| //===-- lib/divsf3.c - Single-precision 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 single-precision soft-float division |
| // with the IEEE-754 default rounding (to nearest, ties to even). |
| // |
| // For simplicity, this implementation currently flushes denormals to zero. |
| // It should be a fairly straightforward exercise to implement gradual |
| // underflow with correct rounding. |
| // |
| //===----------------------------------------------------------------------===// |
| |
| #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); |
| } |
| |
| // 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); } |
| #else |
| COMPILER_RT_ALIAS(__divsf3, __aeabi_fdiv) |
| #endif |
| #endif |