mirror of
https://github.com/ClickHouse/ClickHouse.git
synced 2024-11-16 04:32:33 +00:00
122 lines
4.0 KiB
C
122 lines
4.0 KiB
C
/*
|
|
* Double-precision 2^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 Shift __exp_data.exp2_shift
|
|
#define T __exp_data.tab
|
|
#define C1 __exp_data.exp2_poly[0]
|
|
#define C2 __exp_data.exp2_poly[1]
|
|
#define C3 __exp_data.exp2_poly[2]
|
|
#define C4 __exp_data.exp2_poly[3]
|
|
#define C5 __exp_data.exp2_poly[4]
|
|
|
|
/* 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 1. */
|
|
sbits -= 1ull << 52;
|
|
scale = asdouble(sbits);
|
|
y = 2 * (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 exp2(double x)
|
|
{
|
|
uint32_t abstop;
|
|
uint64_t ki, idx, top, sbits;
|
|
double_t kd, 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_oflow(0);
|
|
else if (asuint64(x) >= asuint64(-1075.0))
|
|
return __math_uflow(0);
|
|
}
|
|
if (2 * asuint64(x) > 2 * asuint64(928.0))
|
|
/* Large x is special cased below. */
|
|
abstop = 0;
|
|
}
|
|
|
|
/* exp2(x) = 2^(k/N) * 2^r, with 2^r in [2^(-1/2N),2^(1/2N)]. */
|
|
/* x = k/N + r, with int k and r in [-1/2N, 1/2N]. */
|
|
kd = eval_as_double(x + Shift);
|
|
ki = asuint64(kd); /* k. */
|
|
kd -= Shift; /* k/N for int k. */
|
|
r = x - kd;
|
|
/* 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;
|
|
/* exp2(x) = 2^(k/N) * 2^r ~= scale + scale * (tail + 2^r - 1). */
|
|
/* Evaluation is optimized assuming superscalar pipelined execution. */
|
|
r2 = r * r;
|
|
/* Without fma the worst case error is 0.5/N ulp larger. */
|
|
/* Worst case error is less than 0.5+0.86/N+(abs poly error * 2^53) ulp. */
|
|
tmp = tail + r * C1 + 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^-65 and scale > 2^-928, so there
|
|
is no spurious underflow here even without fma. */
|
|
return eval_as_double(scale + scale * tmp);
|
|
}
|