1//===-- lib/divsf3.c - Single-precision division ------------------*- C -*-===//
2//
3//                     The LLVM Compiler Infrastructure
4//
5// This file is dual licensed under the MIT and the University of Illinois Open
6// Source Licenses. See LICENSE.TXT for details.
7//
8//===----------------------------------------------------------------------===//
9//
10// This file implements single-precision soft-float division
11// with the IEEE-754 default rounding (to nearest, ties to even).
12//
13// For simplicity, this implementation currently flushes denormals to zero.
14// It should be a fairly straightforward exercise to implement gradual
15// underflow with correct rounding.
16//
17//===----------------------------------------------------------------------===//
18
19#define SINGLE_PRECISION
20#include "fp_lib.h"
21
22ARM_EABI_FNALIAS(fdiv, divsf3)
23
24fp_t __divsf3(fp_t a, fp_t b) {
25
26    const unsigned int aExponent = toRep(a) >> significandBits & maxExponent;
27    const unsigned int bExponent = toRep(b) >> significandBits & maxExponent;
28    const rep_t quotientSign = (toRep(a) ^ toRep(b)) & signBit;
29
30    rep_t aSignificand = toRep(a) & significandMask;
31    rep_t bSignificand = toRep(b) & significandMask;
32    int scale = 0;
33
34    // Detect if a or b is zero, denormal, infinity, or NaN.
35    if (aExponent-1U >= maxExponent-1U || bExponent-1U >= maxExponent-1U) {
36
37        const rep_t aAbs = toRep(a) & absMask;
38        const rep_t bAbs = toRep(b) & absMask;
39
40        // NaN / anything = qNaN
41        if (aAbs > infRep) return fromRep(toRep(a) | quietBit);
42        // anything / NaN = qNaN
43        if (bAbs > infRep) return fromRep(toRep(b) | quietBit);
44
45        if (aAbs == infRep) {
46            // infinity / infinity = NaN
47            if (bAbs == infRep) return fromRep(qnanRep);
48            // infinity / anything else = +/- infinity
49            else return fromRep(aAbs | quotientSign);
50        }
51
52        // anything else / infinity = +/- 0
53        if (bAbs == infRep) return fromRep(quotientSign);
54
55        if (!aAbs) {
56            // zero / zero = NaN
57            if (!bAbs) return fromRep(qnanRep);
58            // zero / anything else = +/- zero
59            else return fromRep(quotientSign);
60        }
61        // anything else / zero = +/- infinity
62        if (!bAbs) return fromRep(infRep | quotientSign);
63
64        // one or both of a or b is denormal, the other (if applicable) is a
65        // normal number.  Renormalize one or both of a and b, and set scale to
66        // include the necessary exponent adjustment.
67        if (aAbs < implicitBit) scale += normalize(&aSignificand);
68        if (bAbs < implicitBit) scale -= normalize(&bSignificand);
69    }
70
71    // Or in the implicit significand bit.  (If we fell through from the
72    // denormal path it was already set by normalize( ), but setting it twice
73    // won't hurt anything.)
74    aSignificand |= implicitBit;
75    bSignificand |= implicitBit;
76    int quotientExponent = aExponent - bExponent + scale;
77
78    // Align the significand of b as a Q31 fixed-point number in the range
79    // [1, 2.0) and get a Q32 approximate reciprocal using a small minimax
80    // polynomial approximation: reciprocal = 3/4 + 1/sqrt(2) - b/2.  This
81    // is accurate to about 3.5 binary digits.
82    uint32_t q31b = bSignificand << 8;
83    uint32_t reciprocal = UINT32_C(0x7504f333) - q31b;
84
85    // Now refine the reciprocal estimate using a Newton-Raphson iteration:
86    //
87    //     x1 = x0 * (2 - x0 * b)
88    //
89    // This doubles the number of correct binary digits in the approximation
90    // with each iteration, so after three iterations, we have about 28 binary
91    // digits of accuracy.
92    uint32_t correction;
93    correction = -((uint64_t)reciprocal * q31b >> 32);
94    reciprocal = (uint64_t)reciprocal * correction >> 31;
95    correction = -((uint64_t)reciprocal * q31b >> 32);
96    reciprocal = (uint64_t)reciprocal * correction >> 31;
97    correction = -((uint64_t)reciprocal * q31b >> 32);
98    reciprocal = (uint64_t)reciprocal * correction >> 31;
99
100    // Exhaustive testing shows that the error in reciprocal after three steps
101    // is in the interval [-0x1.f58108p-31, 0x1.d0e48cp-29], in line with our
102    // expectations.  We bump the reciprocal by a tiny value to force the error
103    // to be strictly positive (in the range [0x1.4fdfp-37,0x1.287246p-29], to
104    // be specific).  This also causes 1/1 to give a sensible approximation
105    // instead of zero (due to overflow).
106    reciprocal -= 2;
107
108    // The numerical reciprocal is accurate to within 2^-28, lies in the
109    // interval [0x1.000000eep-1, 0x1.fffffffcp-1], and is strictly smaller
110    // than the true reciprocal of b.  Multiplying a by this reciprocal thus
111    // gives a numerical q = a/b in Q24 with the following properties:
112    //
113    //    1. q < a/b
114    //    2. q is in the interval [0x1.000000eep-1, 0x1.fffffffcp0)
115    //    3. the error in q is at most 2^-24 + 2^-27 -- the 2^24 term comes
116    //       from the fact that we truncate the product, and the 2^27 term
117    //       is the error in the reciprocal of b scaled by the maximum
118    //       possible value of a.  As a consequence of this error bound,
119    //       either q or nextafter(q) is the correctly rounded
120    rep_t quotient = (uint64_t)reciprocal*(aSignificand << 1) >> 32;
121
122    // Two cases: quotient is in [0.5, 1.0) or quotient is in [1.0, 2.0).
123    // In either case, we are going to compute a residual of the form
124    //
125    //     r = a - q*b
126    //
127    // We know from the construction of q that r satisfies:
128    //
129    //     0 <= r < ulp(q)*b
130    //
131    // if r is greater than 1/2 ulp(q)*b, then q rounds up.  Otherwise, we
132    // already have the correct result.  The exact halfway case cannot occur.
133    // We also take this time to right shift quotient if it falls in the [1,2)
134    // range and adjust the exponent accordingly.
135    rep_t residual;
136    if (quotient < (implicitBit << 1)) {
137        residual = (aSignificand << 24) - quotient * bSignificand;
138        quotientExponent--;
139    } else {
140        quotient >>= 1;
141        residual = (aSignificand << 23) - quotient * bSignificand;
142    }
143
144    const int writtenExponent = quotientExponent + exponentBias;
145
146    if (writtenExponent >= maxExponent) {
147        // If we have overflowed the exponent, return infinity.
148        return fromRep(infRep | quotientSign);
149    }
150
151    else if (writtenExponent < 1) {
152        // Flush denormals to zero.  In the future, it would be nice to add
153        // code to round them correctly.
154        return fromRep(quotientSign);
155    }
156
157    else {
158        const bool round = (residual << 1) > bSignificand;
159        // Clear the implicit bit
160        rep_t absResult = quotient & significandMask;
161        // Insert the exponent
162        absResult |= (rep_t)writtenExponent << significandBits;
163        // Round
164        absResult += round;
165        // Insert the sign and return
166        return fromRep(absResult | quotientSign);
167    }
168}
169