cosmopolitan/libc/math/powf.c

/*
 * Copyright (c) 2017-2018, Arm Limited.
 * SPDX-License-Identifier: MIT
 */

#include "libc/math/math.h"
#include "libc/math/libm.h"
#include "libc/math/exp2f_data.h"
#include "libc/math/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);
}