math/powf.c

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

#include <math.h>
#include <stdint.h>
#include "math_config.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 for better performance on targets with FLT_EVAL_METHOD==2.  */
  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 for better performance on targets with FLT_EVAL_METHOD==2.  */
  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 (unlikely (ix - 0x00800000 >= 0x7f800000 - 0x00800000 || zeroinfnan (iy)))
    {
      /* Either (x < 0x1p-126 or inf or nan) or (y is 0 or inf or nan).  */
      if (unlikely (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 (unlikely (zeroinfnan (ix)))
	{
	  float_t x2 = x * x;
	  if (ix & 0x80000000 && checkint (iy) == 1)
	    {
	      x2 = -x2;
	      sign_bias = 1;
	    }
#if WANT_ERRNO
	  if (2 * ix == 0 && iy & 0x80000000)
	    return __math_divzerof (sign_bias);
#endif
	  /* Without the barrier some versions of clang hoist the 1/x2 and
	     thus division by zero exception can be signaled spuriously.  */
	  return iy & 0x80000000 ? opt_barrier_float (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; /* Note: cannot overflow, y is single prec.  */
  if (unlikely ((asuint64 (ylogx) >> 47 & 0xffff)
		 >= asuint64 (126.0 * POWF_SCALE) >> 47))
    {
      /* |y*log(x)| >= 126.  */
      if (ylogx > 0x1.fffffffd1d571p+6 * POWF_SCALE)
	/* |x^y| > 0x1.ffffffp127.  */
	return __math_oflowf (sign_bias);
      if (WANT_ROUNDING && WANT_ERRNO
	  && ylogx > 0x1.fffffffa3aae2p+6 * POWF_SCALE)
	/* |x^y| > 0x1.fffffep127, check if we round away from 0.  */
	if ((!sign_bias
	     && eval_as_float (1.0f + opt_barrier_float (0x1p-25f)) != 1.0f)
	    || (sign_bias
		&& eval_as_float (-1.0f - opt_barrier_float (0x1p-25f))
		     != -1.0f))
	  return __math_oflowf (sign_bias);
      if (ylogx <= -150.0 * POWF_SCALE)
	return __math_uflowf (sign_bias);
#if WANT_ERRNO_UFLOW
      if (ylogx < -149.0 * POWF_SCALE)
	return __math_may_uflowf (sign_bias);
#endif
    }
  return exp2_inline (ylogx, sign_bias);
}
#if USE_GLIBC_ABI
strong_alias (powf, __powf_finite)
hidden_alias (powf, __ieee754_powf)
#endif