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/*
* Double-precision e^x function.
*
* Copyright (c) 2018-2019, Arm Limited.
* SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
*/
#include <float.h>
#include <math.h>
#include <stdint.h>
#include "math_config.h"
#define N (1 << EXP_TABLE_BITS)
#define InvLn2N __exp_data.invln2N
#define NegLn2hiN __exp_data.negln2hiN
#define NegLn2loN __exp_data.negln2loN
#define Shift __exp_data.shift
#define T __exp_data.tab
#define C2 __exp_data.poly[5 - EXP_POLY_ORDER]
#define C3 __exp_data.poly[6 - EXP_POLY_ORDER]
#define C4 __exp_data.poly[7 - EXP_POLY_ORDER]
#define C5 __exp_data.poly[8 - EXP_POLY_ORDER]
#define C6 __exp_data.poly[9 - EXP_POLY_ORDER]
/* Handle cases that may overflow or underflow when computing the result that
is scale*(1+TMP) without intermediate rounding. The bit representation of
scale is in SBITS, however it has a computed exponent that may have
overflown into the sign bit so that needs to be adjusted before using it as
a double. (int32_t)KI is the k used in the argument reduction and exponent
adjustment of scale, positive k here means the result may overflow and
negative k means the result may underflow. */
static inline double
specialcase (double_t tmp, uint64_t sbits, uint64_t ki)
{
double_t scale, y;
if ((ki & 0x80000000) == 0)
{
/* k > 0, the exponent of scale might have overflowed by <= 460. */
sbits -= 1009ull << 52;
scale = asdouble (sbits);
y = 0x1p1009 * (scale + scale * tmp);
return check_oflow (eval_as_double (y));
}
/* k < 0, need special care in the subnormal range. */
sbits += 1022ull << 52;
scale = asdouble (sbits);
y = scale + scale * tmp;
if (y < 1.0)
{
/* Round y to the right precision before scaling it into the subnormal
range to avoid double rounding that can cause 0.5+E/2 ulp error where
E is the worst-case ulp error outside the subnormal range. So this
is only useful if the goal is better than 1 ulp worst-case error. */
double_t hi, lo;
lo = scale - y + scale * tmp;
hi = 1.0 + y;
lo = 1.0 - hi + y + lo;
y = eval_as_double (hi + lo) - 1.0;
/* Avoid -0.0 with downward rounding. */
if (WANT_ROUNDING && y == 0.0)
y = 0.0;
/* The underflow exception needs to be signaled explicitly. */
force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022);
}
y = 0x1p-1022 * y;
return check_uflow (eval_as_double (y));
}
/* Top 12 bits of a double (sign and exponent bits). */
static inline uint32_t
top12 (double x)
{
return asuint64 (x) >> 52;
}
/* Computes exp(x+xtail) where |xtail| < 2^-8/N and |xtail| <= |x|.
If hastail is 0 then xtail is assumed to be 0 too. */
static inline double
exp_inline (double x, double xtail, int hastail)
{
uint32_t abstop;
uint64_t ki, idx, top, sbits;
/* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
double_t kd, z, r, r2, scale, tail, tmp;
abstop = top12 (x) & 0x7ff;
if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54)))
{
if (abstop - top12 (0x1p-54) >= 0x80000000)
/* Avoid spurious underflow for tiny x. */
/* Note: 0 is common input. */
return WANT_ROUNDING ? 1.0 + x : 1.0;
if (abstop >= top12 (1024.0))
{
if (asuint64 (x) == asuint64 (-INFINITY))
return 0.0;
if (abstop >= top12 (INFINITY))
return 1.0 + x;
if (asuint64 (x) >> 63)
return __math_uflow (0);
else
return __math_oflow (0);
}
/* Large x is special cased below. */
abstop = 0;
}
/* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)]. */
/* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N]. */
z = InvLn2N * x;
#if TOINT_INTRINSICS
kd = roundtoint (z);
ki = converttoint (z);
#elif EXP_USE_TOINT_NARROW
/* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes. */
kd = eval_as_double (z + Shift);
ki = asuint64 (kd) >> 16;
kd = (double_t) (int32_t) ki;
#else
/* z - kd is in [-1, 1] in non-nearest rounding modes. */
kd = eval_as_double (z + Shift);
ki = asuint64 (kd);
kd -= Shift;
#endif
r = x + kd * NegLn2hiN + kd * NegLn2loN;
/* The code assumes 2^-200 < |xtail| < 2^-8/N. */
if (hastail)
r += xtail;
/* 2^(k/N) ~= scale * (1 + tail). */
idx = 2 * (ki % N);
top = ki << (52 - EXP_TABLE_BITS);
tail = asdouble (T[idx]);
/* This is only a valid scale when -1023*N < k < 1024*N. */
sbits = T[idx + 1] + top;
/* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1). */
/* Evaluation is optimized assuming superscalar pipelined execution. */
r2 = r * r;
/* Without fma the worst case error is 0.25/N ulp larger. */
/* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp. */
#if EXP_POLY_ORDER == 4
tmp = tail + r + r2 * C2 + r * r2 * (C3 + r * C4);
#elif EXP_POLY_ORDER == 5
tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
#elif EXP_POLY_ORDER == 6
tmp = tail + r + r2 * (0.5 + r * C3) + r2 * r2 * (C4 + r * C5 + r2 * C6);
#endif
if (unlikely (abstop == 0))
return specialcase (tmp, sbits, ki);
scale = asdouble (sbits);
/* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there
is no spurious underflow here even without fma. */
return eval_as_double (scale + scale * tmp);
}
double
exp (double x)
{
return exp_inline (x, 0, 0);
}
/* May be useful for implementing pow where more than double
precision input is needed. */
double
__exp_dd (double x, double xtail)
{
return exp_inline (x, xtail, 1);
}
#if USE_GLIBC_ABI
strong_alias (exp, __exp_finite)
hidden_alias (exp, __ieee754_exp)
hidden_alias (__exp_dd, __exp1)
# if LDBL_MANT_DIG == 53
long double expl (long double x) { return exp (x); }
# endif
#endif
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