libc/AOR_v20.02/math/powf.c

   1 /*
   2  * Single-precision pow 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 <math.h>
  10 #include <stdint.h>
  11 #include "math_config.h"
  12
  13 /*
  14 POWF_LOG2_POLY_ORDER = 5
  15 EXP2F_TABLE_BITS = 5
  16
  17 ULP error: 0.82 (~ 0.5 + relerr*2^24)
  18 relerr: 1.27 * 2^-26 (Relative error ~= 128*Ln2*relerr_log2 + relerr_exp2)
  19 relerr_log2: 1.83 * 2^-33 (Relative error of logx.)
  20 relerr_exp2: 1.69 * 2^-34 (Relative error of exp2(ylogx).)
  21 */
  22
  23 #define N (1 << POWF_LOG2_TABLE_BITS)
  24 #define T __powf_log2_data.tab
  25 #define A __powf_log2_data.poly
  26 #define OFF 0x3f330000
  27
  28 /* Subnormal input is normalized so ix has negative biased exponent.
  29    Output is multiplied by N (POWF_SCALE) if TOINT_INTRINSICS is set.  */
  30 static inline double_t
  31 log2_inline (uint32_t ix)
  32 {
  33   /* double_t for better performance on targets with FLT_EVAL_METHOD==2.  */
  34   double_t z, r, r2, r4, p, q, y, y0, invc, logc;
  35   uint32_t iz, top, tmp;
  36   int k, i;
  37
  38   /* x = 2^k z; where z is in range [OFF,2*OFF] and exact.
  39      The range is split into N subintervals.
  40      The ith subinterval contains z and c is near its center.  */
  41   tmp = ix - OFF;
  42   i = (tmp >> (23 - POWF_LOG2_TABLE_BITS)) % N;
  43   top = tmp & 0xff800000;
  44   iz = ix - top;
  45   k = (int32_t) top >> (23 - POWF_SCALE_BITS); /* arithmetic shift */
  46   invc = T[i].invc;
  47   logc = T[i].logc;
  48   z = (double_t) asfloat (iz);
  49
  50   /* log2(x) = log1p(z/c-1)/ln2 + log2(c) + k */
  51   r = z * invc - 1;
  52   y0 = logc + (double_t) k;
  53
  54   /* Pipelined polynomial evaluation to approximate log1p(r)/ln2.  */
  55   r2 = r * r;
  56   y = A[0] * r + A[1];
  57   p = A[2] * r + A[3];
  58   r4 = r2 * r2;
  59   q = A[4] * r + y0;
  60   q = p * r2 + q;
  61   y = y * r4 + q;
  62   return y;
  63 }
  64
  65 #undef N
  66 #undef T
  67 #define N (1 << EXP2F_TABLE_BITS)
  68 #define T __exp2f_data.tab
  69 #define SIGN_BIAS (1 << (EXP2F_TABLE_BITS + 11))
  70
  71 /* The output of log2 and thus the input of exp2 is either scaled by N
  72    (in case of fast toint intrinsics) or not.  The unscaled xd must be
  73    in [-1021,1023], sign_bias sets the sign of the result.  */
  74 static inline float
  75 exp2_inline (double_t xd, uint32_t sign_bias)
  76 {
  77   uint64_t ki, ski, t;
  78   /* double_t for better performance on targets with FLT_EVAL_METHOD==2.  */
  79   double_t kd, z, r, r2, y, s;
  80
  81 #if TOINT_INTRINSICS
  82 # define C __exp2f_data.poly_scaled
  83   /* N*x = k + r with r in [-1/2, 1/2] */
  84   kd = roundtoint (xd); /* k */
  85   ki = converttoint (xd);
  86 #else
  87 # define C __exp2f_data.poly
  88 # define SHIFT __exp2f_data.shift_scaled
  89   /* x = k/N + r with r in [-1/(2N), 1/(2N)] */
  90   kd = eval_as_double (xd + SHIFT);
  91   ki = asuint64 (kd);
  92   kd -= SHIFT; /* k/N */
  93 #endif
  94   r = xd - kd;
  95
  96   /* exp2(x) = 2^(k/N) * 2^r ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */
  97   t = T[ki % N];
  98   ski = ki + sign_bias;
  99   t += ski << (52 - EXP2F_TABLE_BITS);
 100   s = asdouble (t);
 101   z = C[0] * r + C[1];
 102   r2 = r * r;
 103   y = C[2] * r + 1;
 104   y = z * r2 + y;
 105   y = y * s;
 106   return eval_as_float (y);
 107 }
 108
 109 /* Returns 0 if not int, 1 if odd int, 2 if even int.  The argument is
 110    the bit representation of a non-zero finite floating-point value.  */
 111 static inline int
 112 checkint (uint32_t iy)
 113 {
 114   int e = iy >> 23 & 0xff;
 115   if (e < 0x7f)
 116     return 0;
 117   if (e > 0x7f + 23)
 118     return 2;
 119   if (iy & ((1 << (0x7f + 23 - e)) - 1))
 120     return 0;
 121   if (iy & (1 << (0x7f + 23 - e)))
 122     return 1;
 123   return 2;
 124 }
 125
 126 static inline int
 127 zeroinfnan (uint32_t ix)
 128 {
 129   return 2 * ix - 1 >= 2u * 0x7f800000 - 1;
 130 }
 131
 132 float
 133 powf (float x, float y)
 134 {
 135   uint32_t sign_bias = 0;
 136   uint32_t ix, iy;
 137
 138   ix = asuint (x);
 139   iy = asuint (y);
 140   if (unlikely (ix - 0x00800000 >= 0x7f800000 - 0x00800000 || zeroinfnan (iy)))
 141     {
 142       /* Either (x < 0x1p-126 or inf or nan) or (y is 0 or inf or nan).  */
 143       if (unlikely (zeroinfnan (iy)))
 144         {
 145           if (2 * iy == 0)
 146             return issignalingf_inline (x) ? x + y : 1.0f;
 147           if (ix == 0x3f800000)
 148             return issignalingf_inline (y) ? x + y : 1.0f;
 149           if (2 * ix > 2u * 0x7f800000 || 2 * iy > 2u * 0x7f800000)
 150             return x + y;
 151           if (2 * ix == 2 * 0x3f800000)
 152             return 1.0f;
 153           if ((2 * ix < 2 * 0x3f800000) == !(iy & 0x80000000))
 154             return 0.0f; /* |x|<1 && y==inf or |x|>1 && y==-inf.  */
 155           return y * y;
 156         }
 157       if (unlikely (zeroinfnan (ix)))
 158         {
 159           float_t x2 = x * x;
 160           if (ix & 0x80000000 && checkint (iy) == 1)
 161             {
 162               x2 = -x2;
 163               sign_bias = 1;
 164             }
 165 #if WANT_ERRNO
 166           if (2 * ix == 0 && iy & 0x80000000)
 167             return __math_divzerof (sign_bias);
 168 #endif
 169           /* Without the barrier some versions of clang hoist the 1/x2 and
 170              thus division by zero exception can be signaled spuriously.  */
 171           return iy & 0x80000000 ? opt_barrier_float (1 / x2) : x2;
 172         }
 173       /* x and y are non-zero finite.  */
 174       if (ix & 0x80000000)
 175         {
 176           /* Finite x < 0.  */
 177           int yint = checkint (iy);
 178           if (yint == 0)
 179             return __math_invalidf (x);
 180           if (yint == 1)
 181             sign_bias = SIGN_BIAS;
 182           ix &= 0x7fffffff;
 183         }
 184       if (ix < 0x00800000)
 185         {
 186           /* Normalize subnormal x so exponent becomes negative.  */
 187           ix = asuint (x * 0x1p23f);
 188           ix &= 0x7fffffff;
 189           ix -= 23 << 23;
 190         }
 191     }
 192   double_t logx = log2_inline (ix);
 193   double_t ylogx = y * logx; /* Note: cannot overflow, y is single prec.  */
 194   if (unlikely ((asuint64 (ylogx) >> 47 & 0xffff)
 195                  >= asuint64 (126.0 * POWF_SCALE) >> 47))
 196     {
 197       /* |y*log(x)| >= 126.  */
 198       if (ylogx > 0x1.fffffffd1d571p+6 * POWF_SCALE)
 199         /* |x^y| > 0x1.ffffffp127.  */
 200         return __math_oflowf (sign_bias);
 201       if (WANT_ROUNDING && WANT_ERRNO
 202           && ylogx > 0x1.fffffffa3aae2p+6 * POWF_SCALE)
 203         /* |x^y| > 0x1.fffffep127, check if we round away from 0.  */
 204         if ((!sign_bias
 205              && eval_as_float (1.0f + opt_barrier_float (0x1p-25f)) != 1.0f)
 206             || (sign_bias
 207                 && eval_as_float (-1.0f - opt_barrier_float (0x1p-25f))
 208                      != -1.0f))
 209           return __math_oflowf (sign_bias);
 210       if (ylogx <= -150.0 * POWF_SCALE)
 211         return __math_uflowf (sign_bias);
 212 #if WANT_ERRNO_UFLOW
 213       if (ylogx < -149.0 * POWF_SCALE)
 214         return __math_may_uflowf (sign_bias);
 215 #endif
 216     }
 217   return exp2_inline (ylogx, sign_bias);
 218 }
 219 #if USE_GLIBC_ABI
 220 strong_alias (powf, __powf_finite)
 221 hidden_alias (powf, __ieee754_powf)
 222 #endif