1 //===-- lib/divdf3.c - Double-precision division ------------------*- C -*-===//
3 // The LLVM Compiler Infrastructure
5 // This file is dual licensed under the MIT and the University of Illinois Open
6 // Source Licenses. See LICENSE.TXT for details.
8 //===----------------------------------------------------------------------===//
10 // This file implements double-precision soft-float division
11 // with the IEEE-754 default rounding (to nearest, ties to even).
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.
17 //===----------------------------------------------------------------------===//
19 #define DOUBLE_PRECISION
22 ARM_EABI_FNALIAS(ddiv
, divdf3
)
24 fp_t
__divdf3(fp_t a
, fp_t b
) {
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
;
30 rep_t aSignificand
= toRep(a
) & significandMask
;
31 rep_t bSignificand
= toRep(b
) & significandMask
;
34 // Detect if a or b is zero, denormal, infinity, or NaN.
35 if (aExponent
-1U >= maxExponent
-1U || bExponent
-1U >= maxExponent
-1U) {
37 const rep_t aAbs
= toRep(a
) & absMask
;
38 const rep_t bAbs
= toRep(b
) & absMask
;
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
);
46 // infinity / infinity = NaN
47 if (bAbs
== infRep
) return fromRep(qnanRep
);
48 // infinity / anything else = +/- infinity
49 else return fromRep(aAbs
| quotientSign
);
52 // anything else / infinity = +/- 0
53 if (bAbs
== infRep
) return fromRep(quotientSign
);
57 if (!bAbs
) return fromRep(qnanRep
);
58 // zero / anything else = +/- zero
59 else return fromRep(quotientSign
);
61 // anything else / zero = +/- infinity
62 if (!bAbs
) return fromRep(infRep
| quotientSign
);
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
);
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
;
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 const uint32_t q31b
= bSignificand
>> 21;
83 uint32_t recip32
= UINT32_C(0x7504f333) - q31b
;
85 // Now refine the reciprocal estimate using a Newton-Raphson iteration:
87 // x1 = x0 * (2 - x0 * b)
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 correction32
;
93 correction32
= -((uint64_t)recip32
* q31b
>> 32);
94 recip32
= (uint64_t)recip32
* correction32
>> 31;
95 correction32
= -((uint64_t)recip32
* q31b
>> 32);
96 recip32
= (uint64_t)recip32
* correction32
>> 31;
97 correction32
= -((uint64_t)recip32
* q31b
>> 32);
98 recip32
= (uint64_t)recip32
* correction32
>> 31;
100 // recip32 might have overflowed to exactly zero in the preceeding
101 // computation if the high word of b is exactly 1.0. This would sabotage
102 // the full-width final stage of the computation that follows, so we adjust
103 // recip32 downward by one bit.
106 // We need to perform one more iteration to get us to 56 binary digits;
107 // The last iteration needs to happen with extra precision.
108 const uint32_t q63blo
= bSignificand
<< 11;
109 uint64_t correction
, reciprocal
;
110 correction
= -((uint64_t)recip32
*q31b
+ ((uint64_t)recip32
*q63blo
>> 32));
111 uint32_t cHi
= correction
>> 32;
112 uint32_t cLo
= correction
;
113 reciprocal
= (uint64_t)recip32
*cHi
+ ((uint64_t)recip32
*cLo
>> 32);
115 // We already adjusted the 32-bit estimate, now we need to adjust the final
116 // 64-bit reciprocal estimate downward to ensure that it is strictly smaller
117 // than the infinitely precise exact reciprocal. Because the computation
118 // of the Newton-Raphson step is truncating at every step, this adjustment
119 // is small; most of the work is already done.
122 // The numerical reciprocal is accurate to within 2^-56, lies in the
123 // interval [0.5, 1.0), and is strictly smaller than the true reciprocal
124 // of b. Multiplying a by this reciprocal thus gives a numerical q = a/b
125 // in Q53 with the following properties:
128 // 2. q is in the interval [0.5, 2.0)
129 // 3. the error in q is bounded away from 2^-53 (actually, we have a
130 // couple of bits to spare, but this is all we need).
132 // We need a 64 x 64 multiply high to compute q, which isn't a basic
133 // operation in C, so we need to be a little bit fussy.
134 rep_t quotient
, quotientLo
;
135 wideMultiply(aSignificand
<< 2, reciprocal
, "ient
, "ientLo
);
137 // Two cases: quotient is in [0.5, 1.0) or quotient is in [1.0, 2.0).
138 // In either case, we are going to compute a residual of the form
142 // We know from the construction of q that r satisfies:
146 // if r is greater than 1/2 ulp(q)*b, then q rounds up. Otherwise, we
147 // already have the correct result. The exact halfway case cannot occur.
148 // We also take this time to right shift quotient if it falls in the [1,2)
149 // range and adjust the exponent accordingly.
151 if (quotient
< (implicitBit
<< 1)) {
152 residual
= (aSignificand
<< 53) - quotient
* bSignificand
;
156 residual
= (aSignificand
<< 52) - quotient
* bSignificand
;
159 const int writtenExponent
= quotientExponent
+ exponentBias
;
161 if (writtenExponent
>= maxExponent
) {
162 // If we have overflowed the exponent, return infinity.
163 return fromRep(infRep
| quotientSign
);
166 else if (writtenExponent
< 1) {
167 // Flush denormals to zero. In the future, it would be nice to add
168 // code to round them correctly.
169 return fromRep(quotientSign
);
173 const bool round
= (residual
<< 1) > bSignificand
;
174 // Clear the implicit bit
175 rep_t absResult
= quotient
& significandMask
;
176 // Insert the exponent
177 absResult
|= (rep_t
)writtenExponent
<< significandBits
;
180 // Insert the sign and return
181 const double result
= fromRep(absResult
| quotientSign
);