1//===-- fp_div_impl.inc - Floating point division -----------------*- C -*-===// 2// 3// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions. 4// See https://llvm.org/LICENSE.txt for license information. 5// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception 6// 7//===----------------------------------------------------------------------===// 8// 9// This file implements soft-float division with the IEEE-754 default 10// rounding (to nearest, ties to even). 11// 12//===----------------------------------------------------------------------===// 13 14#include "fp_lib.h" 15 16// The __divXf3__ function implements Newton-Raphson floating point division. 17// It uses 3 iterations for float32, 4 for float64 and 5 for float128, 18// respectively. Due to number of significant bits being roughly doubled 19// every iteration, the two modes are supported: N full-width iterations (as 20// it is done for float32 by default) and (N-1) half-width iteration plus one 21// final full-width iteration. It is expected that half-width integer 22// operations (w.r.t rep_t size) can be performed faster for some hardware but 23// they require error estimations to be computed separately due to larger 24// computational errors caused by truncating intermediate results. 25 26// Half the bit-size of rep_t 27#define HW (typeWidth / 2) 28// rep_t-sized bitmask with lower half of bits set to ones 29#define loMask (REP_C(-1) >> HW) 30 31#if NUMBER_OF_FULL_ITERATIONS < 1 32#error At least one full iteration is required 33#endif 34 35static __inline fp_t __divXf3__(fp_t a, fp_t b) { 36 37 const unsigned int aExponent = toRep(a) >> significandBits & maxExponent; 38 const unsigned int bExponent = toRep(b) >> significandBits & maxExponent; 39 const rep_t quotientSign = (toRep(a) ^ toRep(b)) & signBit; 40 41 rep_t aSignificand = toRep(a) & significandMask; 42 rep_t bSignificand = toRep(b) & significandMask; 43 int scale = 0; 44 45 // Detect if a or b is zero, denormal, infinity, or NaN. 46 if (aExponent - 1U >= maxExponent - 1U || 47 bExponent - 1U >= maxExponent - 1U) { 48 49 const rep_t aAbs = toRep(a) & absMask; 50 const rep_t bAbs = toRep(b) & absMask; 51 52 // NaN / anything = qNaN 53 if (aAbs > infRep) 54 return fromRep(toRep(a) | quietBit); 55 // anything / NaN = qNaN 56 if (bAbs > infRep) 57 return fromRep(toRep(b) | quietBit); 58 59 if (aAbs == infRep) { 60 // infinity / infinity = NaN 61 if (bAbs == infRep) 62 return fromRep(qnanRep); 63 // infinity / anything else = +/- infinity 64 else 65 return fromRep(aAbs | quotientSign); 66 } 67 68 // anything else / infinity = +/- 0 69 if (bAbs == infRep) 70 return fromRep(quotientSign); 71 72 if (!aAbs) { 73 // zero / zero = NaN 74 if (!bAbs) 75 return fromRep(qnanRep); 76 // zero / anything else = +/- zero 77 else 78 return fromRep(quotientSign); 79 } 80 // anything else / zero = +/- infinity 81 if (!bAbs) 82 return fromRep(infRep | quotientSign); 83 84 // One or both of a or b is denormal. The other (if applicable) is a 85 // normal number. Renormalize one or both of a and b, and set scale to 86 // include the necessary exponent adjustment. 87 if (aAbs < implicitBit) 88 scale += normalize(&aSignificand); 89 if (bAbs < implicitBit) 90 scale -= normalize(&bSignificand); 91 } 92 93 // Set the implicit significand bit. If we fell through from the 94 // denormal path it was already set by normalize( ), but setting it twice 95 // won't hurt anything. 96 aSignificand |= implicitBit; 97 bSignificand |= implicitBit; 98 99 int writtenExponent = (aExponent - bExponent + scale) + exponentBias; 100 101 const rep_t b_UQ1 = bSignificand << (typeWidth - significandBits - 1); 102 103 // Align the significand of b as a UQ1.(n-1) fixed-point number in the range 104 // [1.0, 2.0) and get a UQ0.n approximate reciprocal using a small minimax 105 // polynomial approximation: x0 = 3/4 + 1/sqrt(2) - b/2. 106 // The max error for this approximation is achieved at endpoints, so 107 // abs(x0(b) - 1/b) <= abs(x0(1) - 1/1) = 3/4 - 1/sqrt(2) = 0.04289..., 108 // which is about 4.5 bits. 109 // The initial approximation is between x0(1.0) = 0.9571... and x0(2.0) = 0.4571... 110 111 // Then, refine the reciprocal estimate using a quadratically converging 112 // Newton-Raphson iteration: 113 // x_{n+1} = x_n * (2 - x_n * b) 114 // 115 // Let b be the original divisor considered "in infinite precision" and 116 // obtained from IEEE754 representation of function argument (with the 117 // implicit bit set). Corresponds to rep_t-sized b_UQ1 represented in 118 // UQ1.(W-1). 119 // 120 // Let b_hw be an infinitely precise number obtained from the highest (HW-1) 121 // bits of divisor significand (with the implicit bit set). Corresponds to 122 // half_rep_t-sized b_UQ1_hw represented in UQ1.(HW-1) that is a **truncated** 123 // version of b_UQ1. 124 // 125 // Let e_n := x_n - 1/b_hw 126 // E_n := x_n - 1/b 127 // abs(E_n) <= abs(e_n) + (1/b_hw - 1/b) 128 // = abs(e_n) + (b - b_hw) / (b*b_hw) 129 // <= abs(e_n) + 2 * 2^-HW 130 131 // rep_t-sized iterations may be slower than the corresponding half-width 132 // variant depending on the handware and whether single/double/quad precision 133 // is selected. 134 // NB: Using half-width iterations increases computation errors due to 135 // rounding, so error estimations have to be computed taking the selected 136 // mode into account! 137#if NUMBER_OF_HALF_ITERATIONS > 0 138 // Starting with (n-1) half-width iterations 139 const half_rep_t b_UQ1_hw = bSignificand >> (significandBits + 1 - HW); 140 141 // C is (3/4 + 1/sqrt(2)) - 1 truncated to W0 fractional bits as UQ0.HW 142 // with W0 being either 16 or 32 and W0 <= HW. 143 // That is, C is the aforementioned 3/4 + 1/sqrt(2) constant (from which 144 // b/2 is subtracted to obtain x0) wrapped to [0, 1) range. 145#if defined(SINGLE_PRECISION) 146 // Use 16-bit initial estimation in case we are using half-width iterations 147 // for float32 division. This is expected to be useful for some 16-bit 148 // targets. Not used by default as it requires performing more work during 149 // rounding and would hardly help on regular 32- or 64-bit targets. 150 const half_rep_t C_hw = HALF_REP_C(0x7504); 151#else 152 // HW is at least 32. Shifting into the highest bits if needed. 153 const half_rep_t C_hw = HALF_REP_C(0x7504F333) << (HW - 32); 154#endif 155 156 // b >= 1, thus an upper bound for 3/4 + 1/sqrt(2) - b/2 is about 0.9572, 157 // so x0 fits to UQ0.HW without wrapping. 158 half_rep_t x_UQ0_hw = C_hw - (b_UQ1_hw /* exact b_hw/2 as UQ0.HW */); 159 // An e_0 error is comprised of errors due to 160 // * x0 being an inherently imprecise first approximation of 1/b_hw 161 // * C_hw being some (irrational) number **truncated** to W0 bits 162 // Please note that e_0 is calculated against the infinitely precise 163 // reciprocal of b_hw (that is, **truncated** version of b). 164 // 165 // e_0 <= 3/4 - 1/sqrt(2) + 2^-W0 166 167 // By construction, 1 <= b < 2 168 // f(x) = x * (2 - b*x) = 2*x - b*x^2 169 // f'(x) = 2 * (1 - b*x) 170 // 171 // On the [0, 1] interval, f(0) = 0, 172 // then it increses until f(1/b) = 1 / b, maximum on (0, 1), 173 // then it decreses to f(1) = 2 - b 174 // 175 // Let g(x) = x - f(x) = b*x^2 - x. 176 // On (0, 1/b), g(x) < 0 <=> f(x) > x 177 // On (1/b, 1], g(x) > 0 <=> f(x) < x 178 // 179 // For half-width iterations, b_hw is used instead of b. 180 REPEAT_N_TIMES(NUMBER_OF_HALF_ITERATIONS, { 181 // corr_UQ1_hw can be **larger** than 2 - b_hw*x by at most 1*Ulp 182 // of corr_UQ1_hw. 183 // "0.0 - (...)" is equivalent to "2.0 - (...)" in UQ1.(HW-1). 184 // On the other hand, corr_UQ1_hw should not overflow from 2.0 to 0.0 provided 185 // no overflow occurred earlier: ((rep_t)x_UQ0_hw * b_UQ1_hw >> HW) is 186 // expected to be strictly positive because b_UQ1_hw has its highest bit set 187 // and x_UQ0_hw should be rather large (it converges to 1/2 < 1/b_hw <= 1). 188 half_rep_t corr_UQ1_hw = 0 - ((rep_t)x_UQ0_hw * b_UQ1_hw >> HW); 189 190 // Now, we should multiply UQ0.HW and UQ1.(HW-1) numbers, naturally 191 // obtaining an UQ1.(HW-1) number and proving its highest bit could be 192 // considered to be 0 to be able to represent it in UQ0.HW. 193 // From the above analysis of f(x), if corr_UQ1_hw would be represented 194 // without any intermediate loss of precision (that is, in twice_rep_t) 195 // x_UQ0_hw could be at most [1.]000... if b_hw is exactly 1.0 and strictly 196 // less otherwise. On the other hand, to obtain [1.]000..., one have to pass 197 // 1/b_hw == 1.0 to f(x), so this cannot occur at all without overflow (due 198 // to 1.0 being not representable as UQ0.HW). 199 // The fact corr_UQ1_hw was virtually round up (due to result of 200 // multiplication being **first** truncated, then negated - to improve 201 // error estimations) can increase x_UQ0_hw by up to 2*Ulp of x_UQ0_hw. 202 x_UQ0_hw = (rep_t)x_UQ0_hw * corr_UQ1_hw >> (HW - 1); 203 // Now, either no overflow occurred or x_UQ0_hw is 0 or 1 in its half_rep_t 204 // representation. In the latter case, x_UQ0_hw will be either 0 or 1 after 205 // any number of iterations, so just subtract 2 from the reciprocal 206 // approximation after last iteration. 207 208 // In infinite precision, with 0 <= eps1, eps2 <= U = 2^-HW: 209 // corr_UQ1_hw = 2 - (1/b_hw + e_n) * b_hw + 2*eps1 210 // = 1 - e_n * b_hw + 2*eps1 211 // x_UQ0_hw = (1/b_hw + e_n) * (1 - e_n*b_hw + 2*eps1) - eps2 212 // = 1/b_hw - e_n + 2*eps1/b_hw + e_n - e_n^2*b_hw + 2*e_n*eps1 - eps2 213 // = 1/b_hw + 2*eps1/b_hw - e_n^2*b_hw + 2*e_n*eps1 - eps2 214 // e_{n+1} = -e_n^2*b_hw + 2*eps1/b_hw + 2*e_n*eps1 - eps2 215 // = 2*e_n*eps1 - (e_n^2*b_hw + eps2) + 2*eps1/b_hw 216 // \------ >0 -------/ \-- >0 ---/ 217 // abs(e_{n+1}) <= 2*abs(e_n)*U + max(2*e_n^2 + U, 2 * U) 218 }) 219 // For initial half-width iterations, U = 2^-HW 220 // Let abs(e_n) <= u_n * U, 221 // then abs(e_{n+1}) <= 2 * u_n * U^2 + max(2 * u_n^2 * U^2 + U, 2 * U) 222 // u_{n+1} <= 2 * u_n * U + max(2 * u_n^2 * U + 1, 2) 223 224 // Account for possible overflow (see above). For an overflow to occur for the 225 // first time, for "ideal" corr_UQ1_hw (that is, without intermediate 226 // truncation), the result of x_UQ0_hw * corr_UQ1_hw should be either maximum 227 // value representable in UQ0.HW or less by 1. This means that 1/b_hw have to 228 // be not below that value (see g(x) above), so it is safe to decrement just 229 // once after the final iteration. On the other hand, an effective value of 230 // divisor changes after this point (from b_hw to b), so adjust here. 231 x_UQ0_hw -= 1U; 232 rep_t x_UQ0 = (rep_t)x_UQ0_hw << HW; 233 x_UQ0 -= 1U; 234 235#else 236 // C is (3/4 + 1/sqrt(2)) - 1 truncated to 32 fractional bits as UQ0.n 237 const rep_t C = REP_C(0x7504F333) << (typeWidth - 32); 238 rep_t x_UQ0 = C - b_UQ1; 239 // E_0 <= 3/4 - 1/sqrt(2) + 2 * 2^-32 240#endif 241 242 // Error estimations for full-precision iterations are calculated just 243 // as above, but with U := 2^-W and taking extra decrementing into account. 244 // We need at least one such iteration. 245 246#ifdef USE_NATIVE_FULL_ITERATIONS 247 REPEAT_N_TIMES(NUMBER_OF_FULL_ITERATIONS, { 248 rep_t corr_UQ1 = 0 - ((twice_rep_t)x_UQ0 * b_UQ1 >> typeWidth); 249 x_UQ0 = (twice_rep_t)x_UQ0 * corr_UQ1 >> (typeWidth - 1); 250 }) 251#else 252#if NUMBER_OF_FULL_ITERATIONS != 1 253#error Only a single emulated full iteration is supported 254#endif 255#if !(NUMBER_OF_HALF_ITERATIONS > 0) 256 // Cannot normally reach here: only one full-width iteration is requested and 257 // the total number of iterations should be at least 3 even for float32. 258#error Check NUMBER_OF_HALF_ITERATIONS, NUMBER_OF_FULL_ITERATIONS and USE_NATIVE_FULL_ITERATIONS. 259#endif 260 // Simulating operations on a twice_rep_t to perform a single final full-width 261 // iteration. Using ad-hoc multiplication implementations to take advantage 262 // of particular structure of operands. 263 rep_t blo = b_UQ1 & loMask; 264 // x_UQ0 = x_UQ0_hw * 2^HW - 1 265 // x_UQ0 * b_UQ1 = (x_UQ0_hw * 2^HW) * (b_UQ1_hw * 2^HW + blo) - b_UQ1 266 // 267 // <--- higher half ---><--- lower half ---> 268 // [x_UQ0_hw * b_UQ1_hw] 269 // + [ x_UQ0_hw * blo ] 270 // - [ b_UQ1 ] 271 // = [ result ][.... discarded ...] 272 rep_t corr_UQ1 = 0U - ( (rep_t)x_UQ0_hw * b_UQ1_hw 273 + ((rep_t)x_UQ0_hw * blo >> HW) 274 - REP_C(1)); // account for *possible* carry 275 rep_t lo_corr = corr_UQ1 & loMask; 276 rep_t hi_corr = corr_UQ1 >> HW; 277 // x_UQ0 * corr_UQ1 = (x_UQ0_hw * 2^HW) * (hi_corr * 2^HW + lo_corr) - corr_UQ1 278 x_UQ0 = ((rep_t)x_UQ0_hw * hi_corr << 1) 279 + ((rep_t)x_UQ0_hw * lo_corr >> (HW - 1)) 280 - REP_C(2); // 1 to account for the highest bit of corr_UQ1 can be 1 281 // 1 to account for possible carry 282 // Just like the case of half-width iterations but with possibility 283 // of overflowing by one extra Ulp of x_UQ0. 284 x_UQ0 -= 1U; 285 // ... and then traditional fixup by 2 should work 286 287 // On error estimation: 288 // abs(E_{N-1}) <= (u_{N-1} + 2 /* due to conversion e_n -> E_n */) * 2^-HW 289 // + (2^-HW + 2^-W)) 290 // abs(E_{N-1}) <= (u_{N-1} + 3.01) * 2^-HW 291 292 // Then like for the half-width iterations: 293 // With 0 <= eps1, eps2 < 2^-W 294 // E_N = 4 * E_{N-1} * eps1 - (E_{N-1}^2 * b + 4 * eps2) + 4 * eps1 / b 295 // abs(E_N) <= 2^-W * [ 4 * abs(E_{N-1}) + max(2 * abs(E_{N-1})^2 * 2^W + 4, 8)) ] 296 // abs(E_N) <= 2^-W * [ 4 * (u_{N-1} + 3.01) * 2^-HW + max(4 + 2 * (u_{N-1} + 3.01)^2, 8) ] 297#endif 298 299 // Finally, account for possible overflow, as explained above. 300 x_UQ0 -= 2U; 301 302 // u_n for different precisions (with N-1 half-width iterations): 303 // W0 is the precision of C 304 // u_0 = (3/4 - 1/sqrt(2) + 2^-W0) * 2^HW 305 306 // Estimated with bc: 307 // define half1(un) { return 2.0 * (un + un^2) / 2.0^hw + 1.0; } 308 // define half2(un) { return 2.0 * un / 2.0^hw + 2.0; } 309 // define full1(un) { return 4.0 * (un + 3.01) / 2.0^hw + 2.0 * (un + 3.01)^2 + 4.0; } 310 // define full2(un) { return 4.0 * (un + 3.01) / 2.0^hw + 8.0; } 311 312 // | f32 (0 + 3) | f32 (2 + 1) | f64 (3 + 1) | f128 (4 + 1) 313 // u_0 | < 184224974 | < 2812.1 | < 184224974 | < 791240234244348797 314 // u_1 | < 15804007 | < 242.7 | < 15804007 | < 67877681371350440 315 // u_2 | < 116308 | < 2.81 | < 116308 | < 499533100252317 316 // u_3 | < 7.31 | | < 7.31 | < 27054456580 317 // u_4 | | | | < 80.4 318 // Final (U_N) | same as u_3 | < 72 | < 218 | < 13920 319 320 // Add 2 to U_N due to final decrement. 321 322#if defined(SINGLE_PRECISION) && NUMBER_OF_HALF_ITERATIONS == 2 && NUMBER_OF_FULL_ITERATIONS == 1 323#define RECIPROCAL_PRECISION REP_C(74) 324#elif defined(SINGLE_PRECISION) && NUMBER_OF_HALF_ITERATIONS == 0 && NUMBER_OF_FULL_ITERATIONS == 3 325#define RECIPROCAL_PRECISION REP_C(10) 326#elif defined(DOUBLE_PRECISION) && NUMBER_OF_HALF_ITERATIONS == 3 && NUMBER_OF_FULL_ITERATIONS == 1 327#define RECIPROCAL_PRECISION REP_C(220) 328#elif defined(QUAD_PRECISION) && NUMBER_OF_HALF_ITERATIONS == 4 && NUMBER_OF_FULL_ITERATIONS == 1 329#define RECIPROCAL_PRECISION REP_C(13922) 330#else 331#error Invalid number of iterations 332#endif 333 334 // Suppose 1/b - P * 2^-W < x < 1/b + P * 2^-W 335 x_UQ0 -= RECIPROCAL_PRECISION; 336 // Now 1/b - (2*P) * 2^-W < x < 1/b 337 338 rep_t quotient_UQ1, dummy; 339 wideMultiply(x_UQ0, aSignificand << 1, "ient_UQ1, &dummy); 340 // Now, a/b - 4*P * 2^-W < q < a/b for q=<quotient_UQ1:dummy> in UQ1.(SB+1+W). 341 342 // quotient_UQ1 is in [0.5, 2.0) as UQ1.(SB+1), 343 // adjust it to be in [1.0, 2.0) as UQ1.SB. 344 rep_t residualLo; 345 if (quotient_UQ1 < (implicitBit << 1)) { 346 if (quotient_UQ1 < implicitBit) { 347 // In a rare case where quotient is < 0.5, we can adjust the quotient and 348 // the written exponent, and then treat them the same way as in [0.5, 1.0) 349 quotient_UQ1 <<= 1; 350 writtenExponent -= 1; 351 } 352 // Highest bit is 0, so just reinterpret quotient_UQ1 as UQ1.SB, 353 // effectively doubling its value as well as its error estimation. 354 residualLo = (aSignificand << (significandBits + 1)) - quotient_UQ1 * bSignificand; 355 writtenExponent -= 1; 356 aSignificand <<= 1; 357 } else { 358 // Highest bit is 1 (the UQ1.(SB+1) value is in [1, 2)), convert it 359 // to UQ1.SB by right shifting by 1. Least significant bit is omitted. 360 quotient_UQ1 >>= 1; 361 residualLo = (aSignificand << significandBits) - quotient_UQ1 * bSignificand; 362 } 363 // NB: residualLo is calculated above for the normal result case. 364 // It is re-computed on denormal path that is expected to be not so 365 // performance-sensitive. 366 367 // Now, q cannot be greater than a/b and can differ by at most 8*P * 2^-W + 2^-SB 368 // Each NextAfter() increments the floating point value by at least 2^-SB 369 // (more, if exponent was incremented). 370 // Different cases (<---> is of 2^-SB length, * = a/b that is shown as a midpoint): 371 // q 372 // | | * | | | | | 373 // <---> 2^t 374 // | | | | | * | | 375 // q 376 // To require at most one NextAfter(), an error should be less than 1.5 * 2^-SB. 377 // (8*P) * 2^-W + 2^-SB < 1.5 * 2^-SB 378 // (8*P) * 2^-W < 0.5 * 2^-SB 379 // P < 2^(W-4-SB) 380 // Generally, for at most R NextAfter() to be enough, 381 // P < (2*R - 1) * 2^(W-4-SB) 382 // For f32 (0+3): 10 < 32 (OK) 383 // For f32 (2+1): 32 < 74 < 32 * 3, so two NextAfter() are required 384 // For f64: 220 < 256 (OK) 385 // For f128: 4096 * 3 < 13922 < 4096 * 5 (three NextAfter() are required) 386 387 // If we have overflowed the exponent, return infinity 388 if (writtenExponent >= maxExponent) 389 return fromRep(infRep | quotientSign); 390 391 // Now, quotient_UQ1_SB <= the correctly-rounded result 392 // and may need taking NextAfter() up to 3 times (see error estimates above) 393 // r = a - b * q 394 rep_t absResult; 395 if (writtenExponent > 0) { 396 // Clear the implicit bit 397 absResult = quotient_UQ1 & significandMask; 398 // Insert the exponent 399 absResult |= (rep_t)writtenExponent << significandBits; 400 residualLo <<= 1; 401 } else { 402 // Prevent shift amount from being negative 403 if (significandBits + writtenExponent < 0) 404 return fromRep(quotientSign); 405 406 absResult = quotient_UQ1 >> (-writtenExponent + 1); 407 408 // multiplied by two to prevent shift amount to be negative 409 residualLo = (aSignificand << (significandBits + writtenExponent)) - (absResult * bSignificand << 1); 410 } 411 412 // Round 413 residualLo += absResult & 1; // tie to even 414 // The above line conditionally turns the below LT comparison into LTE 415 absResult += residualLo > bSignificand; 416#if defined(QUAD_PRECISION) || (defined(SINGLE_PRECISION) && NUMBER_OF_HALF_ITERATIONS > 0) 417 // Do not round Infinity to NaN 418 absResult += absResult < infRep && residualLo > (2 + 1) * bSignificand; 419#endif 420#if defined(QUAD_PRECISION) 421 absResult += absResult < infRep && residualLo > (4 + 1) * bSignificand; 422#endif 423 return fromRep(absResult | quotientSign); 424} 425