libc/AOR_v20.02/math/pow.c

   1 /*
   2  * Double-precision x^y function.
   3  *
   4  * Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
   5  * See https://llvm.org/LICENSE.txt for license information.
   6  * SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
   7  */
   8
   9 #include <float.h>
  10 #include <math.h>
  11 #include <stdint.h>
  12 #include "math_config.h"
  13
  14 /*
  15 Worst-case error: 0.54 ULP (~= ulperr_exp + 1024*Ln2*relerr_log*2^53)
  16 relerr_log: 1.3 * 2^-68 (Relative error of log, 1.5 * 2^-68 without fma)
  17 ulperr_exp: 0.509 ULP (ULP error of exp, 0.511 ULP without fma)
  18 */
  19
  20 #define T __pow_log_data.tab
  21 #define A __pow_log_data.poly
  22 #define Ln2hi __pow_log_data.ln2hi
  23 #define Ln2lo __pow_log_data.ln2lo
  24 #define N (1 << POW_LOG_TABLE_BITS)
  25 #define OFF 0x3fe6955500000000
  26
  27 /* Top 12 bits of a double (sign and exponent bits).  */
  28 static inline uint32_t
  29 top12 (double x)
  30 {
  31   return asuint64 (x) >> 52;
  32 }
  33
  34 /* Compute y+TAIL = log(x) where the rounded result is y and TAIL has about
  35    additional 15 bits precision.  IX is the bit representation of x, but
  36    normalized in the subnormal range using the sign bit for the exponent.  */
  37 static inline double_t
  38 log_inline (uint64_t ix, double_t *tail)
  39 {
  40   /* double_t for better performance on targets with FLT_EVAL_METHOD==2.  */
  41   double_t z, r, y, invc, logc, logctail, kd, hi, t1, t2, lo, lo1, lo2, p;
  42   uint64_t iz, tmp;
  43   int k, i;
  44
  45   /* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
  46      The range is split into N subintervals.
  47      The ith subinterval contains z and c is near its center.  */
  48   tmp = ix - OFF;
  49   i = (tmp >> (52 - POW_LOG_TABLE_BITS)) % N;
  50   k = (int64_t) tmp >> 52; /* arithmetic shift */
  51   iz = ix - (tmp & 0xfffULL << 52);
  52   z = asdouble (iz);
  53   kd = (double_t) k;
  54
  55   /* log(x) = k*Ln2 + log(c) + log1p(z/c-1).  */
  56   invc = T[i].invc;
  57   logc = T[i].logc;
  58   logctail = T[i].logctail;
  59
  60   /* Note: 1/c is j/N or j/N/2 where j is an integer in [N,2N) and
  61      |z/c - 1| < 1/N, so r = z/c - 1 is exactly representable.  */
  62 #if HAVE_FAST_FMA
  63   r = fma (z, invc, -1.0);
  64 #else
  65   /* Split z such that rhi, rlo and rhi*rhi are exact and |rlo| <= |r|.  */
  66   double_t zhi = asdouble ((iz + (1ULL << 31)) & (-1ULL << 32));
  67   double_t zlo = z - zhi;
  68   double_t rhi = zhi * invc - 1.0;
  69   double_t rlo = zlo * invc;
  70   r = rhi + rlo;
  71 #endif
  72
  73   /* k*Ln2 + log(c) + r.  */
  74   t1 = kd * Ln2hi + logc;
  75   t2 = t1 + r;
  76   lo1 = kd * Ln2lo + logctail;
  77   lo2 = t1 - t2 + r;
  78
  79   /* Evaluation is optimized assuming superscalar pipelined execution.  */
  80   double_t ar, ar2, ar3, lo3, lo4;
  81   ar = A[0] * r; /* A[0] = -0.5.  */
  82   ar2 = r * ar;
  83   ar3 = r * ar2;
  84   /* k*Ln2 + log(c) + r + A[0]*r*r.  */
  85 #if HAVE_FAST_FMA
  86   hi = t2 + ar2;
  87   lo3 = fma (ar, r, -ar2);
  88   lo4 = t2 - hi + ar2;
  89 #else
  90   double_t arhi = A[0] * rhi;
  91   double_t arhi2 = rhi * arhi;
  92   hi = t2 + arhi2;
  93   lo3 = rlo * (ar + arhi);
  94   lo4 = t2 - hi + arhi2;
  95 #endif
  96   /* p = log1p(r) - r - A[0]*r*r.  */
  97 #if POW_LOG_POLY_ORDER == 8
  98   p = (ar3
  99        * (A[1] + r * A[2] + ar2 * (A[3] + r * A[4] + ar2 * (A[5] + r * A[6]))));
 100 #endif
 101   lo = lo1 + lo2 + lo3 + lo4 + p;
 102   y = hi + lo;
 103   *tail = hi - y + lo;
 104   return y;
 105 }
 106
 107 #undef N
 108 #undef T
 109 #define N (1 << EXP_TABLE_BITS)
 110 #define InvLn2N __exp_data.invln2N
 111 #define NegLn2hiN __exp_data.negln2hiN
 112 #define NegLn2loN __exp_data.negln2loN
 113 #define Shift __exp_data.shift
 114 #define T __exp_data.tab
 115 #define C2 __exp_data.poly[5 - EXP_POLY_ORDER]
 116 #define C3 __exp_data.poly[6 - EXP_POLY_ORDER]
 117 #define C4 __exp_data.poly[7 - EXP_POLY_ORDER]
 118 #define C5 __exp_data.poly[8 - EXP_POLY_ORDER]
 119 #define C6 __exp_data.poly[9 - EXP_POLY_ORDER]
 120
 121 /* Handle cases that may overflow or underflow when computing the result that
 122    is scale*(1+TMP) without intermediate rounding.  The bit representation of
 123    scale is in SBITS, however it has a computed exponent that may have
 124    overflown into the sign bit so that needs to be adjusted before using it as
 125    a double.  (int32_t)KI is the k used in the argument reduction and exponent
 126    adjustment of scale, positive k here means the result may overflow and
 127    negative k means the result may underflow.  */
 128 static inline double
 129 specialcase (double_t tmp, uint64_t sbits, uint64_t ki)
 130 {
 131   double_t scale, y;
 132
 133   if ((ki & 0x80000000) == 0)
 134     {
 135       /* k > 0, the exponent of scale might have overflowed by <= 460.  */
 136       sbits -= 1009ull << 52;
 137       scale = asdouble (sbits);
 138       y = 0x1p1009 * (scale + scale * tmp);
 139       return check_oflow (eval_as_double (y));
 140     }
 141   /* k < 0, need special care in the subnormal range.  */
 142   sbits += 1022ull << 52;
 143   /* Note: sbits is signed scale.  */
 144   scale = asdouble (sbits);
 145   y = scale + scale * tmp;
 146   if (fabs (y) < 1.0)
 147     {
 148       /* Round y to the right precision before scaling it into the subnormal
 149          range to avoid double rounding that can cause 0.5+E/2 ulp error where
 150          E is the worst-case ulp error outside the subnormal range.  So this
 151          is only useful if the goal is better than 1 ulp worst-case error.  */
 152       double_t hi, lo, one = 1.0;
 153       if (y < 0.0)
 154         one = -1.0;
 155       lo = scale - y + scale * tmp;
 156       hi = one + y;
 157       lo = one - hi + y + lo;
 158       y = eval_as_double (hi + lo) - one;
 159       /* Fix the sign of 0.  */
 160       if (y == 0.0)
 161         y = asdouble (sbits & 0x8000000000000000);
 162       /* The underflow exception needs to be signaled explicitly.  */
 163       force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022);
 164     }
 165   y = 0x1p-1022 * y;
 166   return check_uflow (eval_as_double (y));
 167 }
 168
 169 #define SIGN_BIAS (0x800 << EXP_TABLE_BITS)
 170
 171 /* Computes sign*exp(x+xtail) where |xtail| < 2^-8/N and |xtail| <= |x|.
 172    The sign_bias argument is SIGN_BIAS or 0 and sets the sign to -1 or 1.  */
 173 static inline double
 174 exp_inline (double_t x, double_t xtail, uint32_t sign_bias)
 175 {
 176   uint32_t abstop;
 177   uint64_t ki, idx, top, sbits;
 178   /* double_t for better performance on targets with FLT_EVAL_METHOD==2.  */
 179   double_t kd, z, r, r2, scale, tail, tmp;
 180
 181   abstop = top12 (x) & 0x7ff;
 182   if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54)))
 183     {
 184       if (abstop - top12 (0x1p-54) >= 0x80000000)
 185         {
 186           /* Avoid spurious underflow for tiny x.  */
 187           /* Note: 0 is common input.  */
 188           double_t one = WANT_ROUNDING ? 1.0 + x : 1.0;
 189           return sign_bias ? -one : one;
 190         }
 191       if (abstop >= top12 (1024.0))
 192         {
 193           /* Note: inf and nan are already handled.  */
 194           if (asuint64 (x) >> 63)
 195             return __math_uflow (sign_bias);
 196           else
 197             return __math_oflow (sign_bias);
 198         }
 199       /* Large x is special cased below.  */
 200       abstop = 0;
 201     }
 202
 203   /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)].  */
 204   /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N].  */
 205   z = InvLn2N * x;
 206 #if TOINT_INTRINSICS
 207   kd = roundtoint (z);
 208   ki = converttoint (z);
 209 #elif EXP_USE_TOINT_NARROW
 210   /* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes.  */
 211   kd = eval_as_double (z + Shift);
 212   ki = asuint64 (kd) >> 16;
 213   kd = (double_t) (int32_t) ki;
 214 #else
 215   /* z - kd is in [-1, 1] in non-nearest rounding modes.  */
 216   kd = eval_as_double (z + Shift);
 217   ki = asuint64 (kd);
 218   kd -= Shift;
 219 #endif
 220   r = x + kd * NegLn2hiN + kd * NegLn2loN;
 221   /* The code assumes 2^-200 < |xtail| < 2^-8/N.  */
 222   r += xtail;
 223   /* 2^(k/N) ~= scale * (1 + tail).  */
 224   idx = 2 * (ki % N);
 225   top = (ki + sign_bias) << (52 - EXP_TABLE_BITS);
 226   tail = asdouble (T[idx]);
 227   /* This is only a valid scale when -1023*N < k < 1024*N.  */
 228   sbits = T[idx + 1] + top;
 229   /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1).  */
 230   /* Evaluation is optimized assuming superscalar pipelined execution.  */
 231   r2 = r * r;
 232   /* Without fma the worst case error is 0.25/N ulp larger.  */
 233   /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp.  */
 234 #if EXP_POLY_ORDER == 4
 235   tmp = tail + r + r2 * C2 + r * r2 * (C3 + r * C4);
 236 #elif EXP_POLY_ORDER == 5
 237   tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
 238 #elif EXP_POLY_ORDER == 6
 239   tmp = tail + r + r2 * (0.5 + r * C3) + r2 * r2 * (C4 + r * C5 + r2 * C6);
 240 #endif
 241   if (unlikely (abstop == 0))
 242     return specialcase (tmp, sbits, ki);
 243   scale = asdouble (sbits);
 244   /* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there
 245      is no spurious underflow here even without fma.  */
 246   return eval_as_double (scale + scale * tmp);
 247 }
 248
 249 /* Returns 0 if not int, 1 if odd int, 2 if even int.  The argument is
 250    the bit representation of a non-zero finite floating-point value.  */
 251 static inline int
 252 checkint (uint64_t iy)
 253 {
 254   int e = iy >> 52 & 0x7ff;
 255   if (e < 0x3ff)
 256     return 0;
 257   if (e > 0x3ff + 52)
 258     return 2;
 259   if (iy & ((1ULL << (0x3ff + 52 - e)) - 1))
 260     return 0;
 261   if (iy & (1ULL << (0x3ff + 52 - e)))
 262     return 1;
 263   return 2;
 264 }
 265
 266 /* Returns 1 if input is the bit representation of 0, infinity or nan.  */
 267 static inline int
 268 zeroinfnan (uint64_t i)
 269 {
 270   return 2 * i - 1 >= 2 * asuint64 (INFINITY) - 1;
 271 }
 272
 273 double
 274 pow (double x, double y)
 275 {
 276   uint32_t sign_bias = 0;
 277   uint64_t ix, iy;
 278   uint32_t topx, topy;
 279
 280   ix = asuint64 (x);
 281   iy = asuint64 (y);
 282   topx = top12 (x);
 283   topy = top12 (y);
 284   if (unlikely (topx - 0x001 >= 0x7ff - 0x001
 285                 || (topy & 0x7ff) - 0x3be >= 0x43e - 0x3be))
 286     {
 287       /* Note: if |y| > 1075 * ln2 * 2^53 ~= 0x1.749p62 then pow(x,y) = inf/0
 288          and if |y| < 2^-54 / 1075 ~= 0x1.e7b6p-65 then pow(x,y) = +-1.  */
 289       /* Special cases: (x < 0x1p-126 or inf or nan) or
 290          (|y| < 0x1p-65 or |y| >= 0x1p63 or nan).  */
 291       if (unlikely (zeroinfnan (iy)))
 292         {
 293           if (2 * iy == 0)
 294             return issignaling_inline (x) ? x + y : 1.0;
 295           if (ix == asuint64 (1.0))
 296             return issignaling_inline (y) ? x + y : 1.0;
 297           if (2 * ix > 2 * asuint64 (INFINITY)
 298               || 2 * iy > 2 * asuint64 (INFINITY))
 299             return x + y;
 300           if (2 * ix == 2 * asuint64 (1.0))
 301             return 1.0;
 302           if ((2 * ix < 2 * asuint64 (1.0)) == !(iy >> 63))
 303             return 0.0; /* |x|<1 && y==inf or |x|>1 && y==-inf.  */
 304           return y * y;
 305         }
 306       if (unlikely (zeroinfnan (ix)))
 307         {
 308           double_t x2 = x * x;
 309           if (ix >> 63 && checkint (iy) == 1)
 310             {
 311               x2 = -x2;
 312               sign_bias = 1;
 313             }
 314           if (WANT_ERRNO && 2 * ix == 0 && iy >> 63)
 315             return __math_divzero (sign_bias);
 316           /* Without the barrier some versions of clang hoist the 1/x2 and
 317              thus division by zero exception can be signaled spuriously.  */
 318           return iy >> 63 ? opt_barrier_double (1 / x2) : x2;
 319         }
 320       /* Here x and y are non-zero finite.  */
 321       if (ix >> 63)
 322         {
 323           /* Finite x < 0.  */
 324           int yint = checkint (iy);
 325           if (yint == 0)
 326             return __math_invalid (x);
 327           if (yint == 1)
 328             sign_bias = SIGN_BIAS;
 329           ix &= 0x7fffffffffffffff;
 330           topx &= 0x7ff;
 331         }
 332       if ((topy & 0x7ff) - 0x3be >= 0x43e - 0x3be)
 333         {
 334           /* Note: sign_bias == 0 here because y is not odd.  */
 335           if (ix == asuint64 (1.0))
 336             return 1.0;
 337           if ((topy & 0x7ff) < 0x3be)
 338             {
 339               /* |y| < 2^-65, x^y ~= 1 + y*log(x).  */
 340               if (WANT_ROUNDING)
 341                 return ix > asuint64 (1.0) ? 1.0 + y : 1.0 - y;
 342               else
 343                 return 1.0;
 344             }
 345           return (ix > asuint64 (1.0)) == (topy < 0x800) ? __math_oflow (0)
 346                                                          : __math_uflow (0);
 347         }
 348       if (topx == 0)
 349         {
 350           /* Normalize subnormal x so exponent becomes negative.  */
 351           /* Without the barrier some versions of clang evaluate the mul
 352              unconditionally causing spurious overflow exceptions.  */
 353           ix = asuint64 (opt_barrier_double (x) * 0x1p52);
 354           ix &= 0x7fffffffffffffff;
 355           ix -= 52ULL << 52;
 356         }
 357     }
 358
 359   double_t lo;
 360   double_t hi = log_inline (ix, &lo);
 361   double_t ehi, elo;
 362 #if HAVE_FAST_FMA
 363   ehi = y * hi;
 364   elo = y * lo + fma (y, hi, -ehi);
 365 #else
 366   double_t yhi = asdouble (iy & -1ULL << 27);
 367   double_t ylo = y - yhi;
 368   double_t lhi = asdouble (asuint64 (hi) & -1ULL << 27);
 369   double_t llo = hi - lhi + lo;
 370   ehi = yhi * lhi;
 371   elo = ylo * lhi + y * llo; /* |elo| < |ehi| * 2^-25.  */
 372 #endif
 373   return exp_inline (ehi, elo, sign_bias);
 374 }
 375 #if USE_GLIBC_ABI
 376 strong_alias (pow, __pow_finite)
 377 hidden_alias (pow, __ieee754_pow)
 378 # if LDBL_MANT_DIG == 53
 379 long double powl (long double x, long double y) { return pow (x, y); }
 380 # endif
 381 #endif