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/* Correctly-rounded inverse hyperbolic cosine function for binary32 format.
Copyright (c) 2023 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>
#ifdef CORE_MATH_SUPPORT_ERRNO
#include <errno.h>
#endif
#include <fenv.h> // for feraiseexcept, FE_INVALID
// Warning: clang also defines __GNUC__
#if defined(__GNUC__) && !defined(__clang__)
#pragma GCC diagnostic ignored "-Wunknown-pragmas"
#endif
#pragma STDC FENV_ACCESS ON
typedef union {float f; uint32_t u;} b32u32_u;
typedef union {double f; uint64_t u;} b64u64_u;
static __attribute__((noinline)) float as_special(float x){
b32u32_u t = {.f = x};
if(t.u==0x3f800000u) return 0.0f;
if((t.u<<1)>0xff000000u) return x + x; // nan
if(t.u==0x7f800000u) return x; // inf
#ifdef CORE_MATH_SUPPORT_ERRNO
errno = EDOM;
#endif
/* We could return 0.0f/0.0f here, which will return NaN and raise the
invalid exception, without any dependency on __builtin_nanf() nor
on fenv.h, but we prefer to do the following, so that the code can
be compiled with -ffinite-math-only (for which 0/0 would give 1). */
feraiseexcept(FE_INVALID);
return __builtin_nanf("<1");
}
float cr_acoshf(float x) {
static const double ix[] = {
0x1p+0, 0x1.fc07f01fcp-1, 0x1.f81f81f82p-1, 0x1.f44659e4ap-1, 0x1.f07c1f07cp-1,
0x1.ecc07b302p-1, 0x1.e9131abfp-1, 0x1.e573ac902p-1, 0x1.e1e1e1e1ep-1,
0x1.de5d6e3f8p-1, 0x1.dae6076bap-1, 0x1.d77b654b8p-1, 0x1.d41d41d42p-1,
0x1.d0cb58f6ep-1, 0x1.cd8568904p-1, 0x1.ca4b3055ep-1, 0x1.c71c71c72p-1,
0x1.c3f8f01c4p-1, 0x1.c0e070382p-1, 0x1.bdd2b8994p-1, 0x1.bacf914c2p-1,
0x1.b7d6c3ddap-1, 0x1.b4e81b4e8p-1, 0x1.b2036406cp-1, 0x1.af286bca2p-1,
0x1.ac5701ac6p-1, 0x1.a98ef606ap-1, 0x1.a6d01a6dp-1, 0x1.a41a41a42p-1,
0x1.a16d3f97ap-1, 0x1.9ec8e951p-1, 0x1.9c2d14ee4p-1, 0x1.99999999ap-1,
0x1.970e4f80cp-1, 0x1.948b0fcd6p-1, 0x1.920fb49dp-1, 0x1.8f9c18f9cp-1,
0x1.8d3018d3p-1, 0x1.8acb90f6cp-1, 0x1.886e5f0acp-1, 0x1.861861862p-1,
0x1.83c977ab2p-1, 0x1.818181818p-1, 0x1.7f405fd02p-1, 0x1.7d05f417ep-1,
0x1.7ad2208ep-1, 0x1.78a4c8178p-1, 0x1.767dce434p-1, 0x1.745d1745ep-1,
0x1.724287f46p-1, 0x1.702e05c0cp-1, 0x1.6e1f76b44p-1, 0x1.6c16c16c2p-1,
0x1.6a13cd154p-1, 0x1.681681682p-1, 0x1.661ec6a52p-1, 0x1.642c8590cp-1,
0x1.623fa7702p-1, 0x1.605816058p-1, 0x1.5e75bb8dp-1, 0x1.5c9882b94p-1,
0x1.5ac056b02p-1, 0x1.58ed23082p-1, 0x1.571ed3c5p-1, 0x1.555555556p-1,
0x1.5390948f4p-1, 0x1.51d07eae2p-1, 0x1.501501502p-1, 0x1.4e5e0a73p-1,
0x1.4cab88726p-1, 0x1.4afd6a052p-1, 0x1.49539e3b2p-1, 0x1.47ae147aep-1,
0x1.460cbc7f6p-1, 0x1.446f86562p-1, 0x1.42d6625d6p-1, 0x1.414141414p-1,
0x1.3fb013fbp-1, 0x1.3e22cbce4p-1, 0x1.3c995a47cp-1, 0x1.3b13b13b2p-1,
0x1.3991c2c18p-1, 0x1.381381382p-1, 0x1.3698df3dep-1, 0x1.3521cfb2cp-1,
0x1.33ae45b58p-1, 0x1.323e34a2cp-1, 0x1.30d19013p-1, 0x1.2f684bda2p-1,
0x1.2e025c04cp-1, 0x1.2c9fb4d82p-1, 0x1.2b404ad02p-1, 0x1.29e4129e4p-1,
0x1.288b01288p-1, 0x1.27350b882p-1, 0x1.25e22708p-1, 0x1.24924924ap-1,
0x1.23456789ap-1, 0x1.21fb78122p-1, 0x1.20b470c68p-1, 0x1.1f7047dc2p-1,
0x1.1e2ef3b4p-1, 0x1.1cf06ada2p-1, 0x1.1bb4a4046p-1, 0x1.1a7b9611ap-1,
0x1.19453808cp-1, 0x1.181181182p-1, 0x1.16e068942p-1, 0x1.15b1e5f76p-1,
0x1.1485f0e0ap-1, 0x1.135c81136p-1, 0x1.12358e75ep-1, 0x1.111111112p-1,
0x1.0fef010fep-1, 0x1.0ecf56be6p-1, 0x1.0db20a89p-1, 0x1.0c9714fbcp-1,
0x1.0b7e6ec26p-1, 0x1.0a6810a68p-1, 0x1.0953f3902p-1, 0x1.084210842p-1,
0x1.073260a48p-1, 0x1.0624dd2f2p-1, 0x1.05197f7d8p-1, 0x1.041041042p-1,
0x1.03091b52p-1, 0x1.020408102p-1, 0x1.01010101p-1, 0x1p-1 };
static const double lix[] = {
0x0p+0, 0x1.fe02a6b146789p-8, 0x1.fc0a8b0fa03e4p-7, 0x1.7b91b07de311bp-6,
0x1.f829b0e7c33p-6, 0x1.39e87b9fd7d6p-5, 0x1.77458f63edcfcp-5, 0x1.b42dd7117b1bfp-5,
0x1.f0a30c01362a6p-5, 0x1.16536eea7fae1p-4, 0x1.341d7961791d1p-4, 0x1.51b073f07983fp-4,
0x1.6f0d28ae3eb4cp-4, 0x1.8c345d6383b21p-4, 0x1.a926d3a475563p-4, 0x1.c5e548f63a743p-4,
0x1.e27076e28f2e6p-4, 0x1.fec9131dbaabbp-4, 0x1.0d77e7ccf6e59p-3, 0x1.1b72ad52f87ap-3,
0x1.29552f81eb523p-3, 0x1.371fc201f7f74p-3, 0x1.44d2b6ccbfd1ep-3, 0x1.526e5e3a41438p-3,
0x1.5ff3070a613d4p-3, 0x1.6d60fe717221dp-3, 0x1.7ab890212b909p-3, 0x1.87fa065214911p-3,
0x1.9525a9cf296b4p-3, 0x1.a23bc1fe42563p-3, 0x1.af3c94e81bff3p-3, 0x1.bc2867430acd6p-3,
0x1.c8ff7c7989a22p-3, 0x1.d5c216b535b91p-3, 0x1.e27076e2f92e6p-3, 0x1.ef0adcbe0d936p-3,
0x1.fb9186d5ebe2bp-3, 0x1.0402594b51041p-2, 0x1.0a324e27370e3p-2, 0x1.1058bf9ad7ad5p-2,
0x1.1675cabaa660ep-2, 0x1.1c898c16b91fbp-2, 0x1.22941fbcfb966p-2, 0x1.2895a13dd2ea3p-2,
0x1.2e8e2bade7d31p-2, 0x1.347dd9a9afd55p-2, 0x1.3a64c556b05eap-2, 0x1.40430868877e4p-2,
0x1.4618bc219dec2p-2, 0x1.4be5f9579e0a1p-2, 0x1.51aad872c982dp-2, 0x1.5767717432a6cp-2,
0x1.5d1bdbf5669cap-2, 0x1.62c82f2b83795p-2, 0x1.686c81e9964afp-2, 0x1.6e08eaa2929e4p-2,
0x1.739d7f6b95007p-2, 0x1.792a55fdb7fa2p-2, 0x1.7eaf83b82efc3p-2, 0x1.842d1da1ecb17p-2,
0x1.89a3386be825bp-2, 0x1.8f11e87347ac7p-2, 0x1.947941c1f26fbp-2, 0x1.99d958119208bp-2,
0x1.9f323ecbd984cp-2, 0x1.a484090e5eb0ap-2, 0x1.a9cec9a9cf84ap-2, 0x1.af1293245606bp-2,
0x1.b44f77bc98f63p-2, 0x1.b9858969218fbp-2, 0x1.beb4d9da96b7cp-2, 0x1.c3dd7a7d0354dp-2,
0x1.c8ff7c79ada22p-2, 0x1.ce1af0b855bebp-2, 0x1.d32fe7e039bd5p-2, 0x1.d83e72587673ep-2,
0x1.dd46a04c204a1p-2, 0x1.e24881a7cac26p-2, 0x1.e744261d8a788p-2, 0x1.ec399d2457ccp-2,
0x1.f128f5fac86edp-2, 0x1.f6123fa71c8acp-2, 0x1.faf588f76631fp-2, 0x1.ffd2e08580c98p-2,
0x1.02552a5a4f0ffp-1, 0x1.04bdf9da8b6d2p-1, 0x1.0723e5c1b4f4p-1, 0x1.0986f4f589521p-1,
0x1.0be72e423ca83p-1, 0x1.0e44985d0f48cp-1, 0x1.109f39e2be497p-1, 0x1.12f71959283bcp-1,
0x1.154c3d2f4f5eap-1, 0x1.179eabbd9c9a1p-1, 0x1.19ee6b466516fp-1, 0x1.1c3b81f723c25p-1,
0x1.1e85f5e6ec0dp-1, 0x1.20cdcd193f76ep-1, 0x1.23130d7beb743p-1, 0x1.2555bce9887cbp-1,
0x1.2795e1288211bp-1, 0x1.29d37fec2308bp-1, 0x1.2c0e9ed45768cp-1, 0x1.2e47436e5ae68p-1,
0x1.307d7334ff0bep-1, 0x1.32b1339134571p-1, 0x1.34e289d9b39d3p-1, 0x1.37117b5481bb6p-1,
0x1.393e0d3549a1ap-1, 0x1.3b6844a017823p-1, 0x1.3d9026a70eefbp-1, 0x1.3fb5b84cfeb42p-1,
0x1.41d8fe844b2aep-1, 0x1.43f9fe2fb9267p-1, 0x1.4618bc21d86c2p-1, 0x1.48353d1e928dfp-1,
0x1.4a4f85db1debbp-1, 0x1.4c679afcc323ap-1, 0x1.4e7d811b77bb1p-1, 0x1.50913cbff8c6bp-1,
0x1.52a2d265be5abp-1, 0x1.54b2467998498p-1, 0x1.56bf9d5b34b99p-1, 0x1.58cadb5cbe989p-1,
0x1.5ad404c33af2dp-1, 0x1.5cdb1dc6ad765p-1, 0x1.5ee02a9241e75p-1, 0x1.60e32f447a8d9p-1,
0x1.62e42fefa39efp-1};
b32u32_u t = {.f = x};
if(__builtin_expect(t.u<=0x3f800000u, 0)){
return as_special(x);
} else if(__builtin_expect(t.u<=0x3f99db23u, 0)){
float zf = x - 1.0f;
double z = zf, a = __builtin_sqrt(2.0*z);
static const double c[] =
{-0x1.555555555491ep-4, 0x1.333333319c2p-6, -0x1.6db6da26e5e0ep-8, 0x1.f1c698b7100cep-10,
-0x1.6e7ca9e152dc2p-11, 0x1.1b4cf187158bdp-12, -0x1.b5137f11a88e5p-14, 0x1.056c6c57152b2p-15};
double z2 = z*z, z4 = z2*z2;
double f = ((c[0] + z*c[1]) + z2*(c[2] + z*c[3])) + z4*((c[4] + z*c[5]) + z2*(c[6] + z*c[7]));
double r = a + (a*z)*f;
return r;
} else if(__builtin_expect(t.u<0x7f800000u, 1)){
double xd = x, x2 = xd*xd;
b64u64_u tp = {.f = xd + __builtin_sqrt(x2 - 1)};
uint64_t m = tp.u&(~(uint64_t)0>>12);
int j = (m + ((int64_t)1<<(52-8)))>>(52-7);
int e = (tp.u>>52) - 0x3ff;
b64u64_u w = {.u = m | (uint64_t)0x3ff<<52};
double z = w.f * ix[j] - 1.0;
static const double c[] = {0x1.0000000066947p+0, -0x1.00007f053d8cbp-1, 0x1.555280111d914p-2};
double z2 = z*z;
b64u64_u r = {.f = ((lix[128]*e + lix[j]) + z*c[0]) + z2*(c[1] + z*c[2])};
if(__builtin_expect(((r.u+259000)&0xfffffffll) < 260000, 0)){ // accurate path
static const double cp[] =
{0x1p+0, -0x1p-1, 0x1.55555555030bcp-2, -0x1.ffffffff2b4e5p-3, 0x1.999b5076a42f2p-3, -0x1.55570c45a647dp-3};
z2 = z*z;
double c0 = cp[0] + z*cp[1];
double c2 = cp[2] + z*cp[3];
double c4 = cp[4] + z*cp[5];
c0 += z2*(c2 + z2*c4);
const double ln2l = 0x1.7f7d1cf79abcap-20, ln2h = 0x1.62e4p-1;
double Lh = ln2h * e, Ll = ln2l * e;
r.f = (z * c0 + (Ll + lix[j])) + Lh;
if(__builtin_expect((r.u&0xfffffffll) == 0, 0)){
double h = (z * c0 + (Ll + lix[j])) + (Lh - r.f);
r.f = r.f + 64.0*h;
}
}
return r.f;
} else {
return as_special(x);
}
}