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 
ARM_EABI_FNALIAS(fdiv,divsf3)22 ARM_EABI_FNALIAS(fdiv, divsf3)
23 
24 COMPILER_RT_ABI fp_t
25 __divsf3(fp_t a, fp_t b) {
26 
27     const unsigned int aExponent = toRep(a) >> significandBits & maxExponent;
28     const unsigned int bExponent = toRep(b) >> significandBits & maxExponent;
29     const rep_t quotientSign = (toRep(a) ^ toRep(b)) & signBit;
30 
31     rep_t aSignificand = toRep(a) & significandMask;
32     rep_t bSignificand = toRep(b) & significandMask;
33     int scale = 0;
34 
35     // Detect if a or b is zero, denormal, infinity, or NaN.
36     if (aExponent-1U >= maxExponent-1U || bExponent-1U >= maxExponent-1U) {
37 
38         const rep_t aAbs = toRep(a) & absMask;
39         const rep_t bAbs = toRep(b) & absMask;
40 
41         // NaN / anything = qNaN
42         if (aAbs > infRep) return fromRep(toRep(a) | quietBit);
43         // anything / NaN = qNaN
44         if (bAbs > infRep) return fromRep(toRep(b) | quietBit);
45 
46         if (aAbs == infRep) {
47             // infinity / infinity = NaN
48             if (bAbs == infRep) return fromRep(qnanRep);
49             // infinity / anything else = +/- infinity
50             else return fromRep(aAbs | quotientSign);
51         }
52 
53         // anything else / infinity = +/- 0
54         if (bAbs == infRep) return fromRep(quotientSign);
55 
56         if (!aAbs) {
57             // zero / zero = NaN
58             if (!bAbs) return fromRep(qnanRep);
59             // zero / anything else = +/- zero
60             else return fromRep(quotientSign);
61         }
62         // anything else / zero = +/- infinity
63         if (!bAbs) return fromRep(infRep | quotientSign);
64 
65         // one or both of a or b is denormal, the other (if applicable) is a
66         // normal number.  Renormalize one or both of a and b, and set scale to
67         // include the necessary exponent adjustment.
68         if (aAbs < implicitBit) scale += normalize(&aSignificand);
69         if (bAbs < implicitBit) scale -= normalize(&bSignificand);
70     }
71 
72     // Or in the implicit significand bit.  (If we fell through from the
73     // denormal path it was already set by normalize( ), but setting it twice
74     // won't hurt anything.)
75     aSignificand |= implicitBit;
76     bSignificand |= implicitBit;
77     int quotientExponent = aExponent - bExponent + scale;
78 
79     // Align the significand of b as a Q31 fixed-point number in the range
80     // [1, 2.0) and get a Q32 approximate reciprocal using a small minimax
81     // polynomial approximation: reciprocal = 3/4 + 1/sqrt(2) - b/2.  This
82     // is accurate to about 3.5 binary digits.
83     uint32_t q31b = bSignificand << 8;
84     uint32_t reciprocal = UINT32_C(0x7504f333) - q31b;
85 
86     // Now refine the reciprocal estimate using a Newton-Raphson iteration:
87     //
88     //     x1 = x0 * (2 - x0 * b)
89     //
90     // This doubles the number of correct binary digits in the approximation
91     // with each iteration, so after three iterations, we have about 28 binary
92     // digits of accuracy.
93     uint32_t correction;
94     correction = -((uint64_t)reciprocal * q31b >> 32);
95     reciprocal = (uint64_t)reciprocal * correction >> 31;
96     correction = -((uint64_t)reciprocal * q31b >> 32);
97     reciprocal = (uint64_t)reciprocal * correction >> 31;
98     correction = -((uint64_t)reciprocal * q31b >> 32);
99     reciprocal = (uint64_t)reciprocal * correction >> 31;
100 
101     // Exhaustive testing shows that the error in reciprocal after three steps
102     // is in the interval [-0x1.f58108p-31, 0x1.d0e48cp-29], in line with our
103     // expectations.  We bump the reciprocal by a tiny value to force the error
104     // to be strictly positive (in the range [0x1.4fdfp-37,0x1.287246p-29], to
105     // be specific).  This also causes 1/1 to give a sensible approximation
106     // instead of zero (due to overflow).
107     reciprocal -= 2;
108 
109     // The numerical reciprocal is accurate to within 2^-28, lies in the
110     // interval [0x1.000000eep-1, 0x1.fffffffcp-1], and is strictly smaller
111     // than the true reciprocal of b.  Multiplying a by this reciprocal thus
112     // gives a numerical q = a/b in Q24 with the following properties:
113     //
114     //    1. q < a/b
115     //    2. q is in the interval [0x1.000000eep-1, 0x1.fffffffcp0)
116     //    3. the error in q is at most 2^-24 + 2^-27 -- the 2^24 term comes
117     //       from the fact that we truncate the product, and the 2^27 term
118     //       is the error in the reciprocal of b scaled by the maximum
119     //       possible value of a.  As a consequence of this error bound,
120     //       either q or nextafter(q) is the correctly rounded
121     rep_t quotient = (uint64_t)reciprocal*(aSignificand << 1) >> 32;
122 
123     // Two cases: quotient is in [0.5, 1.0) or quotient is in [1.0, 2.0).
124     // In either case, we are going to compute a residual of the form
125     //
126     //     r = a - q*b
127     //
128     // We know from the construction of q that r satisfies:
129     //
130     //     0 <= r < ulp(q)*b
131     //
132     // if r is greater than 1/2 ulp(q)*b, then q rounds up.  Otherwise, we
133     // already have the correct result.  The exact halfway case cannot occur.
134     // We also take this time to right shift quotient if it falls in the [1,2)
135     // range and adjust the exponent accordingly.
136     rep_t residual;
137     if (quotient < (implicitBit << 1)) {
138         residual = (aSignificand << 24) - quotient * bSignificand;
139         quotientExponent--;
140     } else {
141         quotient >>= 1;
142         residual = (aSignificand << 23) - quotient * bSignificand;
143     }
144 
145     const int writtenExponent = quotientExponent + exponentBias;
146 
147     if (writtenExponent >= maxExponent) {
148         // If we have overflowed the exponent, return infinity.
149         return fromRep(infRep | quotientSign);
150     }
151 
152     else if (writtenExponent < 1) {
153         // Flush denormals to zero.  In the future, it would be nice to add
154         // code to round them correctly.
155         return fromRep(quotientSign);
156     }
157 
158     else {
159         const bool round = (residual << 1) > bSignificand;
160         // Clear the implicit bit
161         rep_t absResult = quotient & significandMask;
162         // Insert the exponent
163         absResult |= (rep_t)writtenExponent << significandBits;
164         // Round
165         absResult += round;
166         // Insert the sign and return
167         return fromRep(absResult | quotientSign);
168     }
169 }
170