github.com/afumu/libc@v0.0.6/musl/src/math/exp2.c (about)

     1  /*
     2   * Double-precision 2^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 "exp_data.h"
    12  
    13  #define N (1 << EXP_TABLE_BITS)
    14  #define Shift __exp_data.exp2_shift
    15  #define T __exp_data.tab
    16  #define C1 __exp_data.exp2_poly[0]
    17  #define C2 __exp_data.exp2_poly[1]
    18  #define C3 __exp_data.exp2_poly[2]
    19  #define C4 __exp_data.exp2_poly[3]
    20  #define C5 __exp_data.exp2_poly[4]
    21  
    22  /* Handle cases that may overflow or underflow when computing the result that
    23     is scale*(1+TMP) without intermediate rounding.  The bit representation of
    24     scale is in SBITS, however it has a computed exponent that may have
    25     overflown into the sign bit so that needs to be adjusted before using it as
    26     a double.  (int32_t)KI is the k used in the argument reduction and exponent
    27     adjustment of scale, positive k here means the result may overflow and
    28     negative k means the result may underflow.  */
    29  static inline double specialcase(double_t tmp, uint64_t sbits, uint64_t ki)
    30  {
    31  	double_t scale, y;
    32  
    33  	if ((ki & 0x80000000) == 0) {
    34  		/* k > 0, the exponent of scale might have overflowed by 1.  */
    35  		sbits -= 1ull << 52;
    36  		scale = asdouble(sbits);
    37  		y = 2 * (scale + scale * tmp);
    38  		return eval_as_double(y);
    39  	}
    40  	/* k < 0, need special care in the subnormal range.  */
    41  	sbits += 1022ull << 52;
    42  	scale = asdouble(sbits);
    43  	y = scale + scale * tmp;
    44  	if (y < 1.0) {
    45  		/* Round y to the right precision before scaling it into the subnormal
    46  		   range to avoid double rounding that can cause 0.5+E/2 ulp error where
    47  		   E is the worst-case ulp error outside the subnormal range.  So this
    48  		   is only useful if the goal is better than 1 ulp worst-case error.  */
    49  		double_t hi, lo;
    50  		lo = scale - y + scale * tmp;
    51  		hi = 1.0 + y;
    52  		lo = 1.0 - hi + y + lo;
    53  		y = eval_as_double(hi + lo) - 1.0;
    54  		/* Avoid -0.0 with downward rounding.  */
    55  		if (WANT_ROUNDING && y == 0.0)
    56  			y = 0.0;
    57  		/* The underflow exception needs to be signaled explicitly.  */
    58  		fp_force_eval(fp_barrier(0x1p-1022) * 0x1p-1022);
    59  	}
    60  	y = 0x1p-1022 * y;
    61  	return eval_as_double(y);
    62  }
    63  
    64  /* Top 12 bits of a double (sign and exponent bits).  */
    65  static inline uint32_t top12(double x)
    66  {
    67  	return asuint64(x) >> 52;
    68  }
    69  
    70  double exp2(double x)
    71  {
    72  	uint32_t abstop;
    73  	uint64_t ki, idx, top, sbits;
    74  	double_t kd, r, r2, scale, tail, tmp;
    75  
    76  	abstop = top12(x) & 0x7ff;
    77  	if (predict_false(abstop - top12(0x1p-54) >= top12(512.0) - top12(0x1p-54))) {
    78  		if (abstop - top12(0x1p-54) >= 0x80000000)
    79  			/* Avoid spurious underflow for tiny x.  */
    80  			/* Note: 0 is common input.  */
    81  			return WANT_ROUNDING ? 1.0 + x : 1.0;
    82  		if (abstop >= top12(1024.0)) {
    83  			if (asuint64(x) == asuint64(-INFINITY))
    84  				return 0.0;
    85  			if (abstop >= top12(INFINITY))
    86  				return 1.0 + x;
    87  			if (!(asuint64(x) >> 63))
    88  				return __math_oflow(0);
    89  			else if (asuint64(x) >= asuint64(-1075.0))
    90  				return __math_uflow(0);
    91  		}
    92  		if (2 * asuint64(x) > 2 * asuint64(928.0))
    93  			/* Large x is special cased below.  */
    94  			abstop = 0;
    95  	}
    96  
    97  	/* exp2(x) = 2^(k/N) * 2^r, with 2^r in [2^(-1/2N),2^(1/2N)].  */
    98  	/* x = k/N + r, with int k and r in [-1/2N, 1/2N].  */
    99  	kd = eval_as_double(x + Shift);
   100  	ki = asuint64(kd); /* k.  */
   101  	kd -= Shift; /* k/N for int k.  */
   102  	r = x - kd;
   103  	/* 2^(k/N) ~= scale * (1 + tail).  */
   104  	idx = 2 * (ki % N);
   105  	top = ki << (52 - EXP_TABLE_BITS);
   106  	tail = asdouble(T[idx]);
   107  	/* This is only a valid scale when -1023*N < k < 1024*N.  */
   108  	sbits = T[idx + 1] + top;
   109  	/* exp2(x) = 2^(k/N) * 2^r ~= scale + scale * (tail + 2^r - 1).  */
   110  	/* Evaluation is optimized assuming superscalar pipelined execution.  */
   111  	r2 = r * r;
   112  	/* Without fma the worst case error is 0.5/N ulp larger.  */
   113  	/* Worst case error is less than 0.5+0.86/N+(abs poly error * 2^53) ulp.  */
   114  	tmp = tail + r * C1 + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
   115  	if (predict_false(abstop == 0))
   116  		return specialcase(tmp, sbits, ki);
   117  	scale = asdouble(sbits);
   118  	/* Note: tmp == 0 or |tmp| > 2^-65 and scale > 2^-928, so there
   119  	   is no spurious underflow here even without fma.  */
   120  	return eval_as_double(scale + scale * tmp);
   121  }