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Diffstat (limited to 'lib/builtins/divsf3.c')
| -rw-r--r-- | lib/builtins/divsf3.c | 309 |
1 files changed, 163 insertions, 146 deletions
diff --git a/lib/builtins/divsf3.c b/lib/builtins/divsf3.c index a74917fd1de5..593f93b45ac2 100644 --- a/lib/builtins/divsf3.c +++ b/lib/builtins/divsf3.c @@ -1,9 +1,8 @@ //===-- lib/divsf3.c - Single-precision division ------------------*- C -*-===// // -// The LLVM Compiler Infrastructure -// -// This file is dual licensed under the MIT and the University of Illinois Open -// Source Licenses. See LICENSE.TXT for details. +// 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 // //===----------------------------------------------------------------------===// // @@ -19,159 +18,177 @@ #define SINGLE_PRECISION #include "fp_lib.h" -COMPILER_RT_ABI fp_t -__divsf3(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); - } - - // Or in 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 quotientExponent = aExponent - bExponent + scale; - - // Align the significand of b as a Q31 fixed-point number in the range - // [1, 2.0) and get a Q32 approximate reciprocal using a small minimax - // polynomial approximation: reciprocal = 3/4 + 1/sqrt(2) - b/2. This - // is accurate to about 3.5 binary digits. - uint32_t q31b = bSignificand << 8; - uint32_t reciprocal = UINT32_C(0x7504f333) - q31b; - - // Now refine the reciprocal estimate using a Newton-Raphson iteration: - // - // x1 = x0 * (2 - x0 * b) - // - // This doubles the number of correct binary digits in the approximation - // with each iteration, so after three iterations, we have about 28 binary - // digits of accuracy. - uint32_t correction; - correction = -((uint64_t)reciprocal * q31b >> 32); - reciprocal = (uint64_t)reciprocal * correction >> 31; - correction = -((uint64_t)reciprocal * q31b >> 32); - reciprocal = (uint64_t)reciprocal * correction >> 31; - correction = -((uint64_t)reciprocal * q31b >> 32); - reciprocal = (uint64_t)reciprocal * correction >> 31; - - // Exhaustive testing shows that the error in reciprocal after three steps - // is in the interval [-0x1.f58108p-31, 0x1.d0e48cp-29], in line with our - // expectations. We bump the reciprocal by a tiny value to force the error - // to be strictly positive (in the range [0x1.4fdfp-37,0x1.287246p-29], to - // be specific). This also causes 1/1 to give a sensible approximation - // instead of zero (due to overflow). - reciprocal -= 2; - - // The numerical reciprocal is accurate to within 2^-28, lies in the - // interval [0x1.000000eep-1, 0x1.fffffffcp-1], and is strictly smaller - // than the true reciprocal of b. Multiplying a by this reciprocal thus - // gives a numerical q = a/b in Q24 with the following properties: - // - // 1. q < a/b - // 2. q is in the interval [0x1.000000eep-1, 0x1.fffffffcp0) - // 3. the error in q is at most 2^-24 + 2^-27 -- the 2^24 term comes - // from the fact that we truncate the product, and the 2^27 term - // is the error in the reciprocal of b scaled by the maximum - // possible value of a. As a consequence of this error bound, - // either q or nextafter(q) is the correctly rounded - rep_t quotient = (uint64_t)reciprocal*(aSignificand << 1) >> 32; - - // Two cases: quotient is in [0.5, 1.0) or quotient is in [1.0, 2.0). - // In either case, we are going to compute a residual of the form - // - // r = a - q*b - // - // We know from the construction of q that r satisfies: - // - // 0 <= r < ulp(q)*b - // - // if r is greater than 1/2 ulp(q)*b, then q rounds up. Otherwise, we - // already have the correct result. The exact halfway case cannot occur. - // We also take this time to right shift quotient if it falls in the [1,2) - // range and adjust the exponent accordingly. - rep_t residual; - if (quotient < (implicitBit << 1)) { - residual = (aSignificand << 24) - quotient * bSignificand; - quotientExponent--; - } else { - quotient >>= 1; - residual = (aSignificand << 23) - quotient * bSignificand; +COMPILER_RT_ABI fp_t __divsf3(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); } - const int writtenExponent = quotientExponent + exponentBias; + // anything else / infinity = +/- 0 + if (bAbs == infRep) + return fromRep(quotientSign); - if (writtenExponent >= maxExponent) { - // If we have overflowed the exponent, return infinity. - return fromRep(infRep | quotientSign); - } - - else if (writtenExponent < 1) { - // Flush denormals to zero. In the future, it would be nice to add - // code to round them correctly. + if (!aAbs) { + // zero / zero = NaN + if (!bAbs) + return fromRep(qnanRep); + // zero / anything else = +/- zero + else return fromRep(quotientSign); } - - else { - const bool round = (residual << 1) > bSignificand; - // Clear the implicit bit - rep_t absResult = quotient & significandMask; - // Insert the exponent - absResult |= (rep_t)writtenExponent << significandBits; - // Round - absResult += round; - // Insert the sign and return + // 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 quotientExponent = aExponent - bExponent + scale; + // 0x7504F333 / 2^32 + 1 = 3/4 + 1/sqrt(2) + + // Align the significand of b as a Q31 fixed-point number in the range + // [1, 2.0) and get a Q32 approximate reciprocal using a small minimax + // polynomial approximation: reciprocal = 3/4 + 1/sqrt(2) - b/2. This + // is accurate to about 3.5 binary digits. + uint32_t q31b = bSignificand << 8; + uint32_t reciprocal = UINT32_C(0x7504f333) - q31b; + + // Now refine the reciprocal estimate using a Newton-Raphson iteration: + // + // x1 = x0 * (2 - x0 * b) + // + // This doubles the number of correct binary digits in the approximation + // with each iteration. + uint32_t correction; + correction = -((uint64_t)reciprocal * q31b >> 32); + reciprocal = (uint64_t)reciprocal * correction >> 31; + correction = -((uint64_t)reciprocal * q31b >> 32); + reciprocal = (uint64_t)reciprocal * correction >> 31; + correction = -((uint64_t)reciprocal * q31b >> 32); + reciprocal = (uint64_t)reciprocal * correction >> 31; + + // Adust the final 32-bit reciprocal estimate downward to ensure that it is + // strictly smaller than the infinitely precise exact reciprocal. Because + // the computation of the Newton-Raphson step is truncating at every step, + // this adjustment is small; most of the work is already done. + reciprocal -= 2; + + // The numerical reciprocal is accurate to within 2^-28, lies in the + // interval [0x1.000000eep-1, 0x1.fffffffcp-1], and is strictly smaller + // than the true reciprocal of b. Multiplying a by this reciprocal thus + // gives a numerical q = a/b in Q24 with the following properties: + // + // 1. q < a/b + // 2. q is in the interval [0x1.000000eep-1, 0x1.fffffffcp0) + // 3. The error in q is at most 2^-24 + 2^-27 -- the 2^24 term comes + // from the fact that we truncate the product, and the 2^27 term + // is the error in the reciprocal of b scaled by the maximum + // possible value of a. As a consequence of this error bound, + // either q or nextafter(q) is the correctly rounded. + rep_t quotient = (uint64_t)reciprocal * (aSignificand << 1) >> 32; + + // Two cases: quotient is in [0.5, 1.0) or quotient is in [1.0, 2.0). + // In either case, we are going to compute a residual of the form + // + // r = a - q*b + // + // We know from the construction of q that r satisfies: + // + // 0 <= r < ulp(q)*b + // + // If r is greater than 1/2 ulp(q)*b, then q rounds up. Otherwise, we + // already have the correct result. The exact halfway case cannot occur. + // We also take this time to right shift quotient if it falls in the [1,2) + // range and adjust the exponent accordingly. + rep_t residual; + if (quotient < (implicitBit << 1)) { + residual = (aSignificand << 24) - quotient * bSignificand; + quotientExponent--; + } else { + quotient >>= 1; + residual = (aSignificand << 23) - quotient * bSignificand; + } + + const int writtenExponent = quotientExponent + exponentBias; + + if (writtenExponent >= maxExponent) { + // If we have overflowed the exponent, return infinity. + return fromRep(infRep | quotientSign); + } + + else if (writtenExponent < 1) { + if (writtenExponent == 0) { + // Check whether the rounded result is normal. + const bool round = (residual << 1) > bSignificand; + // Clear the implicit bit. + rep_t absResult = quotient & significandMask; + // Round. + absResult += round; + if (absResult & ~significandMask) { + // The rounded result is normal; return it. return fromRep(absResult | quotientSign); + } } + // Flush denormals to zero. In the future, it would be nice to add + // code to round them correctly. + return fromRep(quotientSign); + } + + else { + const bool round = (residual << 1) > bSignificand; + // Clear the implicit bit. + rep_t absResult = quotient & significandMask; + // Insert the exponent. + absResult |= (rep_t)writtenExponent << significandBits; + // Round. + absResult += round; + // Insert the sign and return. + return fromRep(absResult | quotientSign); + } } #if defined(__ARM_EABI__) #if defined(COMPILER_RT_ARMHF_TARGET) -AEABI_RTABI fp_t __aeabi_fdiv(fp_t a, fp_t b) { - return __divsf3(a, b); -} +AEABI_RTABI fp_t __aeabi_fdiv(fp_t a, fp_t b) { return __divsf3(a, b); } #else -AEABI_RTABI fp_t __aeabi_fdiv(fp_t a, fp_t b) COMPILER_RT_ALIAS(__divsf3); +COMPILER_RT_ALIAS(__divsf3, __aeabi_fdiv) #endif #endif |