//===-- fp_div_impl.inc - Floating point division -----------------*- C -*-===// // // 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 // //===----------------------------------------------------------------------===// // // This file implements soft-float division with the IEEE-754 default // rounding (to nearest, ties to even). // //===----------------------------------------------------------------------===// #include "fp_lib.h" // The __divXf3__ function implements Newton-Raphson floating point division. // It uses 3 iterations for float32, 4 for float64 and 5 for float128, // respectively. Due to number of significant bits being roughly doubled // every iteration, the two modes are supported: N full-width iterations (as // it is done for float32 by default) and (N-1) half-width iteration plus one // final full-width iteration. It is expected that half-width integer // operations (w.r.t rep_t size) can be performed faster for some hardware but // they require error estimations to be computed separately due to larger // computational errors caused by truncating intermediate results. // Half the bit-size of rep_t #define HW (typeWidth / 2) // rep_t-sized bitmask with lower half of bits set to ones #define loMask (REP_C(-1) >> HW) #if NUMBER_OF_FULL_ITERATIONS < 1 #error At least one full iteration is required #endif static __inline fp_t __divXf3__(fp_t a, fp_t b) { const unsigned int aExponent = toRep(a) >> significandBits & maxExponent; const unsigned int bExponent = toRep(b) >> significandBits & maxExponent; const rep_t quotientSign = (toRep(a) ^ toRep(b)) & signBit; rep_t aSignificand = toRep(a) & significandMask; rep_t bSignificand = toRep(b) & significandMask; int scale = 0; // Detect if a or b is zero, denormal, infinity, or NaN. if (aExponent - 1U >= maxExponent - 1U || bExponent - 1U >= maxExponent - 1U) { const rep_t aAbs = toRep(a) & absMask; const rep_t bAbs = toRep(b) & absMask; // NaN / anything = qNaN if (aAbs > infRep) return fromRep(toRep(a) | quietBit); // anything / NaN = qNaN if (bAbs > infRep) return fromRep(toRep(b) | quietBit); if (aAbs == infRep) { // infinity / infinity = NaN if (bAbs == infRep) return fromRep(qnanRep); // infinity / anything else = +/- infinity else return fromRep(aAbs | quotientSign); } // anything else / infinity = +/- 0 if (bAbs == infRep) return fromRep(quotientSign); if (!aAbs) { // zero / zero = NaN if (!bAbs) return fromRep(qnanRep); // zero / anything else = +/- zero else return fromRep(quotientSign); } // anything else / zero = +/- infinity if (!bAbs) return fromRep(infRep | quotientSign); // One or both of a or b is denormal. The other (if applicable) is a // normal number. Renormalize one or both of a and b, and set scale to // include the necessary exponent adjustment. if (aAbs < implicitBit) scale += normalize(&aSignificand); if (bAbs < implicitBit) scale -= normalize(&bSignificand); } // Set the implicit significand bit. If we fell through from the // denormal path it was already set by normalize( ), but setting it twice // won't hurt anything. aSignificand |= implicitBit; bSignificand |= implicitBit; int writtenExponent = (aExponent - bExponent + scale) + exponentBias; const rep_t b_UQ1 = bSignificand << (typeWidth - significandBits - 1); // Align the significand of b as a UQ1.(n-1) fixed-point number in the range // [1.0, 2.0) and get a UQ0.n approximate reciprocal using a small minimax // polynomial approximation: x0 = 3/4 + 1/sqrt(2) - b/2. // The max error for this approximation is achieved at endpoints, so // abs(x0(b) - 1/b) <= abs(x0(1) - 1/1) = 3/4 - 1/sqrt(2) = 0.04289..., // which is about 4.5 bits. // The initial approximation is between x0(1.0) = 0.9571... and x0(2.0) = 0.4571... // Then, refine the reciprocal estimate using a quadratically converging // Newton-Raphson iteration: // x_{n+1} = x_n * (2 - x_n * b) // // Let b be the original divisor considered "in infinite precision" and // obtained from IEEE754 representation of function argument (with the // implicit bit set). Corresponds to rep_t-sized b_UQ1 represented in // UQ1.(W-1). // // Let b_hw be an infinitely precise number obtained from the highest (HW-1) // bits of divisor significand (with the implicit bit set). Corresponds to // half_rep_t-sized b_UQ1_hw represented in UQ1.(HW-1) that is a **truncated** // version of b_UQ1. // // Let e_n := x_n - 1/b_hw // E_n := x_n - 1/b // abs(E_n) <= abs(e_n) + (1/b_hw - 1/b) // = abs(e_n) + (b - b_hw) / (b*b_hw) // <= abs(e_n) + 2 * 2^-HW // rep_t-sized iterations may be slower than the corresponding half-width // variant depending on the handware and whether single/double/quad precision // is selected. // NB: Using half-width iterations increases computation errors due to // rounding, so error estimations have to be computed taking the selected // mode into account! #if NUMBER_OF_HALF_ITERATIONS > 0 // Starting with (n-1) half-width iterations const half_rep_t b_UQ1_hw = bSignificand >> (significandBits + 1 - HW); // C is (3/4 + 1/sqrt(2)) - 1 truncated to W0 fractional bits as UQ0.HW // with W0 being either 16 or 32 and W0 <= HW. // That is, C is the aforementioned 3/4 + 1/sqrt(2) constant (from which // b/2 is subtracted to obtain x0) wrapped to [0, 1) range. #if defined(SINGLE_PRECISION) // Use 16-bit initial estimation in case we are using half-width iterations // for float32 division. This is expected to be useful for some 16-bit // targets. Not used by default as it requires performing more work during // rounding and would hardly help on regular 32- or 64-bit targets. const half_rep_t C_hw = HALF_REP_C(0x7504); #else // HW is at least 32. Shifting into the highest bits if needed. const half_rep_t C_hw = HALF_REP_C(0x7504F333) << (HW - 32); #endif // b >= 1, thus an upper bound for 3/4 + 1/sqrt(2) - b/2 is about 0.9572, // so x0 fits to UQ0.HW without wrapping. half_rep_t x_UQ0_hw = C_hw - (b_UQ1_hw /* exact b_hw/2 as UQ0.HW */); // An e_0 error is comprised of errors due to // * x0 being an inherently imprecise first approximation of 1/b_hw // * C_hw being some (irrational) number **truncated** to W0 bits // Please note that e_0 is calculated against the infinitely precise // reciprocal of b_hw (that is, **truncated** version of b). // // e_0 <= 3/4 - 1/sqrt(2) + 2^-W0 // By construction, 1 <= b < 2 // f(x) = x * (2 - b*x) = 2*x - b*x^2 // f'(x) = 2 * (1 - b*x) // // On the [0, 1] interval, f(0) = 0, // then it increses until f(1/b) = 1 / b, maximum on (0, 1), // then it decreses to f(1) = 2 - b // // Let g(x) = x - f(x) = b*x^2 - x. // On (0, 1/b), g(x) < 0 <=> f(x) > x // On (1/b, 1], g(x) > 0 <=> f(x) < x // // For half-width iterations, b_hw is used instead of b. REPEAT_N_TIMES(NUMBER_OF_HALF_ITERATIONS, { // corr_UQ1_hw can be **larger** than 2 - b_hw*x by at most 1*Ulp // of corr_UQ1_hw. // "0.0 - (...)" is equivalent to "2.0 - (...)" in UQ1.(HW-1). // On the other hand, corr_UQ1_hw should not overflow from 2.0 to 0.0 provided // no overflow occurred earlier: ((rep_t)x_UQ0_hw * b_UQ1_hw >> HW) is // expected to be strictly positive because b_UQ1_hw has its highest bit set // and x_UQ0_hw should be rather large (it converges to 1/2 < 1/b_hw <= 1). half_rep_t corr_UQ1_hw = 0 - ((rep_t)x_UQ0_hw * b_UQ1_hw >> HW); // Now, we should multiply UQ0.HW and UQ1.(HW-1) numbers, naturally // obtaining an UQ1.(HW-1) number and proving its highest bit could be // considered to be 0 to be able to represent it in UQ0.HW. // From the above analysis of f(x), if corr_UQ1_hw would be represented // without any intermediate loss of precision (that is, in twice_rep_t) // x_UQ0_hw could be at most [1.]000... if b_hw is exactly 1.0 and strictly // less otherwise. On the other hand, to obtain [1.]000..., one have to pass // 1/b_hw == 1.0 to f(x), so this cannot occur at all without overflow (due // to 1.0 being not representable as UQ0.HW). // The fact corr_UQ1_hw was virtually round up (due to result of // multiplication being **first** truncated, then negated - to improve // error estimations) can increase x_UQ0_hw by up to 2*Ulp of x_UQ0_hw. x_UQ0_hw = (rep_t)x_UQ0_hw * corr_UQ1_hw >> (HW - 1); // Now, either no overflow occurred or x_UQ0_hw is 0 or 1 in its half_rep_t // representation. In the latter case, x_UQ0_hw will be either 0 or 1 after // any number of iterations, so just subtract 2 from the reciprocal // approximation after last iteration. // In infinite precision, with 0 <= eps1, eps2 <= U = 2^-HW: // corr_UQ1_hw = 2 - (1/b_hw + e_n) * b_hw + 2*eps1 // = 1 - e_n * b_hw + 2*eps1 // x_UQ0_hw = (1/b_hw + e_n) * (1 - e_n*b_hw + 2*eps1) - eps2 // = 1/b_hw - e_n + 2*eps1/b_hw + e_n - e_n^2*b_hw + 2*e_n*eps1 - eps2 // = 1/b_hw + 2*eps1/b_hw - e_n^2*b_hw + 2*e_n*eps1 - eps2 // e_{n+1} = -e_n^2*b_hw + 2*eps1/b_hw + 2*e_n*eps1 - eps2 // = 2*e_n*eps1 - (e_n^2*b_hw + eps2) + 2*eps1/b_hw // \------ >0 -------/ \-- >0 ---/ // abs(e_{n+1}) <= 2*abs(e_n)*U + max(2*e_n^2 + U, 2 * U) }) // For initial half-width iterations, U = 2^-HW // Let abs(e_n) <= u_n * U, // then abs(e_{n+1}) <= 2 * u_n * U^2 + max(2 * u_n^2 * U^2 + U, 2 * U) // u_{n+1} <= 2 * u_n * U + max(2 * u_n^2 * U + 1, 2) // Account for possible overflow (see above). For an overflow to occur for the // first time, for "ideal" corr_UQ1_hw (that is, without intermediate // truncation), the result of x_UQ0_hw * corr_UQ1_hw should be either maximum // value representable in UQ0.HW or less by 1. This means that 1/b_hw have to // be not below that value (see g(x) above), so it is safe to decrement just // once after the final iteration. On the other hand, an effective value of // divisor changes after this point (from b_hw to b), so adjust here. x_UQ0_hw -= 1U; rep_t x_UQ0 = (rep_t)x_UQ0_hw << HW; x_UQ0 -= 1U; #else // C is (3/4 + 1/sqrt(2)) - 1 truncated to 32 fractional bits as UQ0.n const rep_t C = REP_C(0x7504F333) << (typeWidth - 32); rep_t x_UQ0 = C - b_UQ1; // E_0 <= 3/4 - 1/sqrt(2) + 2 * 2^-32 #endif // Error estimations for full-precision iterations are calculated just // as above, but with U := 2^-W and taking extra decrementing into account. // We need at least one such iteration. #ifdef USE_NATIVE_FULL_ITERATIONS REPEAT_N_TIMES(NUMBER_OF_FULL_ITERATIONS, { rep_t corr_UQ1 = 0 - ((twice_rep_t)x_UQ0 * b_UQ1 >> typeWidth); x_UQ0 = (twice_rep_t)x_UQ0 * corr_UQ1 >> (typeWidth - 1); }) #else #if NUMBER_OF_FULL_ITERATIONS != 1 #error Only a single emulated full iteration is supported #endif #if !(NUMBER_OF_HALF_ITERATIONS > 0) // Cannot normally reach here: only one full-width iteration is requested and // the total number of iterations should be at least 3 even for float32. #error Check NUMBER_OF_HALF_ITERATIONS, NUMBER_OF_FULL_ITERATIONS and USE_NATIVE_FULL_ITERATIONS. #endif // Simulating operations on a twice_rep_t to perform a single final full-width // iteration. Using ad-hoc multiplication implementations to take advantage // of particular structure of operands. rep_t blo = b_UQ1 & loMask; // x_UQ0 = x_UQ0_hw * 2^HW - 1 // x_UQ0 * b_UQ1 = (x_UQ0_hw * 2^HW) * (b_UQ1_hw * 2^HW + blo) - b_UQ1 // // <--- higher half ---><--- lower half ---> // [x_UQ0_hw * b_UQ1_hw] // + [ x_UQ0_hw * blo ] // - [ b_UQ1 ] // = [ result ][.... discarded ...] rep_t corr_UQ1 = 0U - ( (rep_t)x_UQ0_hw * b_UQ1_hw + ((rep_t)x_UQ0_hw * blo >> HW) - REP_C(1)); // account for *possible* carry rep_t lo_corr = corr_UQ1 & loMask; rep_t hi_corr = corr_UQ1 >> HW; // x_UQ0 * corr_UQ1 = (x_UQ0_hw * 2^HW) * (hi_corr * 2^HW + lo_corr) - corr_UQ1 x_UQ0 = ((rep_t)x_UQ0_hw * hi_corr << 1) + ((rep_t)x_UQ0_hw * lo_corr >> (HW - 1)) - REP_C(2); // 1 to account for the highest bit of corr_UQ1 can be 1 // 1 to account for possible carry // Just like the case of half-width iterations but with possibility // of overflowing by one extra Ulp of x_UQ0. x_UQ0 -= 1U; // ... and then traditional fixup by 2 should work // On error estimation: // abs(E_{N-1}) <= (u_{N-1} + 2 /* due to conversion e_n -> E_n */) * 2^-HW // + (2^-HW + 2^-W)) // abs(E_{N-1}) <= (u_{N-1} + 3.01) * 2^-HW // Then like for the half-width iterations: // With 0 <= eps1, eps2 < 2^-W // E_N = 4 * E_{N-1} * eps1 - (E_{N-1}^2 * b + 4 * eps2) + 4 * eps1 / b // abs(E_N) <= 2^-W * [ 4 * abs(E_{N-1}) + max(2 * abs(E_{N-1})^2 * 2^W + 4, 8)) ] // abs(E_N) <= 2^-W * [ 4 * (u_{N-1} + 3.01) * 2^-HW + max(4 + 2 * (u_{N-1} + 3.01)^2, 8) ] #endif // Finally, account for possible overflow, as explained above. x_UQ0 -= 2U; // u_n for different precisions (with N-1 half-width iterations): // W0 is the precision of C // u_0 = (3/4 - 1/sqrt(2) + 2^-W0) * 2^HW // Estimated with bc: // define half1(un) { return 2.0 * (un + un^2) / 2.0^hw + 1.0; } // define half2(un) { return 2.0 * un / 2.0^hw + 2.0; } // define full1(un) { return 4.0 * (un + 3.01) / 2.0^hw + 2.0 * (un + 3.01)^2 + 4.0; } // define full2(un) { return 4.0 * (un + 3.01) / 2.0^hw + 8.0; } // | f32 (0 + 3) | f32 (2 + 1) | f64 (3 + 1) | f128 (4 + 1) // u_0 | < 184224974 | < 2812.1 | < 184224974 | < 791240234244348797 // u_1 | < 15804007 | < 242.7 | < 15804007 | < 67877681371350440 // u_2 | < 116308 | < 2.81 | < 116308 | < 499533100252317 // u_3 | < 7.31 | | < 7.31 | < 27054456580 // u_4 | | | | < 80.4 // Final (U_N) | same as u_3 | < 72 | < 218 | < 13920 // Add 2 to U_N due to final decrement. #if defined(SINGLE_PRECISION) && NUMBER_OF_HALF_ITERATIONS == 2 && NUMBER_OF_FULL_ITERATIONS == 1 #define RECIPROCAL_PRECISION REP_C(74) #elif defined(SINGLE_PRECISION) && NUMBER_OF_HALF_ITERATIONS == 0 && NUMBER_OF_FULL_ITERATIONS == 3 #define RECIPROCAL_PRECISION REP_C(10) #elif defined(DOUBLE_PRECISION) && NUMBER_OF_HALF_ITERATIONS == 3 && NUMBER_OF_FULL_ITERATIONS == 1 #define RECIPROCAL_PRECISION REP_C(220) #elif defined(QUAD_PRECISION) && NUMBER_OF_HALF_ITERATIONS == 4 && NUMBER_OF_FULL_ITERATIONS == 1 #define RECIPROCAL_PRECISION REP_C(13922) #else #error Invalid number of iterations #endif // Suppose 1/b - P * 2^-W < x < 1/b + P * 2^-W x_UQ0 -= RECIPROCAL_PRECISION; // Now 1/b - (2*P) * 2^-W < x < 1/b // FIXME Is x_UQ0 still >= 0.5? rep_t quotient_UQ1, dummy; wideMultiply(x_UQ0, aSignificand << 1, "ient_UQ1, &dummy); // Now, a/b - 4*P * 2^-W < q < a/b for q= in UQ1.(SB+1+W). // quotient_UQ1 is in [0.5, 2.0) as UQ1.(SB+1), // adjust it to be in [1.0, 2.0) as UQ1.SB. rep_t residualLo; if (quotient_UQ1 < (implicitBit << 1)) { // Highest bit is 0, so just reinterpret quotient_UQ1 as UQ1.SB, // effectively doubling its value as well as its error estimation. residualLo = (aSignificand << (significandBits + 1)) - quotient_UQ1 * bSignificand; writtenExponent -= 1; aSignificand <<= 1; } else { // Highest bit is 1 (the UQ1.(SB+1) value is in [1, 2)), convert it // to UQ1.SB by right shifting by 1. Least significant bit is omitted. quotient_UQ1 >>= 1; residualLo = (aSignificand << significandBits) - quotient_UQ1 * bSignificand; } // NB: residualLo is calculated above for the normal result case. // It is re-computed on denormal path that is expected to be not so // performance-sensitive. // Now, q cannot be greater than a/b and can differ by at most 8*P * 2^-W + 2^-SB // Each NextAfter() increments the floating point value by at least 2^-SB // (more, if exponent was incremented). // Different cases (<---> is of 2^-SB length, * = a/b that is shown as a midpoint): // q // | | * | | | | | // <---> 2^t // | | | | | * | | // q // To require at most one NextAfter(), an error should be less than 1.5 * 2^-SB. // (8*P) * 2^-W + 2^-SB < 1.5 * 2^-SB // (8*P) * 2^-W < 0.5 * 2^-SB // P < 2^(W-4-SB) // Generally, for at most R NextAfter() to be enough, // P < (2*R - 1) * 2^(W-4-SB) // For f32 (0+3): 10 < 32 (OK) // For f32 (2+1): 32 < 74 < 32 * 3, so two NextAfter() are required // For f64: 220 < 256 (OK) // For f128: 4096 * 3 < 13922 < 4096 * 5 (three NextAfter() are required) // If we have overflowed the exponent, return infinity if (writtenExponent >= maxExponent) return fromRep(infRep | quotientSign); // Now, quotient_UQ1_SB <= the correctly-rounded result // and may need taking NextAfter() up to 3 times (see error estimates above) // r = a - b * q rep_t absResult; if (writtenExponent > 0) { // Clear the implicit bit absResult = quotient_UQ1 & significandMask; // Insert the exponent absResult |= (rep_t)writtenExponent << significandBits; residualLo <<= 1; } else { // Prevent shift amount from being negative if (significandBits + writtenExponent < 0) return fromRep(quotientSign); absResult = quotient_UQ1 >> (-writtenExponent + 1); // multiplied by two to prevent shift amount to be negative residualLo = (aSignificand << (significandBits + writtenExponent)) - (absResult * bSignificand << 1); } // Round residualLo += absResult & 1; // tie to even // The above line conditionally turns the below LT comparison into LTE absResult += residualLo > bSignificand; #if defined(QUAD_PRECISION) || (defined(SINGLE_PRECISION) && NUMBER_OF_HALF_ITERATIONS > 0) // Do not round Infinity to NaN absResult += absResult < infRep && residualLo > (2 + 1) * bSignificand; #endif #if defined(QUAD_PRECISION) absResult += absResult < infRep && residualLo > (4 + 1) * bSignificand; #endif return fromRep(absResult | quotientSign); }