static inline double sum(double xh, double xl, double ch, double cl, double *l){
double sl, sh = twosum(xh,ch, &sl);
*l = (xl + cl) + sl;
return sh;
}
static inline double muldd(double xh, double xl, double ch, double cl, double *l){
double ahhh = ch*xh;
*l = (cl*xh + ch*xl) + __builtin_fma(ch, xh, -ahhh);
return ahhh;
}
static inline double polydd(double xh, double xl, int n, const double c[][2], double *l){
int i = n-1;
double ch = fasttwosum(c[i][0], *l, l), cl = c[i][1] + *l;
while(--i>=0){
ch = muldd(xh,xl, ch,cl, &cl);
ch = fastsum(c[i][0],c[i][1], ch,cl, &cl);
}
*l = cl;
return ch;
}
static double __attribute__((noinline,cold)) as_acos_refine(double, double);
double cr_acos (double x){
// coefficients of a polynomial approximation of asin(x):
// asin(x) = x*(cc[j][0] + cc[j][1] + t*P(t, cc[j] + 2))
// where t = x^2 - j/128
static const double cc[][8] = {
{ 1, 0, 0x1.5555555555555p-3, 0x1.33333333333e4p-4,
0x1.6db6db6d31f82p-5, 0x1.f1c71f6889397p-6, 0x1.6e874b7045b46p-6, 0x1.1f753132271e2p-6},
{0x1.0055a27e0d033p+0, -0x1.d9ba10494c062p-54, 0x1.57c00cb5d6c4dp-3, 0x1.37881f5649a75p-4,
0x1.759af49d494ddp-5, 0x1.002e1864dda2ep-5, 0x1.7c2d5d468cdd9p-6, 0x1.292834c025357p-6},
{0x1.00abe0c129e1ep+0, 0x1.7ceb0ee49d42ap-57, 0x1.5a3385d5c7ba5p-3, 0x1.3bf51056f6637p-4,
0x1.7dba76b124b37p-5, 0x1.07be4b02e94c4p-5, 0x1.8a6bb92513f01p-6, 0x1.36afd4c615aecp-6},
{0x1.0102bcffd6acdp+0, -0x1.c2294c65d2e86p-55, 0x1.5caff17351901p-3, 0x1.407abbc04feb3p-4,
0x1.86179b8005949p-5, 0x1.0f97520bd4e72p-5, 0x1.9950c5c89f3dfp-6, 0x1.44f2344e7b664p-6},
{0x1.015a397cf0f1cp+0, -0x1.eebd6ccfe3ee3p-55, 0x1.5f3581be7b08bp-3, 0x1.4519ddf1ae531p-4,
0x1.8eb4b6ee35e92p-5, 0x1.17bc85414cd46p-5, 0x1.a8e5895e3fcf9p-6, 0x1.53fafdc629400p-6}, {0x1.01b2588811eebp+0, 0x1.7193e5d0a915fp-59, 0x1.61c46a67205d2p-3, 0x1.49d33a6eeae0bp-4,
0x1.979438563c014p-5, 0x1.20316ae977f05p-5, 0x1.b9339afb53aa4p-6, 0x1.63d6b02c42d0ap-6},
{0x1.020b1c7df0575p+0, -0x1.dd547e329c1e5p-55, 0x1.645ce0ab901bbp-3, 0x1.4ea79c34fc7a6p-4,
0x1.a0b8ac08940ecp-5, 0x1.28f9babd0629bp-5, 0x1.ca452cf90a55ep-6, 0x1.7492b016730efp-6},
{0x1.026487c8c5d71p+0, -0x1.5fd9b68dc3b6ep-54, 0x1.66ff1b67d5d70p-3, 0x1.5397d613373ebp-4,
0x1.aa24bce3aeb4ap-5, 0x1.3219610b150acp-5, 0x1.dc251825103f1p-6, 0x1.863d5a3932532p-6},
{0x1.02be9ce0b87cdp+0, 0x1.e5d09da2e0f04p-56, 0x1.69ab5325bc359p-3, 0x1.58a4c3097aab3p-4,
0x1.b3db3605f46f2p-5, 0x1.3b94821742cabp-5, 0x1.eedee7da72a15p-6, 0x1.98e6179a3e9a0p-6},
{0x1.03195e4c483f1p+0, -0x1.5db10ad66eacbp-54, 0x1.6c61c22d908f0p-3, 0x1.5dcf46ab9f2cbp-4,
0x1.bddf049bb1f4dp-5, 0x1.456f7db6ac768p-5, 0x1.013f738bd7bb3p-5, 0x1.ac9d739783d21p-6},
{0x1.0374cea0c0c9fp+0, -0x1.917bff5241c76p-54, 0x1.6f22a497b2ec0p-3, 0x1.63184d8a79db5p-4,
0x1.c83339caf946ep-5, 0x1.4faef331019d4p-5, 0x1.0b8917547d678p-5, 0x1.c17533f147e1cp-6},
{0x1.03d0f082afcc8p+0, -0x1.018bbcddb49ebp-54, 0x1.71ee385efdf06p-3, 0x1.6880cda2d3885p-4,
0x1.d2db0cbfae54dp-5, 0x1.5a57c56b50c5ep-5, 0x1.16535a40098b2p-5, 0x1.d780730b8ebb8p-6},
{0x1.042dc6a65ffbfp+0, -0x1.c7ea28dce95d1p-55, 0x1.74c4bd7412f9ep-3, 0x1.6e09c6d2b72bcp-4,
0x1.ddd9dcda253dep-5, 0x1.656f1f62b5001p-5, 0x1.21a5ae2ac77eep-5, 0x1.eed3bca067f0ep-6},
{0x1.048b53d05907bp+0, 0x1.634fffed6e2a6p-54, 0x1.77a675d1978bep-3, 0x1.73b4435583415p-4,
0x1.e9333402ebbf3p-5, 0x1.70fa78fd9f73fp-5, 0x1.2d8804d934fe1p-5, 0x1.03c29691a281cp-5},
{0x1.04e99ad5e4bcdp+0, -0x1.e97a72fe827e0p-54, 0x1.7a93a5917200cp-3, 0x1.7981584731c05p-4,
0x1.f4eac9268fae2p-5, 0x1.7cff9c3b19721p-5, 0x1.3a02d9c1e0145p-5, 0x1.10d64a0e56953p-5},
{0x1.05489e9d99995p+0, 0x1.d177637ec6a2bp-55, 0x1.7d8c930314681p-3, 0x1.7f72262f532e4p-4,
0x1.0082416e39013p-4, 0x1.8984aac80ddf4p-5, 0x1.471f3caf18eb8p-5, 0x1.1eb1cce6dd570p-5},
{0x1.05a8621feb16bp+0, -0x1.e5b33b1407c5fp-56, 0x1.809186c2e57ddp-3, 0x1.8587d99442dc8p-4,
0x1.06c23d1dfcb7fp-4, 0x1.969024036dd22p-5, 0x1.54e6dd4d2af33p-5, 0x1.2d62f439f2a31p-5},
{0x1.0608e867bff30p+0, 0x1.cbef5d8580027p-55, 0x1.83a2cbd2d8ba2p-3, 0x1.8bc3ab9724c6ep-4,
0x1.0d377ef1e0c39p-4, 0x1.a428eb7addf84p-5, 0x1.636417bc01ff2p-5, 0x1.3cf8acc7eb2a0p-5},
{0x1.066a34930ec8dp+0, -0x1.480f445fedad1p-54, 0x1.86c0afb447a74p-3, 0x1.9226e29948d9cp-4,
0x1.13e44a9b5a3a6p-4, 0x1.b2564fea8b3fep-5, 0x1.72a2023d92458p-5, 0x1.4d8313cec3485p-5},
{0x1.06cc49d38146cp+0, -0x1.b55394f4fc07bp-55, 0x1.89eb82831feedp-3, 0x1.98b2d2eb9bb23p-4,
0x1.1acb01e9ab414p-4, 0x1.c12012cbd00c6p-5, 0x1.82ac7c1d15c38p-5, 0x1.5f13925c6edcap-5},
{0x1.072f2b6f1e601p+0, -0x1.2dcbb05419970p-54, 0x1.8d2397127aebbp-3, 0x1.9f68df88da51dp-4,
0x1.21ee26a4f62a1p-4, 0x1.d08e707f7ae6fp-5, 0x1.93903dee3feb0p-5, 0x1.71bcfb5c57b59p-5},
{0x1.0792dcc0fbd20p+0, -0x1.5bf23ee4f9d54p-56, 0x1.9069430ab508ap-3, 0x1.a64a7adb4cd85p-4,
0x1.29505c8b48349p-4, 0x1.e0aa2921cfa60p-5, 0x1.a55aeb46f4322p-5, 0x1.8593acad3becep-5},
{0x1.07f76139f761dp+0, 0x1.fa1046481bb82p-54, 0x1.93bcdf091cca6p-3, 0x1.ad59278edc42fp-4,
0x1.30f46b7261652p-4, 0x1.f17c8a17c843ep-5, 0x1.b81b2619e15b5p-5, 0x1.9aadb395f5ae4p-5},
{0x1.085cbc61783c1p+0, 0x1.0a6e9efa20176p-54, 0x1.971ec6c1531e4p-3, 0x1.b496797068912p-4,
0x1.38dd419140184p-4, 0x1.0187bc3357bbbp-4, 0x1.cbe0a3dcafe26p-5, 0x1.b122f4fa499d0p-5},
{0x1.08c2f1d638e4cp+0, 0x1.b47c159534a3dp-56, 0x1.9a8f592078624p-3, 0x1.bc04165b57ab2p-4,
0x1.410df5f4bed1dp-4, 0x1.0ab6bdf478c71p-4, 0x1.e0bc44a945c64p-5, 0x1.c90d59bcd5701p-5},
{0x1.092a054f1a2fcp+0, -0x1.2f657224e9830p-54, 0x1.9e0ef87243a2cp-3, 0x1.c3a3b7366a278p-4,
0x1.4989cb22e2175p-4, 0x1.1450e5ba7ad39p-4, 0x1.f6c02c8f0ef93p-5, 0x1.e288ffc8d182cp-5},
{0x1.0991fa9bffbf4p+0, -0x1.ca1140a1abbf4p-58, 0x1.a19e0a8823b80p-3, 0x1.cb772900f9c24p-4,
0x1.525431f0cbb2ep-4, 0x1.1e5c2d06804e1p-4, 0x1.06ffefa7aa6b8p-4, 0x1.fdb4704dca347p-5},
{0x1.09fad5a6b68f9p+0, 0x1.aa1f06e92964ep-56, 0x1.a53cf8e28c50ep-3, 0x1.d3804df1de350p-4,
0x1.5b70cc8fa98dcp-4, 0x1.28def298c979bp-4, 0x1.13482f6347eebp-4, 0x1.0d586de48358cp-4},
{0x1.0a649a73e61f2p+0, 0x1.74ac0d817e9c7p-55, 0x1.a8ec30dc93891p-3, 0x1.dbc11ea950625p-4,
0x1.64e371d5616d3p-4, 0x1.33e0023936249p-4, 0x1.204426263066ap-4, 0x1.1cd12e4629723p-4},
{0x1.0acf4d240ccc4p+0, 0x1.da890f3b40bd3p-54, 0x1.acac23da07797p-3, 0x1.e43bab7741a98p-4,
0x1.6eb030c631819p-4, 0x1.3f669d2eb516ep-4, 0x1.2e0006ae505aep-4, 0x1.2d58204457c82p-4},
{0x1.0b3af1f4880bbp+0, 0x1.f450fb78d32bap-56, 0x1.b07d4778263afp-3, 0x1.ecf21db7be0efp-4,
0x1.78db5465013e4p-4, 0x1.4b7a8376f0996p-4, 0x1.3c88f9f2ef221p-4, 0x1.3f02ad9eb9753p-4},
{0x1.0ba78d40a9260p+0, -0x1.57b07a441e242p-54, 0x1.b46015c126262p-3, 0x1.f5e6b94713f3dp-4,
0x1.836967d0afecfp-4, 0x1.5823fdd1707b9p-4, 0x1.4bed355269dc2p-4, 0x1.51e83065121cfp-4},
{0x1.0c152382d7366p+0, -0x1.ee6913347c2a6p-54, 0x1.b8550d62bfb6ep-3, 0x1.ff1bde0fa3cadp-4,
0x1.8e5f3ab550989p-4, 0x1.656be8b38ebafp-4, 0x1.5c3c13008a099p-4, 0x1.662225a1b4f77p-4},
};
b64u64_u ix = {.f = x};
u64 ax = ix.u<<1;
double t,z,zl,jd,f0h,f0l;
if(ax>0x7fc0000000000000ull){ // |x|>0.5
static const double off[][2] = {{0,0}, {0x1.921fb54442d18p+1, 0x1.1a62633145c07p-53}};
i64 k = ix.u>>63;
f0h = off[k][0];
f0l = off[k][1];
if(__builtin_expect(ax>=0x7fe0000000000000ull, 0)){ // |x| >= 1
if(ax==0x7fe0000000000000ull) return f0h + f0l; // |x| = 1
if(ax>0xffe0000000000000ull) return x + x; // nan
#ifdef CORE_MATH_SUPPORT_ERRNO errno = EDOM;
#endif
return 0./0.; // |x|>1
}
// for x>0.5 we use range reduction for double angle formula
// acos(x) = 2*asin((1-x)/2) and for x<-0.5 acos(x) = pi -
// 2*asin((1-x)/2)
t = 2 - 2*__builtin_fabs(x);
jd = roundeven_finite(t*0x1p5);
z = __builtin_copysign(__builtin_sqrt(t), x);
zl = __builtin_fma(z,z,-t)*((-0.5/t)*z);
t = 0.25*t - jd*0x1p-7;
} else { // |x|<=0.5
f0h = 0x1.921fb54442d18p+0;
f0l = 0x1.1a62633145c07p-54;
// for |x| <= 0x1.cb3b399d747f2p-55, acos(x) rounds to pi/2 to nearest
// this avoids a spurious underflow exception with the code below
if(__builtin_expect(ax <= 0x7919676733ae8fe4, 0)) return f0h + f0l;
// for |x|<=0.5 we use acos(x) = pi/2 - asin(x) so the argument
// range for asin is the same for both branches to reuse the lookup
// tables.
t = x*x;
jd = roundeven_finite(t*0x1p7);
t = __builtin_fma(x,x,-0x1p-7*jd);
z = -x;
zl = 0;
}
// asin(xh+xl) = (xh + xl)*(cc[j][0] + (cc[j][1] + t*Poly(t, cc[j]+2)))
// where t = xh^2 - j/128 and j = round(128*xh^2)
int64_t j = jd;
const double *c = cc[j];
double t2 = t*t, d = t*((c[2] + t*c[3]) + t2*((c[4] + t*c[5]) + t2*(c[6] + t*c[7])));
double fh = c[0], fl = c[1] + d;
fh = muldd(z,zl, fh,fl, &fl);
fh = fastsum(f0h,f0l, fh,fl, &fl);
double eps = __builtin_fabs(z*t)*0x1.8bp-52 + 0x1p-105; // all arguments in [-0x1.1a93e5d11dac2p-1, -0x1.1a86cd0e3b2c2p-1] were checked
double lb = fh + (fl - eps), ub = fh + (fl + eps);
if(__builtin_expect(lb!=ub, 0)) return as_acos_refine(x, lb);
return lb;
}
__attribute__((noinline,cold))
static
double as_acos_refine(double x, double phi){
// Consider x as sin(phi) then cos(phi) is ch + cl = sqrt(1-x^2)
// Using angle rotation formula bring the argument close to zero
// where the asin Taylor expansion works well.
double s2 = x*x, dx2 = __builtin_fma(x,x,-s2);
double c2l, c2h = fasttwosub(1.0,s2,&c2l);
c2l -= dx2;
c2h = fasttwosum(c2h,c2l,&c2l);
double c2f = __builtin_fma(x,-x,1);
double ch = __builtin_sqrt(c2f);
double cl = (c2l - __builtin_fma(ch,ch,-c2f))*((0.5/c2f)*ch);
int64_t jf = roundeven_finite(__builtin_fabs(phi - 0x1.921fb54442d18p+0) * 0x1.45f306dc9c883p+4);
// sin(pi/64*j) in the double-double format
static const double s[33][2] = {
{0x0p+0, 0x0p+0}, {-0x1.912bd0d569a9p-61, 0x1.91f65f10dd814p-5},
{-0x1.e2718d26ed688p-60, 0x1.917a6bc29b42cp-4}, {0x1.13000a89a11ep-58, 0x1.2c8106e8e613ap-3},
{-0x1.26d19b9ff8d82p-57, 0x1.8f8b83c69a60bp-3}, {-0x1.42deef11da2c4p-57, 0x1.f19f97b215f1bp-3},
{-0x1.5d28da2c4612dp-56, 0x1.294062ed59f06p-2}, {-0x1.efdc0d58cf62p-62, 0x1.58f9a75ab1fddp-2},
{-0x1.72cedd3d5a61p-57, 0x1.87de2a6aea963p-2}, {0x1.5b362cb974183p-57, 0x1.b5d1009e15ccp-2},
{0x1.e0d891d3c6841p-58, 0x1.e2b5d3806f63bp-2}, {-0x1.a5a014347406cp-55, 0x1.073879922ffeep-1},
{0x1.b25dd267f66p-55, 0x1.1c73b39ae68c8p-1}, {-0x1.efcc626f74a6fp-57, 0x1.30ff7fce17035p-1},
{0x1.8076a2cfdc6b3p-57, 0x1.44cf325091dd6p-1}, {-0x1.75720992bfbb2p-55, 0x1.57d69348cecap-1},
{-0x1.bdd3413b26456p-55, 0x1.6a09e667f3bcdp-1}, {-0x1.0f537acdf0ad7p-56, 0x1.7b5df226aafafp-1},
{-0x1.2c5e12ed1336dp-55, 0x1.8bc806b151741p-1}, {-0x1.30ee286712474p-55, 0x1.9b3e047f38741p-1}, {0x1.9f630e8b6dac8p-60, 0x1.a9b66290ea1a3p-1}, {-0x1.bc69f324e6d61p-55, 0x1.b728345196e3ep-1},
{-0x1.6e0b1757c8d07p-56, 0x1.c38b2f180bdb1p-1}, {-0x1.e7b6bb5ab58aep-58, 0x1.ced7af43cc773p-1},
{0x1.457e610231ac2p-56, 0x1.d906bcf328d46p-1}, {-0x1.014c76c126527p-55, 0x1.e212104f686e5p-1},
{0x1.760b1e2e3f81ep-55, 0x1.e9f4156c62ddap-1}, {0x1.52c7adc6b4989p-56, 0x1.f0a7efb9230d7p-1},
{0x1.562172a361fd3p-56, 0x1.f6297cff75cbp-1}, {-0x1.7a0a8ca13571fp-55, 0x1.fa7557f08a517p-1},
{-0x1.87df6378811c7p-55, 0x1.fd88da3d12526p-1}, {-0x1.c57bc2e24aa15p-57, 0x1.ff621e3796d7ep-1},
{0x0p+0, 0x1p+0}
};
// 0 <= jf <= 32
double Ch = s[32-jf][1], Cl = s[32-jf][0], Sh = s[jf][1], Sl = s[jf][0];
double ax = __builtin_fabs(x);
double dsh = ax - Sh, dsl = -Sl;
double dch = ch - Ch, dcl = cl - Cl;
double Sc = __builtin_fma(Sh, dch, 0x1.8p-4) - 0x1.8p-4;
double dSc = __builtin_fma(Sh, dch, -Sc);
double Cs = __builtin_fma(Ch, dsh, 0x1.8p-4) - 0x1.8p-4;
double dCs = __builtin_fma(Ch, dsh, -Cs);
double v = Cs - Sc;
double dv = (Ch*dsl + Cl*dsh) - (Sh*dcl + Sl*dch) - (dSc - dCs);
v = fasttwosum(v,dv,&dv);
double sgn = __builtin_copysign(1.0, x), jt = 32 - jf*sgn;
// 0 <= jt <= 64
static const double c[][2] = {
{0x1p+0, -0x1.fc2c76456515bp-108}, {0x1.5555555555555p-3, 0x1.5555555623513p-57},
{0x1.3333333333333p-4, 0x1.9997e3427441bp-59}, {0x1.6db6db6db6db7p-5, -0x1.cb95ff08658e6p-62},
{0x1.f1c71c71c6d5bp-6, 0x1.b125bccdcc89ep-60}};
static const double ct[] = {0x1.6e8ba2ec8cb69p-6, 0x1.1c4ea7a15c997p-6, 0x1.ca8355d39bb67p-7};
double dv2, v2 = muldd(v,dv, v,dv, &dv2);
v *= -sgn;
dv *= -sgn;
double fl = v2*(ct[0] + v2*(ct[1] + v2*ct[2])), fh = polydd(v2,dv2, 5,c, &fl);
fh = muldd(v,dv, fh,fl, &fl);
double ph = jt * 0x1.921fb54442dp-5, pl = 0x1.8469898cc518p-53*jt, ps = -0x1.fc8f8cbb5bf6cp-102*jt;
// since 0 <= jt <= 64, ph and pl are exact
pl = sum(fh,fl, pl,ps, &ps);
ph = fasttwosum(ph,pl, &pl);
pl = fasttwosum(pl,ps, &ps);
ph = fasttwosum(ph,pl, &pl);
pl = fasttwosum(pl,ps, &ps);
b64u64_u t = {.f = pl};
i64 e = ((t.u>>52)&0x7ff) - 1023;
e = 52-(107+e);
e = e<0?0:e;
e = e>52?52:e;
u64 m = ((u64)1<<52)-((u64)1<<e);
e = (e == 0) ? 64 : e;
if(__builtin_expect(!((t.u+((u64)1<<(e-1)))&m), 0)){
if(x==-0x1.771164bfd1f84p-3 ) return 0x1.c14601daaf657p+0 - 0x1p-54;
if(x==-0x1.4510ee8eb4e67p-1 ) return 0x1.211c0e2c2559ep+1 - 0x1p-53;
if(x==-0x1.011c543f23a17p-2 ) return 0x1.d318c90d9e8b7p+0 - 0x1p-54;
if(x== 0x1.ffffffffffdc0p-1 ) return 0x1.8000000000024p-22 + 0x1p-76;
if(x== 0x1.53ea6c7255e88p-4 ) return 0x1.7cdacb6bbe707p+0 + 0x1p-54;
if(x== 0x1.fd737be914578p-11) return 0x1.91e006d41d8d8p+0 + 0x1.8p-53;
if(x== 0x1.fffffffffff70p-1 ) return 0x1.8000000000009p-23 + 0x1p-77;
b64u64_u w = {.f = ps};
if((w.u^t.u)>>63)
t.u--;
else
t.u++;
pl = t.f;
}
return ph + pl;
}