asinh.c

/* Correctly-rounded inverse hyperbolic sine function for binary64.

Copyright (c) 2023-2025 Alexei Sibidanov <sibid@uvic.ca>.

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

// Warning: clang also defines __GNUC__
#if defined(__GNUC__) && !defined(__clang__)
#pragma GCC diagnostic ignored "-Wunknown-pragmas"
#endif

#pragma STDC FENV_ACCESS ON

typedef uint64_t u64;
typedef union {double f; u64 u;} b64u64_u;
typedef uint16_t ushort;

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 adddd(double xh, double xl, double ch, double cl, double *l) {
  double s = xh + ch, d = s - xh;
  *l = ((ch - d) + (xh + (d - s))) + (xl + cl);
  return s;
}

static inline double muldd_acc(double xh, double xl, double ch, double cl, double *l){
  double ahlh = ch*xl, alhh = cl*xh, ahhh = ch*xh, ahhl = __builtin_fma(ch, xh, -ahhh);
  ahhl += alhh + ahlh;
  ch = ahhh + ahhl;
  *l = (ahhh - ch) + ahhl;
  return ch;
}

static inline double mulddd(double xh, double xl, double ch, double *l){
  double ahlh = ch*xl, ahhh = ch*xh, ahhl = __builtin_fma(ch, xh, -ahhh);
  ahhl += ahlh;
  ch = ahhh + ahhl;
  *l = (ahhh - ch) + ahhl;
  return ch;
}

static inline double polydd(double xh, double xl, int n, const double c[][2], double *l){
  int i = n-1;
  double ch = c[i][0] + *l, cl = ((c[i][0] - ch) + *l) + c[i][1];
  while(--i>=0){
    ch = muldd_acc(xh, xl, ch, cl, &cl);
    double th = ch + c[i][0], tl = (c[i][0] - th) + ch;
    ch = th;
    cl += tl + c[i][1];
  }
  *l = cl;
  return ch;
}

static double __attribute__((noinline)) as_asinh_refine(double, double, double, double);
static double __attribute__((noinline)) as_asinh_zero(double x, double x2h, double x2l){
  static const double ch[][2] = {
    {-0x1.5555555555555p-3, -0x1.5555555555555p-57}, {0x1.3333333333333p-4, 0x1.99999999949dfp-59},
    {-0x1.6db6db6db6db7p-5, 0x1.2492496091b0cp-60}, {0x1.f1c71c71c71c7p-6, 0x1.c71a35cfa0671p-62},
    {-0x1.6e8ba2e8ba2e9p-6, 0x1.17f937248cf81p-60}, {0x1.1c4ec4ec4ec4fp-6, -0x1.74e3c1dfd4c3dp-60},
    {-0x1.c999999999977p-7, -0x1.38e7a467ecc55p-61}, {0x1.7a87878786c7ep-7, 0x1.a83c7bace55ebp-61},
    {-0x1.3fde50d764083p-7, -0x1.d024df7fa0542p-61}, {0x1.12ef3ceae4d12p-7, -0x1.ba9c13deb261fp-61},
    {-0x1.df3bd104aa267p-8, -0x1.546da9bc5b32ap-62}, {0x1.a685fc5de7a04p-8, 0x1.40d284a1d67f9p-62}};
  static const double cl[] = {
    -0x1.7828d553ec8p-8, 0x1.51712f7bee368p-8, -0x1.2e6d98527bcc6p-8, 0x1.0095da47b392cp-8, -0x1.3b92d6368192cp-9};
  double y2 = x2h * (cl[0] + x2h * (cl[1] + x2h * (cl[2] + x2h * (cl[3] + x2h * (cl[4])))));
  double y1 = polydd(x2h, x2l, 12, ch, &y2);

  y1 = muldd_acc(y1,y2, x2h,x2l, &y2);
  y1 = mulddd(y1,y2, x, &y2);
  double y0 = fasttwosum(x, y1, &y1);
  y1 = fasttwosum(y1,y2,&y2);
  
  b64u64_u t = {.f = y1};
  if(__builtin_expect(!(t.u&(~(u64)0>>12)), 0)){
    b64u64_u w = {.f = y2};
    if((w.u^t.u)>>63)
      t.u--;
    else
      t.u++;
    y1 = t.f;
  }
  return y0 + y1;
}

static const struct {ushort c0; short c1;} B[] = {
  {301, 27565}, {7189, 24786}, {13383, 22167}, {18923, 19696}, {23845, 17361}, {28184, 15150},
  {31969, 13054}, {35231, 11064}, {37996, 9173}, {40288, 7372}, {42129, 5657}, {43542, 4020},
  {44546, 2457}, {45160, 962}, {45399, -468}, {45281, -1838}, {44821, -3151}, {44032, -4412},
  {42929, -5622}, {41522, -6786}, {39825, -7905}, {37848, -8982}, {35602, -10020}, {33097, -11020},
  {30341, -11985}, {27345, -12916}, {24115, -13816}, {20661, -14685}, {16989, -15526}, {13107, -16339},
  {9022, -17126}, {4740, -17889}};
static const double r1[] = 
  {0x1p+0, 0x1.f5076p-1, 0x1.ea4bp-1, 0x1.dfc98p-1, 0x1.d5818p-1,
   0x1.cb72p-1, 0x1.c199cp-1, 0x1.b7f76p-1, 0x1.ae8ap-1,
   0x1.a5504p-1, 0x1.9c492p-1, 0x1.93738p-1, 0x1.8ace6p-1,
   0x1.8258ap-1, 0x1.7a114p-1, 0x1.71f76p-1, 0x1.6a09ep-1,
   0x1.6247ep-1, 0x1.5ab08p-1, 0x1.5342cp-1, 0x1.4bfdap-1,
   0x1.44e08p-1, 0x1.3dea6p-1, 0x1.371a8p-1, 0x1.306fep-1,
   0x1.29e9ep-1, 0x1.2387ap-1, 0x1.1d488p-1, 0x1.172b8p-1,
   0x1.11302p-1, 0x1.0b558p-1, 0x1.059bp-1, 0x1p-1};
static const double r2[] = 
  {0x1p+0, 0x1.ffa74p-1, 0x1.ff4eap-1, 0x1.fef62p-1, 0x1.fe9dap-1,
   0x1.fe452p-1, 0x1.fdeccp-1, 0x1.fd946p-1, 0x1.fd3c2p-1,
   0x1.fce3ep-1, 0x1.fc8bcp-1, 0x1.fc33ap-1, 0x1.fbdbap-1,
   0x1.fb83ap-1, 0x1.fb2bcp-1, 0x1.fad3ep-1, 0x1.fa7c2p-1,
   0x1.fa246p-1, 0x1.f9ccap-1, 0x1.f975p-1, 0x1.f91d8p-1,
   0x1.f8c6p-1, 0x1.f86e8p-1, 0x1.f8172p-1, 0x1.f7bfep-1,
   0x1.f768ap-1, 0x1.f7116p-1, 0x1.f6ba4p-1, 0x1.f6632p-1,
   0x1.f60c2p-1, 0x1.f5b52p-1, 0x1.f55e4p-1, 0x1.f5076p-1};
static const double l1[][2] = {
  {0x0p+0, 0x0p+0}, {-0x1.269e2038315b3p-46, 0x1.62e4eacd4p-6},
  {-0x1.3f2558bddfc47p-45, 0x1.62e3ce7218p-5}, {0x1.07ea13c34efb5p-45, 0x1.0a2ab6d3ecp-4},
  {0x1.8f3e77084d3bap-44, 0x1.62e4a86d8cp-4}, {-0x1.8d92a005f1a7ep-46, 0x1.bb9db7062cp-4},
  {0x1.58239e799bfe5p-44, 0x1.0a2b1a22ccp-3}, {-0x1.a93fcf5f593b7p-44, 0x1.3687f0a298p-3},
  {-0x1.db4cac32fd2b5p-46, 0x1.62e4116b64p-3}, {-0x1.0e65a92ee0f3bp-46, 0x1.8f409e4df6p-3},
  {-0x1.8261383d475f1p-44, 0x1.bb9d15001cp-3}, {-0x1.359886207513bp-44, 0x1.e7f9a8c94p-3},
  {0x1.811f87496ceb7p-44, 0x1.0a2b052ddbp-2}, {0x1.4991ec6cb435cp-44, 0x1.205955ef73p-2},
  {-0x1.4581abfeb8927p-44, 0x1.3687bd9121p-2}, {0x1.cab48f6942703p-44, 0x1.4cb5e8f2b5p-2},
  {-0x1.df2c452fde132p-47, 0x1.62e4420e2p-2}, {0x1.6109f4fdb74bdp-45, 0x1.791292c46ap-2},
  {-0x1.6b95fbdac7696p-44, 0x1.8f40af84e7p-2}, {0x1.7394fa880cbdap-46, 0x1.a56ed8f865p-2},
  {-0x1.50b06a94eccabp-46, 0x1.bb9d6505b4p-2}, {-0x1.be2abf0b38989p-44, 0x1.d1cb91e728p-2},
  {-0x1.7d6bf1e34da04p-44, 0x1.e7f9d139e2p-2}, {-0x1.423c1e14de6edp-44, 0x1.fe27db9b0ep-2},
  {0x1.c46f1a0efbbc2p-44, 0x1.0a2b25060a8p-1}, {0x1.834fe4e3e6018p-45, 0x1.154244482ap-1},
  {0x1.6a03d0f02b65p-46, 0x1.20597312988p-1}, {0x1.d437056526f3p-44, 0x1.2b707145dep-1},
  {-0x1.a0233728405c5p-45, 0x1.3687b0e0b28p-1}, {-0x1.4dbdda10d2bf1p-45, 0x1.419ec5d3f68p-1},
  {0x1.f7d0a25d154f2p-44, 0x1.4cb5f9fc02p-1}, {0x1.15ede4d803b18p-44, 0x1.57cd28421a8p-1},
  {0x1.ef35793c7673p-45, 0x1.62e42fefa38p-1}};
static const double l2[][2] = {
  {0x0p+0, 0x0p+0}, {0x1.5abdac3638e99p-44, 0x1.631ec81ep-11},
  {-0x1.16b8be9bbe239p-45, 0x1.62fd8127p-10}, {-0x1.364c6315542ebp-44, 0x1.0a2520508p-9},
  {0x1.734abe459c9p-45, 0x1.62dadc1dp-9}, {0x1.0cf8a761431bfp-44, 0x1.bb9ff94dp-9},
  {0x1.da2718eb78708p-45, 0x1.0a2a2def8p-8}, {0x1.34ada62c59b93p-44, 0x1.368c0fae4p-8},
  {0x1.d09ab376682d4p-44, 0x1.62e58e4f8p-8}, {-0x1.3cb7b94329211p-45, 0x1.8f46bd28cp-8},
  {-0x1.eec5c297c41dp-45, 0x1.bb9f8312p-8}, {-0x1.6411b9395d15p-44, 0x1.e7fff8f3p-8},
  {-0x1.1c0e59a43053cp-44, 0x1.0a2c0006ep-7}, {0x1.6506596e077b6p-46, 0x1.205bdb6fp-7},
  {0x1.e256bce6faa27p-44, 0x1.36877c86ep-7}, {0x1.bd42467b0c8d1p-51, 0x1.4cb6f5578p-7},
  {-0x1.c4f92132ff0fp-44, 0x1.62e230e8cp-7}, {-0x1.80be08bfab39p-44, 0x1.7911440f6p-7},
  {-0x1.f0b1319ceb1f7p-44, 0x1.8f443020ap-7}, {0x1.a65fcfb8de99bp-45, 0x1.a572dbef4p-7},
  {0x1.4233885d3779cp-46, 0x1.bb9d449a6p-7}, {0x1.f46a59e646edbp-44, 0x1.d1cb8491cp-7},
  {-0x1.c3d2f11c11446p-44, 0x1.e7fd9d2aap-7}, {0x1.7763f78a1e0ccp-45, 0x1.fe2b6f978p-7},
  {0x1.b4c37fc60c043p-44, 0x1.0a2a7c7a5p-6}, {-0x1.5b8a822859be3p-46, 0x1.15412ca86p-6},
  {-0x1.f2d8c9fc064p-44, 0x1.2059c9005p-6}, {-0x1.e80e79c20378dp-44, 0x1.2b703f49bp-6},
  {0x1.68256e4329bdbp-44, 0x1.3688a1a8dp-6}, {0x1.7e9741da248c3p-44, 0x1.419edc7bap-6},
  {0x1.e330dccce602bp-45, 0x1.4cb7034fap-6}, {0x1.2f32b5d18eefbp-49, 0x1.57cd01187p-6},
  {-0x1.269e2038315b3p-46, 0x1.62e4eacd4p-6}};
static const double c[] = {-0x1p-1, 0x1.555555555553p-2, -0x1.fffffffffffap-3, 0x1.99999e33a6366p-3, -0x1.555559ef9525fp-3};

double cr_asinh(double x){
  double ax = __builtin_fabs(x);
  b64u64_u ix = {.f = ax};
  u64 u = ix.u;
  if(__builtin_expect(u<0x3fbb000000000000ull, 0)){ // |x| < 0x1.bp-4
    // for |x| < 0x1.7137449123ef7p-26, asinh(x) rounds to x to nearest
    // for |x| < 0x1p-1022 we have underflow but not for 0x1p-1022 (to nearest)
    if(__builtin_expect(u<0x3e57137449123ef7ull, 0)){ // |x| < 0x1.7137449123ef7p-26
      /* we need to check x=0 separately because fma(-2^-60,-0,-0) yields +0
         for rounding to nearest */
      if(__builtin_expect(!u, 0)) return x;
      double res = __builtin_fma(-0x1p-60,x,x);
#ifdef CORE_MATH_SUPPORT_ERRNO
      if (__builtin_fabs (res) < 0x1p-1022)
        errno = ERANGE; // underflow
#endif
      return res;
    }
    double x2h = x*x, x2l = __builtin_fma(x, x, -x2h);
    double x3h = x2h*x, sl;
    if(__builtin_expect(u<0x3f93000000000000ull, 0)){ // |x| < 0x1.3p-6
      if(__builtin_expect(u<0x3f30000000000000ull, 0)){ // |x| < 0x1p-12
	if(__builtin_expect(u<0x3e5a000000000000ull, 0)){ // |x| < 0x1.ap-26
	  static const double cl[] = {-0x1.5555555555555p-3};
	  sl = x3h*cl[0];
	} else {
	  static const double cl[] = {-0x1.5555555555555p-3, 0x1.3333327c57c6p-4};
	  sl = x3h*(cl[0] + x2h*cl[1]);
	}
      } else {
	static const double cl[] = {-0x1.5555555555555p-3, 0x1.333333332f2ffp-4, -0x1.6db6d9a665159p-5, 0x1.f186866d775fp-6};
	sl = x3h*(cl[0] + x2h*(cl[1] + x2h*(cl[2] + x2h*cl[3])));
      }
    } else { // 0x1.3p-6 <= |x| < 0x1.bp-4
      /* p = x + cl[0]*x^3 + ... + cl[6]*x^15 is a minimax polynomial
         with relative error < 2^-63.091 on [0x1.3p-6, 0x1.bp-4].
         This branch (0x1.3p-6 <= x < 0x1.bp-4) was tested exhaustively
         by Vincenzo Innocente (both with/without FMA) with revision 702a447.
         All found failures were added to asinh.wc. */
      static const double cl[] = {-0x1.5555555555555p-3, 0x1.333333333331p-4, -0x1.6db6db6da466cp-5, 0x1.f1c71c2ea7be4p-6,
				 -0x1.6e8b651b09d72p-6, 0x1.1c309fc0e69c2p-6, -0x1.bab7833c1ep-7};
      double c1 = cl[1] + x2h*cl[2];
      double c3 = cl[3] + x2h*cl[4];
      double c5 = cl[5] + x2h*cl[6];
      double x4 = x2h*x2h;
      sl = x3h*(cl[0] + x2h*(c1 + x4*(c3 + x4*c5)));
    }
    double eps = 0x1.79p-53*x3h;
    // revision 03523e1 fails with 0.999*eps and x=0x1.019bcf56d16f7p-4 (rndz)
    double lb = x + (sl - eps), ub = x + (sl + eps);
    if(lb == ub) return lb;
    return as_asinh_zero(x,x2h,x2l);
  }
  // |x| >= 0x1.bp-4
  double x2h = 0, x2l = 0;
  double ah, al;
  int off = 0x3ff;
  if(__builtin_expect(u<0x4190000000000000ull, 1)){ // |x| < 0x1p+26
    double th, tl;
    x2h = x * x;
    x2l = __builtin_fma(x, x, -x2h);
    if(__builtin_expect(u<0x3ff0000000000000ull, 0)){
      th = fasttwosum(1, x2h, &tl);
    } else {
      th = fasttwosum(x2h, 1, &tl);
    }
    tl += x2l;
    ah = __builtin_sqrt(th); double rs = 0.5/th;
    al = (tl - __builtin_fma(ah,ah,-th))*(rs*ah);
    ah = fasttwosum(ah, ax, &tl);
    al += tl;
  } else if(u<0x4330000000000000ull){ // |x| < 0x1p+52
    /* this branch was tested exhaustively with/without FMA by Vincenzo
       Innocente from 2^51 to 2^52 (commit 1bd85b8) */
    ah = 2*ax;
    al = 0.5/ax;
  } else { // |x| >= 0x1p+52
    if(__builtin_expect(u>=(u64)0x7ff0000000000000ull, 0)) return x + x; // +-inf or nan
    off = 0x3fe;
    ah = ax;
    al = 0;
  }

  b64u64_u t = {.f = ah};
  int ex = t.u>>52, e = ex - off;
  t.u &= ~(u64)0>>12;
  double ed = e;
  u64 i = t.u>>(52-5);
  int64_t d = t.u & (~(u64)0>>17);
  u64 j = (t.u + ((u64)B[i].c0<<33) + ((int64_t)B[i].c1*(d>>16)))>>(52-10);
  t.u |= (u64)0x3ff<<52;
  int i1 = j>>5, i2 = j&0x1f;
  double r = r1[i1]*r2[i2], dx = __builtin_fma(r, t.f, -1), dx2 = dx*dx;
  double f = dx2*((c[0] + dx*c[1]) + dx2*((c[2] + dx*c[3]) + dx2*c[4]));
  const double l2h = 0x1.62e42fefa38p-1, l2l = 0x1.ef35793c7673p-45;
  double lh = l2h*ed + (l1[i1][1] + l2[i2][1]);
  double ll = l2l*ed + l1[i1][0] + l2[i2][0] + al/ah + f;
  ll += dx;
  lh *= __builtin_copysign(1, x);
  ll *= __builtin_copysign(1, x);
  double eps = 1.63e-19;
  // revision 03523e1 fails with eps=0.997*eps, x=0x1.eece8f7802fbp+468 and rndz
  double lb = lh + (ll - eps), ub = lh + (ll + eps);
  if (lb == ub) return lb;
  if(ax<0x1p-2) return as_asinh_zero(x,x2h,x2l);
  return as_asinh_refine(x, ah, al, 0x1.71547652b82fep+0*__builtin_fabs(lb));
}

static __attribute__((noinline)) double
as_asinh_database(double x, double f){
  static const double db[][3] = {
    {0x1.00f9476450863p-2, 0x1.fcb35067f343cp-3, 0x1p-57},
    {0x1.1f0a79315b287p-2, 0x1.1b68aae88febap-2, 0x1p-56},
    {0x1.2b9618ff7acb7p-2, 0x1.27781d9aa4e25p-2, -0x1p-56},
    {0x1.389ef683f3aa7p-2, 0x1.33f52db6df1afp-2, 0x1p-56},
    {0x1.3b07e0c779ddap-2, 0x1.364303e1ad8f6p-2, 0x1p-56},
    {0x1.48441df33b6d3p-2, 0x1.42e385800f0a4p-2, 0x1p-56},
    {0x1.687bd068c1c1ep-2, 0x1.616cc75d49226p-2, -0x1p-56},
    {0x1.8740c4453a056p-2, 0x1.7e4f2ad132a1dp-2, 0x1p-56},
    {0x1.891acda11167ep-2, 0x1.8009d924a3ffdp-2, 0x1p-56},
    {0x1.bafc3479fc9ccp-2, 0x1.ae3773250e7d2p-2, 0x1p-56},
    {0x1.c59869f17b483p-2, 0x1.b7efa91915c95p-2, 0x1p-56},
    {0x1.c8be879787986p-2, 0x1.bad0485e0fe0ap-2, -0x1p-56},
    {0x1.e73b46abb01e1p-2, 0x1.d68039861ab53p-2, 0x1p-56},
    {0x1.ed6236da268bp-2, 0x1.dc0cb8f638126p-2, 0x1p-56},
    {0x1.f399ebafc1951p-2, 0x1.e1a4f519fab77p-2, -0x1p-56},
    {0x1.f70975ab0d471p-2, 0x1.e4bae8bcd6ea6p-2, 0x1p-56},
    {0x1.fbdd4a37760b7p-2, 0x1.e90f16eb88c09p-2, 0x1p-56},
    {0x1.fee72efb4bfddp-2, 0x1.ebc791a88bed8p-2, 0x1p-56},
    {0x1.02339d6bdb741p-1, 0x1.f0b2264e34555p-2, 0x1p-56},
    {0x1.09e7c831b1a23p-1, 0x1.fe694c3c89138p-2, 0x1p-56},
    {0x1.16d32c862fc3bp-1, 0x1.0a9c9334066dbp-1, -0x1p-55},
    {0x1.857954132083dp-1, 0x1.67425fe575c88p-1, -0x1p-55},
    {0x1.8a5c3b60f7e11p-1, 0x1.6b23ad4415a17p-1, -0x1p-55},
    {0x1.9740eb419dd04p-1, 0x1.754ab7535d47dp-1, 0x1p-55},
    {0x1.a16d9cc06011ap-1, 0x1.7d3755d851062p-1, -0x1p-55},
    {0x1.bb635be2213d1p-1, 0x1.91167cae3cfa9p-1, 0x1p-55},
    {0x1.d4b21ebf542fp-1, 0x1.a3fc7e4dd47d1p-1, -0x1p-55},
    {0x1.7b8516ffd2406p+0, 0x1.2f5d3b178914ap+0, 0x1p-54},
    {0x1.9295b9116e2e2p+0, 0x1.3bffa8863976p+0, 0x1p-54},
    {0x1.fedc65e32714p+0, 0x1.710f91e844f9bp+0, 0x1p-54},
    {0x1.57e377b3f0b4bp+1, 0x1.b6e2c73f41415p+0, 0x1p-54},
    {0x1.6056b06a21918p+3, 0x1.8c0a26d055288p+1, 0x1p-53},
    {0x1.843e1b5e5979cp+4, 0x1.f0f978201eb84p+1, 0x1p-53},
    {0x1.fee8f69c4cd25p+10, 0x1.0a19aebb51e9p+3, -0x1p-51},
    {0x1.0fbc6c02b1c9p+24, 0x1.16369cd53bb69p+4, 0x1p-50},
  };
  double ax = __builtin_fabs(x);
  int a = 0, b = sizeof(db)/sizeof(db[0]) - 1, m = (a + b)/2;
  while (a <= b) { // binary search
    if (db[m][0] < ax)
      a = m + 1;
    else if (db[m][0] == ax) {
      double sgn = __builtin_copysign(1,x);
      f = sgn*db[m][1] + sgn*db[m][2];
      break;
    } else
      b = m - 1;
    m = (a + b)/2;
  }
  return f;
}

static double as_asinh_refine(double x, double zh, double zl, double a){
  static const double t1[] = {
    0x1p+0, 0x1.ea4afap-1, 0x1.d5818ep-1, 0x1.c199bep-1, 0x1.ae89f98p-1, 0x1.9c4918p-1,
    0x1.8ace54p-1, 0x1.7a1147p-1, 0x1.6a09e68p-1, 0x1.5ab07ep-1, 0x1.4bfdad8p-1,
    0x1.3dea65p-1, 0x1.306fe08p-1, 0x1.2387a7p-1, 0x1.172b84p-1, 0x1.0b5587p-1, 0x1p-1};
  static const double t2[] = {
    0x1p+0, 0x1.fe9d968p-1, 0x1.fd3c228p-1, 0x1.fbdba38p-1, 0x1.fa7c18p-1, 0x1.f91d8p-1,
    0x1.f7bfdbp-1, 0x1.f663278p-1, 0x1.f507658p-1, 0x1.f3ac948p-1, 0x1.f252b38p-1,
    0x1.f0f9c2p-1, 0x1.efa1bfp-1, 0x1.ee4aaap-1, 0x1.ecf483p-1, 0x1.eb9f488p-1};
  static const double t3[] = {
    0x1p+0, 0x1.ffe9d2p-1, 0x1.ffd3a58p-1, 0x1.ffbd798p-1, 0x1.ffa74e8p-1, 0x1.ff91248p-1,
    0x1.ff7afb8p-1, 0x1.ff64d38p-1, 0x1.ff4eac8p-1, 0x1.ff38868p-1, 0x1.ff22618p-1,
    0x1.ff0c3dp-1, 0x1.fef61ap-1, 0x1.fedff78p-1, 0x1.fec9d68p-1, 0x1.feb3b6p-1};
  static const double t4[] = {
    0x1p+0, 0x1.fffe9dp-1, 0x1.fffd3ap-1, 0x1.fffbd78p-1, 0x1.fffa748p-1, 0x1.fff9118p-1,
    0x1.fff7ae8p-1, 0x1.fff64cp-1, 0x1.fff4e9p-1, 0x1.fff386p-1, 0x1.fff2238p-1,
    0x1.fff0c08p-1, 0x1.ffef5d8p-1, 0x1.ffedfa8p-1, 0x1.ffec98p-1, 0x1.ffeb35p-1};
  static const double LL[4][17][3] = {
    {{0x0p+0, 0x0p+0, 0x0p+0},
     {0x1.62e432b24p-6, -0x1.745af34bb54b8p-42, -0x1.17e3ec05cde7p-97},
     {0x1.62e42e4a8p-5, 0x1.111a4eadf312p-44, 0x1.cff3027abb119p-93},
     {0x1.0a2b233f1p-4, -0x1.88ac4ec78af8p-42, 0x1.4fa087ca75dfdp-93},
     {0x1.62e43056cp-4, 0x1.6bd65e8b0b7p-46, -0x1.b18e160362c24p-95},
     {0x1.bb9d3cbd6p-4, 0x1.de14aa55ec2bp-42, -0x1.c6ac3f1862a6bp-94},
     {0x1.0a2b244dap-3, 0x1.94def487fea7p-42, -0x1.dead1a4581acfp-94},
     {0x1.3687aa9b78p-3, 0x1.9cec9a50db22p-43, 0x1.34a70684f8e0ep-93},
     {0x1.62e42fabap-3, -0x1.d69047a3aebp-44, -0x1.4e061f79144e2p-95},
     {0x1.8f40b56d28p-3, 0x1.de7d755fd2e2p-42, 0x1.bdc7ecf001489p-94},
     {0x1.bb9d3b61fp-3, 0x1.c14f1445b12p-46, 0x1.a1d78cbdc5b58p-93},
     {0x1.e7f9c11f08p-3, -0x1.6e3e0000dae7p-43, 0x1.6a4559fadde98p-94},
     {0x1.0a2b242ec4p-2, 0x1.bb7cf852a5fe8p-42, 0x1.a6aef11ee43bdp-93},
     {0x1.205966c764p-2, 0x1.ad3a5f214294p-45, 0x1.5cc344fa10652p-93},
     {0x1.3687a98aacp-2, 0x1.1623671842fp-45, -0x1.0b428fe1f9e43p-94},
     {0x1.4cb5ec93f4p-2, 0x1.3d50980ea513p-42, 0x1.67f0ea083b1c4p-93},
     {0x1.62e42fefa4p-2, -0x1.8432a1b0e264p-44, 0x1.803f2f6af40f3p-93},
    },
    {{0x0p+0, 0x0p+0, 0x0p+0},
     {0x1.62e462b4p-10, 0x1.061d003b97318p-42, 0x1.d7faee66a2e1ep-93},
     {0x1.62e44c92p-9, 0x1.95a7bff5e239p-42, -0x1.f7e788a87135p-95},
     {0x1.0a2b1e33p-8, 0x1.2a3a1a65aa3ap-43, -0x1.54599c9605442p-93},
     {0x1.62e4367cp-8, -0x1.4a995b6d9ddcp-45, -0x1.56bb79b254f33p-100},
     {0x1.bb9d449ap-8, 0x1.8a119c42e9bcp-42, -0x1.8ecf7d8d661f1p-93},
     {0x1.0a2b1f19p-7, 0x1.8863771bd10a8p-42, 0x1.e9731de7f0155p-94},
     {0x1.3687ad11p-7, 0x1.e026a347ca1c8p-42, 0x1.fadc62522444dp-97},
     {0x1.62e436f28p-7, 0x1.25b84f71b70b8p-42, -0x1.fcb3f98612d27p-96},
     {0x1.8f40b7b38p-7, -0x1.62a0a4fd4758p-43, 0x1.3cb3c35d9f6a1p-93},
     {0x1.bb9d3abbp-7, -0x1.0ec48f94d786p-42, -0x1.6b47d410e4cc7p-93},
     {0x1.e7f9bb23p-7, 0x1.e4415cbc97ap-43, -0x1.3729fdb677231p-93},
     {0x1.0a2b22478p-6, -0x1.cb73f4505b03p-42, -0x1.1b3b3a3bc370ap-93},
     {0x1.2059691e8p-6, -0x1.abcc3412f264p-43, -0x1.fe6e998e48673p-95},
     {0x1.3687a768p-6, -0x1.43901e5c97a9p-42, 0x1.b54cdd52a5d88p-96},
     {0x1.4cb5eb5d8p-6, -0x1.8f106f00f13b8p-42, -0x1.8f793f5fce148p-93},
     {0x1.62e432b24p-6, -0x1.745af34bb54b8p-42, -0x1.17e3ec05cde7p-97},
    },
    {{0x0p+0, 0x0p+0, 0x0p+0},
     {0x1.62e7bp-14, -0x1.868625640a68p-44, -0x1.34bf0db910f65p-93},
     {0x1.62e35f6p-13, -0x1.2ee3d96b696ap-43, 0x1.a2948cd558655p-94},
     {0x1.0a2b4b2p-12, 0x1.53edbcf1165p-47, -0x1.cfc26ccf6d0e4p-97},
     {0x1.62e4be1p-12, 0x1.783e334614p-52, -0x1.04b96da30e63ap-93},
     {0x1.bb9e085p-12, -0x1.60785f20acb2p-43, -0x1.f33369bf7dff1p-96},
     {0x1.0a2b94dp-11, 0x1.fd4b3a273353p-42, -0x1.685a35575eff1p-96},
     {0x1.368810f8p-11, 0x1.7ded26dc813p-47, -0x1.4c4d1abca79bfp-96},
     {0x1.62e47878p-11, 0x1.7d2bee9a1f63p-42, 0x1.860233b7ad13p-93},
     {0x1.8f40cb48p-11, -0x1.af034eaf471cp-42, 0x1.ae748822d57b7p-94},
     {0x1.bb9d094p-11, -0x1.7a223013a20fp-42, -0x1.1e499087075b6p-93},
     {0x1.e7fa32c8p-11, -0x1.b2e67b1b59bdp-43, -0x1.54a41eda30fa6p-93},
     {0x1.0a2b237p-10, -0x1.7ad97ff4ac7ap-44, 0x1.f932da91371ddp-93},
     {0x1.2059a338p-10, -0x1.96422d90df4p-44, -0x1.90800fbbf2ed3p-94},
     {0x1.36879824p-10, 0x1.0f9054001812p-44, 0x1.9567e01e48f9ap-93},
     {0x1.4cb602cp-10, -0x1.0d709a5ec0b5p-43, 0x1.253dfd44635d2p-94},
     {0x1.62e462b4p-10, 0x1.061d003b97318p-42, 0x1.d7faee66a2e1ep-93},
    },
    {{0x0p+0, 0x0p+0, 0x0p+0},
     {0x1.63007cp-18, -0x1.db0e38e5aaaap-43, 0x1.259a7b94815b9p-93},
     {0x1.6300f6p-17, 0x1.2b1c75580438p-44, 0x1.78cabba01e3e4p-93},
     {0x1.0a2115p-16, -0x1.5ff223730759p-42, 0x1.8074feacfe49dp-95},
     {0x1.62e1ecp-16, -0x1.85d6f6487ce4p-45, 0x1.05485074b9276p-93},
     {0x1.bba301p-16, -0x1.af5d58a7c921p-43, -0x1.30a8c0fd2ff5fp-93},
     {0x1.0a32298p-15, 0x1.590faa0883bdp-43, 0x1.95e9bda999947p-93},
     {0x1.3682f1p-15, 0x1.f0224376efaf8p-42, -0x1.5843c0db50d1p-93},
     {0x1.62e3d8p-15, -0x1.142c13daed4ap-43, 0x1.c68a61183ce87p-93},
     {0x1.8f44dd8p-15, -0x1.aa489f399931p-43, 0x1.11c5c376854eap-94},
     {0x1.bb9601p-15, 0x1.9904d8b6a3638p-42, 0x1.8c89554493c8fp-93},
     {0x1.e7f744p-15, 0x1.5785ddbe7cba8p-42, 0x1.e7ff3cde7d70cp-94},
     {0x1.0a2c53p-14, -0x1.6d9e8780d0d5p-43, 0x1.ad9c178106693p-94},
     {0x1.205d134p-14, -0x1.214a2e893fccp-43, 0x1.548a9500c9822p-93},
     {0x1.3685e28p-14, 0x1.e23588646103p-43, 0x1.2a97b26da2d88p-94},
     {0x1.4cb6c18p-14, 0x1.2b7cfcea9e0d8p-42, -0x1.5095048a6b824p-93},
     {0x1.62e7bp-14, -0x1.868625640a68p-44, -0x1.34bf0db910f65p-93},
    },
  };
  static const double ch[][2] = {
    {0x1p-1, 0x1.24b67ee516e3bp-111}, {-0x1p-2, -0x1.932ce43199a8dp-110}, {0x1.5555555555555p-3, 0x1.55540c15cf91fp-57},
  };
  static const double cl[3] = {-0x1p-3, 0x1.9999999a0754fp-4,-0x1.55555555c3157p-4};
  b64u64_u t = {.f = zh};
  
  int ex = t.u>>52, e = ex - 0x3ff + (zl==0.0);
  t.u &= ~(u64)0>>12;
  t.u |= (u64)0x3ff<<52;
  double ed = e;
  b64u64_u v = {.f = a - ed + 0x1.00008p+0};
  u64 i = (v.u - ((u64)0x3ff<<52))>>(52-16);
  int i1 = (i>>12)&0x1f, i2 = (i>>8)&0xf, i3 = (i>>4)&0xf, i4 = i&0xf;
  const double l20 = 0x1.62e42fefa38p-2, l21 = 0x1.ef35793c768p-46, l22 = -0x1.9ff0342542fc3p-91;
  double el2 = l22*ed, el1 = l21*ed, el0 = l20*ed;
  double L[3];
  L[0] = LL[0][i1][0] + LL[1][i2][0] + (LL[2][i3][0] + LL[3][i4][0]);
  L[1] = LL[0][i1][1] + LL[1][i2][1] + (LL[2][i3][1] + LL[3][i4][1]);
  L[2] = LL[0][i1][2] + LL[1][i2][2] + (LL[2][i3][2] + LL[3][i4][2]);
  L[0] += el0;
  double t12 = t1[i1]*t2[i2], t34 = t3[i3]*t4[i4];
  double th = t12*t34, tl = __builtin_fma(t12,t34,-th);
  double dh = th*t.f, dl = __builtin_fma(th,t.f,-dh);
  double sh = tl*t.f, sl = __builtin_fma(tl,t.f,-sh);
  double xl, xh = fasttwosum(dh-1, dl, &xl);
  if(zl != 0.0){
    t.f = zl;
    t.u -= (int64_t)e<<52;
    xl += th*t.f;
  }
  xh = adddd(xh, xl, sh, sl, &xl);
  sl = xh*(cl[0] + xh*(cl[1] + xh*cl[2]));
  sh = polydd(xh, xl, 3, ch, &sl);
  sh = muldd_acc(xh, xl, sh, sl, &sl);
  sh = adddd(sh, sl, el1, el2, &sl);
  sh = adddd(sh, sl, L[1], L[2], &sl);
  double v2, v0 = fasttwosum(L[0], sh, &v2);
  double v1 = fasttwosum(v2, sl, &v2);
  v0 = fasttwosum(v0,v1, &v1);
  v1 = fasttwosum(v1,v2, &v2);
  v0 *= __builtin_copysign(2,x);
  v1 *= __builtin_copysign(2,x);
  v2 *= __builtin_copysign(2,x);
  t.f = v1;
  if(__builtin_expect(!(t.u&(~(u64)0>>12)), 0)){
    b64u64_u w = {.f = v2};
    if((w.u^t.u)>>63)
      t.u--;
    else
      t.u++;
    v1 = t.f;
  }
  b64u64_u t0 = {.f = v0};
  uint64_t er = ((t.u + 41) & (~(u64)0>>12)), de = ((t0.u>>52)&0x7ff) - ((t.u>>52)&0x7ff);
  double res = v0 + v1;
  if(__builtin_expect(de>99 || er<80, 0)) return as_asinh_database(x,res);
  return res;
}