/* * Single-precision pow function. * * Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions. * See https://llvm.org/LICENSE.txt for license information. * SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception */ #include #include #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_INTRINSICS 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