/* * Double-precision 2^x function. * * Copyright (c) 2018, Arm Limited. * SPDX-License-Identifier: MIT */ #include #include #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); }