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//===-- 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.
//
//===----------------------------------------------------------------------===//
//
// 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"
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;
}
const int writtenExponent = quotientExponent + exponentBias;
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.
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);
}
}
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