mirror of
https://github.com/ClickHouse/ClickHouse.git
synced 2024-12-12 09:22:05 +00:00
135 lines
4.4 KiB
C
135 lines
4.4 KiB
C
|
/*
|
||
|
* Double-precision e^x function.
|
||
|
*
|
||
|
* Copyright (c) 2018, Arm Limited.
|
||
|
* SPDX-License-Identifier: MIT
|
||
|
*/
|
||
|
|
||
|
#include <math.h>
|
||
|
#include <stdint.h>
|
||
|
#include "libm.h"
|
||
|
#include "exp_data.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]
|
||
|
|
||
|
/* 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 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. */
|
||
|
fp_force_eval(fp_barrier(0x1p-1022) * 0x1p-1022);
|
||
|
}
|
||
|
y = 0x1p-1022 * y;
|
||
|
return 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;
|
||
|
}
|
||
|
|
||
|
double exp(double x)
|
||
|
{
|
||
|
uint32_t abstop;
|
||
|
uint64_t ki, idx, top, sbits;
|
||
|
double_t kd, z, r, r2, scale, tail, tmp;
|
||
|
|
||
|
abstop = top12(x) & 0x7ff;
|
||
|
if (predict_false(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;
|
||
|
/* 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. */
|
||
|
tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
|
||
|
if (predict_false(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);
|
||
|
}
|