/*
* Copyright (c) 2017-2018, Arm Limited.
* SPDX-License-Identifier: MIT
*/
#include <math.h>
#include <stdint.h>
#include "libm.h"
#include "exp2f_data.h"
#include "powf_data.h"
/*
POWF_LOG2_POLY_ORDER = 5
EXP2F_TABLE_BITS = 5
ULP error: 0.82 (~ 0.5 + relerr*2^24)
relerr: 1.27 * 2^-26 (Relative error ~= 128*Ln2*relerr_log2 + relerr_exp2)
relerr_log2: 1.83 * 2^-33 (Relative error of logx.)
relerr_exp2: 1.69 * 2^-34 (Relative error of exp2(ylogx).)
*/
#define N (1 << POWF_LOG2_TABLE_BITS)
#define T __powf_log2_data.tab
#define A __powf_log2_data.poly
#define OFF 0x3f330000
/* Subnormal input is normalized so ix has negative biased exponent.
Output is multiplied by N (POWF_SCALE) if TOINT_INTRINICS is set. */
static inline double_t log2_inline(uint32_t ix)
{
double_t z, r, r2, r4, p, q, y, y0, invc, logc;
uint32_t iz, top, tmp;
int k, i;
/* x = 2^k z; where z is in range [OFF,2*OFF] and exact.
The range is split into N subintervals.
The ith subinterval contains z and c is near its center. */
tmp = ix - OFF;
i = (tmp >> (23 - POWF_LOG2_TABLE_BITS)) % N;
top = tmp & 0xff800000;
iz = ix - top;
k = (int32_t)top >> (23 - POWF_SCALE_BITS); /* arithmetic shift */
invc = T[i].invc;
logc = T[i].logc;
z = (double_t)asfloat(iz);
/* log2(x) = log1p(z/c-1)/ln2 + log2(c) + k */
r = z * invc - 1;
y0 = logc + (double_t)k;
/* Pipelined polynomial evaluation to approximate log1p(r)/ln2. */
r2 = r * r;
y = A[0] * r + A[1];
p = A[2] * r + A[3];
r4 = r2 * r2;
q = A[4] * r + y0;
q = p * r2 + q;
y = y * r4 + q;
return y;
}
#undef N
#undef T
#define N (1 << EXP2F_TABLE_BITS)
#define T __exp2f_data.tab
#define SIGN_BIAS (1 << (EXP2F_TABLE_BITS + 11))
/* The output of log2 and thus the input of exp2 is either scaled by N
(in case of fast toint intrinsics) or not. The unscaled xd must be
in [-1021,1023], sign_bias sets the sign of the result. */
static inline float exp2_inline(double_t xd, uint32_t sign_bias)
{
uint64_t ki, ski, t;
double_t kd, z, r, r2, y, s;
#if TOINT_INTRINSICS
#define C __exp2f_data.poly_scaled
/* N*x = k + r with r in [-1/2, 1/2] */
kd = roundtoint(xd); /* k */
ki = converttoint(xd);
#else
#define C __exp2f_data.poly
#define SHIFT __exp2f_data.shift_scaled
/* x = k/N + r with r in [-1/(2N), 1/(2N)] */
kd = eval_as_double(xd + SHIFT);
ki = asuint64(kd);
kd -= SHIFT; /* k/N */
#endif
r = xd - kd;
/* exp2(x) = 2^(k/N) * 2^r ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */
t = T[ki % N];
ski = ki + sign_bias;
t += ski << (52 - EXP2F_TABLE_BITS);
s = asdouble(t);
z = C[0] * r + C[1];
r2 = r * r;
y = C[2] * r + 1;
y = z * r2 + y;
y = y * s;
return eval_as_float(y);
}
/* Returns 0 if not int, 1 if odd int, 2 if even int. The argument is
the bit representation of a non-zero finite floating-point value. */
static inline int checkint(uint32_t iy)
{
int e = iy >> 23 & 0xff;
if (e < 0x7f)
return 0;
if (e > 0x7f + 23)
return 2;
if (iy & ((1 << (0x7f + 23 - e)) - 1))
return 0;
if (iy & (1 << (0x7f + 23 - e)))
return 1;
return 2;
}
static inline int zeroinfnan(uint32_t ix)
{
return 2 * ix - 1 >= 2u * 0x7f800000 - 1;
}
float powf(float x, float y)
{
uint32_t sign_bias = 0;
uint32_t ix, iy;
ix = asuint(x);
iy = asuint(y);
if (predict_false(ix - 0x00800000 >= 0x7f800000 - 0x00800000 ||
zeroinfnan(iy))) {
/* Either (x < 0x1p-126 or inf or nan) or (y is 0 or inf or nan). */
if (predict_false(zeroinfnan(iy))) {
if (2 * iy == 0)
return issignalingf_inline(x) ? x + y : 1.0f;
if (ix == 0x3f800000)
return issignalingf_inline(y) ? x + y : 1.0f;
if (2 * ix > 2u * 0x7f800000 ||
2 * iy > 2u * 0x7f800000)
return x + y;
if (2 * ix == 2 * 0x3f800000)
return 1.0f;
if ((2 * ix < 2 * 0x3f800000) == !(iy & 0x80000000))
return 0.0f; /* |x|<1 && y==inf or |x|>1 && y==-inf. */
return y * y;
}
if (predict_false(zeroinfnan(ix))) {
float_t x2 = x * x;
if (ix & 0x80000000 && checkint(iy) == 1)
x2 = -x2;
/* Without the barrier some versions of clang hoist the 1/x2 and
thus division by zero exception can be signaled spuriously. */
return iy & 0x80000000 ? fp_barrierf(1 / x2) : x2;
}
/* x and y are non-zero finite. */
if (ix & 0x80000000) {
/* Finite x < 0. */
int yint = checkint(iy);
if (yint == 0)
return __math_invalidf(x);
if (yint == 1)
sign_bias = SIGN_BIAS;
ix &= 0x7fffffff;
}
if (ix < 0x00800000) {
/* Normalize subnormal x so exponent becomes negative. */
ix = asuint(x * 0x1p23f);
ix &= 0x7fffffff;
ix -= 23 << 23;
}
}
double_t logx = log2_inline(ix);
double_t ylogx = y * logx; /* cannot overflow, y is single prec. */
if (predict_false((asuint64(ylogx) >> 47 & 0xffff) >=
asuint64(126.0 * POWF_SCALE) >> 47)) {
/* |y*log(x)| >= 126. */
if (ylogx > 0x1.fffffffd1d571p+6 * POWF_SCALE)
return __math_oflowf(sign_bias);
if (ylogx <= -150.0 * POWF_SCALE)
return __math_uflowf(sign_bias);
}
return exp2_inline(ylogx, sign_bias);
}
