github.com/afumu/libc@v0.0.6/musl/src/math/log.c (about)

     1  /*
     2   * Double-precision log(x) function.
     3   *
     4   * Copyright (c) 2018, Arm Limited.
     5   * SPDX-License-Identifier: MIT
     6   */
     7  
     8  #include <math.h>
     9  #include <stdint.h>
    10  #include "libm.h"
    11  #include "log_data.h"
    12  
    13  #define T __log_data.tab
    14  #define T2 __log_data.tab2
    15  #define B __log_data.poly1
    16  #define A __log_data.poly
    17  #define Ln2hi __log_data.ln2hi
    18  #define Ln2lo __log_data.ln2lo
    19  #define N (1 << LOG_TABLE_BITS)
    20  #define OFF 0x3fe6000000000000
    21  
    22  /* Top 16 bits of a double.  */
    23  static inline uint32_t top16(double x)
    24  {
    25  	return asuint64(x) >> 48;
    26  }
    27  
    28  double log(double x)
    29  {
    30  	double_t w, z, r, r2, r3, y, invc, logc, kd, hi, lo;
    31  	uint64_t ix, iz, tmp;
    32  	uint32_t top;
    33  	int k, i;
    34  
    35  	ix = asuint64(x);
    36  	top = top16(x);
    37  #define LO asuint64(1.0 - 0x1p-4)
    38  #define HI asuint64(1.0 + 0x1.09p-4)
    39  	if (predict_false(ix - LO < HI - LO)) {
    40  		/* Handle close to 1.0 inputs separately.  */
    41  		/* Fix sign of zero with downward rounding when x==1.  */
    42  		if (WANT_ROUNDING && predict_false(ix == asuint64(1.0)))
    43  			return 0;
    44  		r = x - 1.0;
    45  		r2 = r * r;
    46  		r3 = r * r2;
    47  		y = r3 *
    48  		    (B[1] + r * B[2] + r2 * B[3] +
    49  		     r3 * (B[4] + r * B[5] + r2 * B[6] +
    50  			   r3 * (B[7] + r * B[8] + r2 * B[9] + r3 * B[10])));
    51  		/* Worst-case error is around 0.507 ULP.  */
    52  		w = r * 0x1p27;
    53  		double_t rhi = r + w - w;
    54  		double_t rlo = r - rhi;
    55  		w = rhi * rhi * B[0]; /* B[0] == -0.5.  */
    56  		hi = r + w;
    57  		lo = r - hi + w;
    58  		lo += B[0] * rlo * (rhi + r);
    59  		y += lo;
    60  		y += hi;
    61  		return eval_as_double(y);
    62  	}
    63  	if (predict_false(top - 0x0010 >= 0x7ff0 - 0x0010)) {
    64  		/* x < 0x1p-1022 or inf or nan.  */
    65  		if (ix * 2 == 0)
    66  			return __math_divzero(1);
    67  		if (ix == asuint64(INFINITY)) /* log(inf) == inf.  */
    68  			return x;
    69  		if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
    70  			return __math_invalid(x);
    71  		/* x is subnormal, normalize it.  */
    72  		ix = asuint64(x * 0x1p52);
    73  		ix -= 52ULL << 52;
    74  	}
    75  
    76  	/* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
    77  	   The range is split into N subintervals.
    78  	   The ith subinterval contains z and c is near its center.  */
    79  	tmp = ix - OFF;
    80  	i = (tmp >> (52 - LOG_TABLE_BITS)) % N;
    81  	k = (int64_t)tmp >> 52; /* arithmetic shift */
    82  	iz = ix - (tmp & 0xfffULL << 52);
    83  	invc = T[i].invc;
    84  	logc = T[i].logc;
    85  	z = asdouble(iz);
    86  
    87  	/* log(x) = log1p(z/c-1) + log(c) + k*Ln2.  */
    88  	/* r ~= z/c - 1, |r| < 1/(2*N).  */
    89  #if __FP_FAST_FMA
    90  	/* rounding error: 0x1p-55/N.  */
    91  	r = __builtin_fma(z, invc, -1.0);
    92  #else
    93  	/* rounding error: 0x1p-55/N + 0x1p-66.  */
    94  	r = (z - T2[i].chi - T2[i].clo) * invc;
    95  #endif
    96  	kd = (double_t)k;
    97  
    98  	/* hi + lo = r + log(c) + k*Ln2.  */
    99  	w = kd * Ln2hi + logc;
   100  	hi = w + r;
   101  	lo = w - hi + r + kd * Ln2lo;
   102  
   103  	/* log(x) = lo + (log1p(r) - r) + hi.  */
   104  	r2 = r * r; /* rounding error: 0x1p-54/N^2.  */
   105  	/* Worst case error if |y| > 0x1p-5:
   106  	   0.5 + 4.13/N + abs-poly-error*2^57 ULP (+ 0.002 ULP without fma)
   107  	   Worst case error if |y| > 0x1p-4:
   108  	   0.5 + 2.06/N + abs-poly-error*2^56 ULP (+ 0.001 ULP without fma).  */
   109  	y = lo + r2 * A[0] +
   110  	    r * r2 * (A[1] + r * A[2] + r2 * (A[3] + r * A[4])) + hi;
   111  	return eval_as_double(y);
   112  }