169 lines
6.9 KiB
C
169 lines
6.9 KiB
C
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//===-- lib/divsf3.c - Single-precision division ------------------*- C -*-===//
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//
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// The LLVM Compiler Infrastructure
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//
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// This file is dual licensed under the MIT and the University of Illinois Open
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// Source Licenses. See LICENSE.TXT for details.
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//
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//===----------------------------------------------------------------------===//
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//
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// This file implements single-precision soft-float division
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// with the IEEE-754 default rounding (to nearest, ties to even).
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//
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// For simplicity, this implementation currently flushes denormals to zero.
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// It should be a fairly straightforward exercise to implement gradual
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// underflow with correct rounding.
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//
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//===----------------------------------------------------------------------===//
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#define SINGLE_PRECISION
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#include "fp_lib.h"
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ARM_EABI_FNALIAS(fdiv, divsf3)
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fp_t __divsf3(fp_t a, fp_t b) {
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const unsigned int aExponent = toRep(a) >> significandBits & maxExponent;
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const unsigned int bExponent = toRep(b) >> significandBits & maxExponent;
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const rep_t quotientSign = (toRep(a) ^ toRep(b)) & signBit;
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rep_t aSignificand = toRep(a) & significandMask;
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rep_t bSignificand = toRep(b) & significandMask;
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int scale = 0;
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// Detect if a or b is zero, denormal, infinity, or NaN.
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if (aExponent-1U >= maxExponent-1U || bExponent-1U >= maxExponent-1U) {
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const rep_t aAbs = toRep(a) & absMask;
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const rep_t bAbs = toRep(b) & absMask;
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// NaN / anything = qNaN
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if (aAbs > infRep) return fromRep(toRep(a) | quietBit);
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// anything / NaN = qNaN
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if (bAbs > infRep) return fromRep(toRep(b) | quietBit);
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if (aAbs == infRep) {
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// infinity / infinity = NaN
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if (bAbs == infRep) return fromRep(qnanRep);
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// infinity / anything else = +/- infinity
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else return fromRep(aAbs | quotientSign);
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}
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// anything else / infinity = +/- 0
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if (bAbs == infRep) return fromRep(quotientSign);
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if (!aAbs) {
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// zero / zero = NaN
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if (!bAbs) return fromRep(qnanRep);
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// zero / anything else = +/- zero
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else return fromRep(quotientSign);
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}
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// anything else / zero = +/- infinity
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if (!bAbs) return fromRep(infRep | quotientSign);
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// one or both of a or b is denormal, the other (if applicable) is a
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// normal number. Renormalize one or both of a and b, and set scale to
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// include the necessary exponent adjustment.
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if (aAbs < implicitBit) scale += normalize(&aSignificand);
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if (bAbs < implicitBit) scale -= normalize(&bSignificand);
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}
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// Or in the implicit significand bit. (If we fell through from the
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// denormal path it was already set by normalize( ), but setting it twice
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// won't hurt anything.)
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aSignificand |= implicitBit;
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bSignificand |= implicitBit;
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int quotientExponent = aExponent - bExponent + scale;
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// Align the significand of b as a Q31 fixed-point number in the range
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// [1, 2.0) and get a Q32 approximate reciprocal using a small minimax
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// polynomial approximation: reciprocal = 3/4 + 1/sqrt(2) - b/2. This
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// is accurate to about 3.5 binary digits.
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uint32_t q31b = bSignificand << 8;
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uint32_t reciprocal = UINT32_C(0x7504f333) - q31b;
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// Now refine the reciprocal estimate using a Newton-Raphson iteration:
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//
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// x1 = x0 * (2 - x0 * b)
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//
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// This doubles the number of correct binary digits in the approximation
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// with each iteration, so after three iterations, we have about 28 binary
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// digits of accuracy.
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uint32_t correction;
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correction = -((uint64_t)reciprocal * q31b >> 32);
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reciprocal = (uint64_t)reciprocal * correction >> 31;
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correction = -((uint64_t)reciprocal * q31b >> 32);
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reciprocal = (uint64_t)reciprocal * correction >> 31;
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correction = -((uint64_t)reciprocal * q31b >> 32);
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reciprocal = (uint64_t)reciprocal * correction >> 31;
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// Exhaustive testing shows that the error in reciprocal after three steps
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// is in the interval [-0x1.f58108p-31, 0x1.d0e48cp-29], in line with our
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// expectations. We bump the reciprocal by a tiny value to force the error
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// to be strictly positive (in the range [0x1.4fdfp-37,0x1.287246p-29], to
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// be specific). This also causes 1/1 to give a sensible approximation
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// instead of zero (due to overflow).
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reciprocal -= 2;
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// The numerical reciprocal is accurate to within 2^-28, lies in the
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// interval [0x1.000000eep-1, 0x1.fffffffcp-1], and is strictly smaller
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// than the true reciprocal of b. Multiplying a by this reciprocal thus
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// gives a numerical q = a/b in Q24 with the following properties:
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//
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// 1. q < a/b
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// 2. q is in the interval [0x1.000000eep-1, 0x1.fffffffcp0)
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// 3. the error in q is at most 2^-24 + 2^-27 -- the 2^24 term comes
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// from the fact that we truncate the product, and the 2^27 term
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// is the error in the reciprocal of b scaled by the maximum
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// possible value of a. As a consequence of this error bound,
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// either q or nextafter(q) is the correctly rounded
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rep_t quotient = (uint64_t)reciprocal*(aSignificand << 1) >> 32;
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// Two cases: quotient is in [0.5, 1.0) or quotient is in [1.0, 2.0).
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// In either case, we are going to compute a residual of the form
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//
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// r = a - q*b
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//
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// We know from the construction of q that r satisfies:
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//
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// 0 <= r < ulp(q)*b
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//
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// if r is greater than 1/2 ulp(q)*b, then q rounds up. Otherwise, we
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// already have the correct result. The exact halfway case cannot occur.
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// We also take this time to right shift quotient if it falls in the [1,2)
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// range and adjust the exponent accordingly.
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rep_t residual;
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if (quotient < (implicitBit << 1)) {
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residual = (aSignificand << 24) - quotient * bSignificand;
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quotientExponent--;
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} else {
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quotient >>= 1;
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residual = (aSignificand << 23) - quotient * bSignificand;
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}
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const int writtenExponent = quotientExponent + exponentBias;
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if (writtenExponent >= maxExponent) {
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// If we have overflowed the exponent, return infinity.
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return fromRep(infRep | quotientSign);
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}
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else if (writtenExponent < 1) {
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// Flush denormals to zero. In the future, it would be nice to add
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// code to round them correctly.
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return fromRep(quotientSign);
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}
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else {
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const bool round = (residual << 1) > bSignificand;
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// Clear the implicit bit
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rep_t absResult = quotient & significandMask;
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// Insert the exponent
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absResult |= (rep_t)writtenExponent << significandBits;
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// Round
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absResult += round;
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// Insert the sign and return
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return fromRep(absResult | quotientSign);
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}
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}
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