1 /* 2 * Single-precision vector powf function. 3 * 4 * Copyright (c) 2019-2023, Arm Limited. 5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6 */ 7 8 #include "v_math.h" 9 10 #define Min v_u32 (0x00800000) 11 #define Max v_u32 (0x7f800000) 12 #define Thresh v_u32 (0x7f000000) /* Max - Min. */ 13 #define MantissaMask v_u32 (0x007fffff) 14 15 #define A data.log2_poly 16 #define C data.exp2f_poly 17 18 /* 2.6 ulp ~ 0.5 + 2^24 (128*Ln2*relerr_log2 + relerr_exp2). */ 19 #define Off v_u32 (0x3f35d000) 20 21 #define V_POWF_LOG2_TABLE_BITS 5 22 #define V_EXP2F_TABLE_BITS 5 23 #define Log2IdxMask v_u32 ((1 << V_POWF_LOG2_TABLE_BITS) - 1) 24 #define Scale ((double) (1 << V_EXP2F_TABLE_BITS)) 25 26 static const struct 27 { 28 struct 29 { 30 double invc, logc; 31 } log2_tab[1 << V_POWF_LOG2_TABLE_BITS]; 32 double log2_poly[4]; 33 uint64_t exp2f_tab[1 << V_EXP2F_TABLE_BITS]; 34 double exp2f_poly[3]; 35 } data = { 36 .log2_tab = {{0x1.6489890582816p+0, -0x1.e960f97b22702p-2 * Scale}, 37 {0x1.5cf19b35e3472p+0, -0x1.c993406cd4db6p-2 * Scale}, 38 {0x1.55aac0e956d65p+0, -0x1.aa711d9a7d0f3p-2 * Scale}, 39 {0x1.4eb0022977e01p+0, -0x1.8bf37bacdce9bp-2 * Scale}, 40 {0x1.47fcccda1dd1fp+0, -0x1.6e13b3519946ep-2 * Scale}, 41 {0x1.418ceabab68c1p+0, -0x1.50cb8281e4089p-2 * Scale}, 42 {0x1.3b5c788f1edb3p+0, -0x1.341504a237e2bp-2 * Scale}, 43 {0x1.3567de48e9c9ap+0, -0x1.17eaab624ffbbp-2 * Scale}, 44 {0x1.2fabc80fd19bap+0, -0x1.f88e708f8c853p-3 * Scale}, 45 {0x1.2a25200ce536bp+0, -0x1.c24b6da113914p-3 * Scale}, 46 {0x1.24d108e0152e3p+0, -0x1.8d02ee397cb1dp-3 * Scale}, 47 {0x1.1facd8ab2fbe1p+0, -0x1.58ac1223408b3p-3 * Scale}, 48 {0x1.1ab614a03efdfp+0, -0x1.253e6fd190e89p-3 * Scale}, 49 {0x1.15ea6d03af9ffp+0, -0x1.e5641882c12ffp-4 * Scale}, 50 {0x1.1147b994bb776p+0, -0x1.81fea712926f7p-4 * Scale}, 51 {0x1.0ccbf650593aap+0, -0x1.203e240de64a3p-4 * Scale}, 52 {0x1.0875408477302p+0, -0x1.8029b86a78281p-5 * Scale}, 53 {0x1.0441d42a93328p+0, -0x1.85d713190fb9p-6 * Scale}, 54 {0x1p+0, 0x0p+0 * Scale}, 55 {0x1.f1d006c855e86p-1, 0x1.4c1cc07312997p-5 * Scale}, 56 {0x1.e28c3341aa301p-1, 0x1.5e1848ccec948p-4 * Scale}, 57 {0x1.d4bdf9aa64747p-1, 0x1.04cfcb7f1196fp-3 * Scale}, 58 {0x1.c7b45a24e5803p-1, 0x1.582813d463c21p-3 * Scale}, 59 {0x1.bb5f5eb2ed60ap-1, 0x1.a936fa68760ccp-3 * Scale}, 60 {0x1.afb0bff8fe6b4p-1, 0x1.f81bc31d6cc4ep-3 * Scale}, 61 {0x1.a49badf7ab1f5p-1, 0x1.2279a09fae6b1p-2 * Scale}, 62 {0x1.9a14a111fc4c9p-1, 0x1.47ec0b6df5526p-2 * Scale}, 63 {0x1.901131f5b2fdcp-1, 0x1.6c71762280f1p-2 * Scale}, 64 {0x1.8687f73f6d865p-1, 0x1.90155070798dap-2 * Scale}, 65 {0x1.7d7067eb77986p-1, 0x1.b2e23b1d3068cp-2 * Scale}, 66 {0x1.74c2c1cf97b65p-1, 0x1.d4e21b0daa86ap-2 * Scale}, 67 {0x1.6c77f37cff2a1p-1, 0x1.f61e2a2f67f3fp-2 * Scale},}, 68 .log2_poly = { /* rel err: 1.5 * 2^-30. */ 69 -0x1.6ff5daa3b3d7cp-2 * Scale, 0x1.ec81d03c01aebp-2 * Scale, 70 -0x1.71547bb43f101p-1 * Scale, 0x1.7154764a815cbp0 * Scale,}, 71 .exp2f_tab = {0x3ff0000000000000, 0x3fefd9b0d3158574, 0x3fefb5586cf9890f, 72 0x3fef9301d0125b51, 0x3fef72b83c7d517b, 0x3fef54873168b9aa, 73 0x3fef387a6e756238, 0x3fef1e9df51fdee1, 0x3fef06fe0a31b715, 74 0x3feef1a7373aa9cb, 0x3feedea64c123422, 0x3feece086061892d, 75 0x3feebfdad5362a27, 0x3feeb42b569d4f82, 0x3feeab07dd485429, 76 0x3feea47eb03a5585, 0x3feea09e667f3bcd, 0x3fee9f75e8ec5f74, 77 0x3feea11473eb0187, 0x3feea589994cce13, 0x3feeace5422aa0db, 78 0x3feeb737b0cdc5e5, 0x3feec49182a3f090, 0x3feed503b23e255d, 79 0x3feee89f995ad3ad, 0x3feeff76f2fb5e47, 0x3fef199bdd85529c, 80 0x3fef3720dcef9069, 0x3fef5818dcfba487, 0x3fef7c97337b9b5f, 81 0x3fefa4afa2a490da, 0x3fefd0765b6e4540,}, 82 .exp2f_poly = { /* rel err: 1.69 * 2^-34. */ 83 0x1.c6af84b912394p-5 / Scale / Scale / Scale, 84 0x1.ebfce50fac4f3p-3 / Scale / Scale, 85 0x1.62e42ff0c52d6p-1 / Scale}}; 86 87 static float32x4_t VPCS_ATTR NOINLINE 88 special_case (float32x4_t x, float32x4_t y, float32x4_t ret, uint32x4_t cmp) 89 { 90 return v_call2_f32 (powf, x, y, ret, cmp); 91 } 92 93 float32x4_t VPCS_ATTR V_NAME_F2 (pow) (float32x4_t x, float32x4_t y) 94 { 95 uint32x4_t u = vreinterpretq_u32_f32 (x); 96 uint32x4_t cmp = vcgeq_u32 (vsubq_u32 (u, Min), Thresh); 97 uint32x4_t tmp = vsubq_u32 (u, Off); 98 uint32x4_t i = vandq_u32 (vshrq_n_u32 (tmp, (23 - V_POWF_LOG2_TABLE_BITS)), 99 Log2IdxMask); 100 uint32x4_t top = vbicq_u32 (tmp, MantissaMask); 101 uint32x4_t iz = vsubq_u32 (u, top); 102 int32x4_t k = vshrq_n_s32 (vreinterpretq_s32_u32 (top), 103 23 - V_EXP2F_TABLE_BITS); /* arithmetic shift. */ 104 105 float32x4_t ret; 106 for (int lane = 0; lane < 4; lane++) 107 { 108 /* Use double precision for each lane. */ 109 double invc = data.log2_tab[i[lane]].invc; 110 double logc = data.log2_tab[i[lane]].logc; 111 double z = (double) asfloat (iz[lane]); 112 113 /* log2(x) = log1p(z/c-1)/ln2 + log2(c) + k. */ 114 double r = __builtin_fma (z, invc, -1.0); 115 double y0 = logc + (double) k[lane]; 116 117 /* Polynomial to approximate log1p(r)/ln2. */ 118 double logx = A[0]; 119 logx = r * logx + A[1]; 120 logx = r * logx + A[2]; 121 logx = r * logx + A[3]; 122 logx = r * logx + y0; 123 double ylogx = y[lane] * logx; 124 cmp[lane] = (asuint64 (ylogx) >> 47 & 0xffff) 125 >= asuint64 (126.0 * (1 << V_EXP2F_TABLE_BITS)) >> 47 126 ? 1 127 : cmp[lane]; 128 129 /* N*x = k + r with r in [-1/2, 1/2]. */ 130 double kd = round (ylogx); 131 uint64_t ki = lround (ylogx); 132 r = ylogx - kd; 133 134 /* exp2(x) = 2^(k/N) * 2^r ~= s * (C0*r^3 + C1*r^2 + C2*r + 1). */ 135 uint64_t t = data.exp2f_tab[ki % (1 << V_EXP2F_TABLE_BITS)]; 136 t += ki << (52 - V_EXP2F_TABLE_BITS); 137 double s = asdouble (t); 138 double p = C[0]; 139 p = __builtin_fma (p, r, C[1]); 140 p = __builtin_fma (p, r, C[2]); 141 p = __builtin_fma (p, s * r, s); 142 143 ret[lane] = p; 144 } 145 if (unlikely (v_any_u32 (cmp))) 146 return special_case (x, y, ret, cmp); 147 return ret; 148 } 149