xref: /llvm-project/flang/lib/Evaluate/real.cpp (revision e619c07d168dff1d27f90cef84222a68064c35ea) 
1 //===-- lib/Evaluate/real.cpp ---------------------------------------------===//
2 //
3 // Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
4 // See https://llvm.org/LICENSE.txt for license information.
5 // SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
6 //
7 //===----------------------------------------------------------------------===//
8 
9 #include "flang/Evaluate/real.h"
10 #include "int-power.h"
11 #include "flang/Common/idioms.h"
12 #include "flang/Decimal/decimal.h"
13 #include "flang/Parser/characters.h"
14 #include "llvm/Support/raw_ostream.h"
15 #include <limits>
16 
17 namespace Fortran::evaluate::value {
18 
19 template <typename W, int P> Relation Real<W, P>::Compare(const Real &y) const {
20   if (IsNotANumber() || y.IsNotANumber()) { // NaN vs x, x vs NaN
21     return Relation::Unordered;
22   } else if (IsInfinite()) {
23     if (y.IsInfinite()) {
24       if (IsNegative()) { // -Inf vs +/-Inf
25         return y.IsNegative() ? Relation::Equal : Relation::Less;
26       } else { // +Inf vs +/-Inf
27         return y.IsNegative() ? Relation::Greater : Relation::Equal;
28       }
29     } else { // +/-Inf vs finite
30       return IsNegative() ? Relation::Less : Relation::Greater;
31     }
32   } else if (y.IsInfinite()) { // finite vs +/-Inf
33     return y.IsNegative() ? Relation::Greater : Relation::Less;
34   } else { // two finite numbers
35     bool isNegative{IsNegative()};
36     if (isNegative != y.IsNegative()) {
37       if (word_.IOR(y.word_).IBCLR(bits - 1).IsZero()) {
38         return Relation::Equal; // +/-0.0 == -/+0.0
39       } else {
40         return isNegative ? Relation::Less : Relation::Greater;
41       }
42     } else {
43       // same sign
44       Ordering order{evaluate::Compare(Exponent(), y.Exponent())};
45       if (order == Ordering::Equal) {
46         order = GetSignificand().CompareUnsigned(y.GetSignificand());
47       }
48       if (isNegative) {
49         order = Reverse(order);
50       }
51       return RelationFromOrdering(order);
52     }
53   }
54 }
55 
56 template <typename W, int P>
57 ValueWithRealFlags<Real<W, P>> Real<W, P>::Add(
58     const Real &y, Rounding rounding) const {
59   ValueWithRealFlags<Real> result;
60   if (IsNotANumber() || y.IsNotANumber()) {
61     result.value = NotANumber(); // NaN + x -> NaN
62     if (IsSignalingNaN() || y.IsSignalingNaN()) {
63       result.flags.set(RealFlag::InvalidArgument);
64     }
65     return result;
66   }
67   bool isNegative{IsNegative()};
68   bool yIsNegative{y.IsNegative()};
69   if (IsInfinite()) {
70     if (y.IsInfinite()) {
71       if (isNegative == yIsNegative) {
72         result.value = *this; // +/-Inf + +/-Inf -> +/-Inf
73       } else {
74         result.value = NotANumber(); // +/-Inf + -/+Inf -> NaN
75         result.flags.set(RealFlag::InvalidArgument);
76       }
77     } else {
78       result.value = *this; // +/-Inf + x -> +/-Inf
79     }
80     return result;
81   }
82   if (y.IsInfinite()) {
83     result.value = y; // x + +/-Inf -> +/-Inf
84     return result;
85   }
86   int exponent{Exponent()};
87   int yExponent{y.Exponent()};
88   if (exponent < yExponent) {
89     // y is larger in magnitude; simplify by reversing operands
90     return y.Add(*this, rounding);
91   }
92   if (exponent == yExponent && isNegative != yIsNegative) {
93     Ordering order{GetSignificand().CompareUnsigned(y.GetSignificand())};
94     if (order == Ordering::Less) {
95       // Same exponent, opposite signs, and y is larger in magnitude
96       return y.Add(*this, rounding);
97     }
98     if (order == Ordering::Equal) {
99       // x + (-x) -> +0.0 unless rounding is directed downwards
100       if (rounding.mode == common::RoundingMode::Down) {
101         result.value.word_ = result.value.word_.IBSET(bits - 1); // -0.0
102       }
103       return result;
104     }
105   }
106   // Our exponent is greater than y's, or the exponents match and y is not
107   // of the opposite sign and greater magnitude.  So (x+y) will have the
108   // same sign as x.
109   Fraction fraction{GetFraction()};
110   Fraction yFraction{y.GetFraction()};
111   int rshift = exponent - yExponent;
112   if (exponent > 0 && yExponent == 0) {
113     --rshift; // correct overshift when only y is subnormal
114   }
115   RoundingBits roundingBits{yFraction, rshift};
116   yFraction = yFraction.SHIFTR(rshift);
117   bool carry{false};
118   if (isNegative != yIsNegative) {
119     // Opposite signs: subtract via addition of two's complement of y and
120     // the rounding bits.
121     yFraction = yFraction.NOT();
122     carry = roundingBits.Negate();
123   }
124   auto sum{fraction.AddUnsigned(yFraction, carry)};
125   fraction = sum.value;
126   if (isNegative == yIsNegative && sum.carry) {
127     roundingBits.ShiftRight(sum.value.BTEST(0));
128     fraction = fraction.SHIFTR(1).IBSET(fraction.bits - 1);
129     ++exponent;
130   }
131   NormalizeAndRound(
132       result, isNegative, exponent, fraction, rounding, roundingBits);
133   return result;
134 }
135 
136 template <typename W, int P>
137 ValueWithRealFlags<Real<W, P>> Real<W, P>::Multiply(
138     const Real &y, Rounding rounding) const {
139   ValueWithRealFlags<Real> result;
140   if (IsNotANumber() || y.IsNotANumber()) {
141     result.value = NotANumber(); // NaN * x -> NaN
142     if (IsSignalingNaN() || y.IsSignalingNaN()) {
143       result.flags.set(RealFlag::InvalidArgument);
144     }
145   } else {
146     bool isNegative{IsNegative() != y.IsNegative()};
147     if (IsInfinite() || y.IsInfinite()) {
148       if (IsZero() || y.IsZero()) {
149         result.value = NotANumber(); // 0 * Inf -> NaN
150         result.flags.set(RealFlag::InvalidArgument);
151       } else {
152         result.value = Infinity(isNegative);
153       }
154     } else {
155       auto product{GetFraction().MultiplyUnsigned(y.GetFraction())};
156       std::int64_t exponent{CombineExponents(y, false)};
157       if (exponent < 1) {
158         int rshift = 1 - exponent;
159         exponent = 1;
160         bool sticky{false};
161         if (rshift >= product.upper.bits + product.lower.bits) {
162           sticky = !product.lower.IsZero() || !product.upper.IsZero();
163         } else if (rshift >= product.lower.bits) {
164           sticky = !product.lower.IsZero() ||
165               !product.upper
166                    .IAND(product.upper.MASKR(rshift - product.lower.bits))
167                    .IsZero();
168         } else {
169           sticky = !product.lower.IAND(product.lower.MASKR(rshift)).IsZero();
170         }
171         product.lower = product.lower.SHIFTRWithFill(product.upper, rshift);
172         product.upper = product.upper.SHIFTR(rshift);
173         if (sticky) {
174           product.lower = product.lower.IBSET(0);
175         }
176       }
177       int leadz{product.upper.LEADZ()};
178       if (leadz >= product.upper.bits) {
179         leadz += product.lower.LEADZ();
180       }
181       int lshift{leadz};
182       if (lshift > exponent - 1) {
183         lshift = exponent - 1;
184       }
185       exponent -= lshift;
186       product.upper = product.upper.SHIFTLWithFill(product.lower, lshift);
187       product.lower = product.lower.SHIFTL(lshift);
188       RoundingBits roundingBits{product.lower, product.lower.bits};
189       NormalizeAndRound(result, isNegative, exponent, product.upper, rounding,
190           roundingBits, true /*multiply*/);
191     }
192   }
193   return result;
194 }
195 
196 template <typename W, int P>
197 ValueWithRealFlags<Real<W, P>> Real<W, P>::Divide(
198     const Real &y, Rounding rounding) const {
199   ValueWithRealFlags<Real> result;
200   if (IsNotANumber() || y.IsNotANumber()) {
201     result.value = NotANumber(); // NaN / x -> NaN, x / NaN -> NaN
202     if (IsSignalingNaN() || y.IsSignalingNaN()) {
203       result.flags.set(RealFlag::InvalidArgument);
204     }
205   } else {
206     bool isNegative{IsNegative() != y.IsNegative()};
207     if (IsInfinite()) {
208       if (y.IsInfinite()) {
209         result.value = NotANumber(); // Inf/Inf -> NaN
210         result.flags.set(RealFlag::InvalidArgument);
211       } else { // Inf/x -> Inf,  Inf/0 -> Inf
212         result.value = Infinity(isNegative);
213       }
214     } else if (y.IsZero()) {
215       if (IsZero()) { // 0/0 -> NaN
216         result.value = NotANumber();
217         result.flags.set(RealFlag::InvalidArgument);
218       } else { // x/0 -> Inf, Inf/0 -> Inf
219         result.value = Infinity(isNegative);
220         result.flags.set(RealFlag::DivideByZero);
221       }
222     } else if (IsZero() || y.IsInfinite()) { // 0/x, x/Inf -> 0
223       if (isNegative) {
224         result.value.word_ = result.value.word_.IBSET(bits - 1);
225       }
226     } else {
227       // dividend and divisor are both finite and nonzero numbers
228       Fraction top{GetFraction()}, divisor{y.GetFraction()};
229       std::int64_t exponent{CombineExponents(y, true)};
230       Fraction quotient;
231       bool msb{false};
232       if (!top.BTEST(top.bits - 1) || !divisor.BTEST(divisor.bits - 1)) {
233         // One or two subnormals
234         int topLshift{top.LEADZ()};
235         top = top.SHIFTL(topLshift);
236         int divisorLshift{divisor.LEADZ()};
237         divisor = divisor.SHIFTL(divisorLshift);
238         exponent += divisorLshift - topLshift;
239       }
240       for (int j{1}; j <= quotient.bits; ++j) {
241         if (NextQuotientBit(top, msb, divisor)) {
242           quotient = quotient.IBSET(quotient.bits - j);
243         }
244       }
245       bool guard{NextQuotientBit(top, msb, divisor)};
246       bool round{NextQuotientBit(top, msb, divisor)};
247       bool sticky{msb || !top.IsZero()};
248       RoundingBits roundingBits{guard, round, sticky};
249       if (exponent < 1) {
250         std::int64_t rshift{1 - exponent};
251         for (; rshift > 0; --rshift) {
252           roundingBits.ShiftRight(quotient.BTEST(0));
253           quotient = quotient.SHIFTR(1);
254         }
255         exponent = 1;
256       }
257       NormalizeAndRound(
258           result, isNegative, exponent, quotient, rounding, roundingBits);
259     }
260   }
261   return result;
262 }
263 
264 template <typename W, int P>
265 ValueWithRealFlags<Real<W, P>> Real<W, P>::SQRT(Rounding rounding) const {
266   ValueWithRealFlags<Real> result;
267   if (IsNotANumber()) {
268     result.value = NotANumber();
269     if (IsSignalingNaN()) {
270       result.flags.set(RealFlag::InvalidArgument);
271     }
272   } else if (IsNegative()) {
273     if (IsZero()) {
274       // SQRT(-0) == -0 in IEEE-754.
275       result.value.word_ = result.value.word_.IBSET(bits - 1);
276     } else {
277       result.value = NotANumber();
278     }
279   } else if (IsInfinite()) {
280     // SQRT(+Inf) == +Inf
281     result.value = Infinity(false);
282   } else {
283     int expo{UnbiasedExponent()};
284     if (expo < -1 || expo > 1) {
285       // Reduce the range to [0.5 .. 4.0) by dividing by an integral power
286       // of four to avoid trouble with very large and very small values
287       // (esp. truncation of subnormals).
288       // SQRT(2**(2a) * x) = SQRT(2**(2a)) * SQRT(x) = 2**a * SQRT(x)
289       Real scaled;
290       int adjust{expo / 2};
291       scaled.Normalize(false, expo - 2 * adjust + exponentBias, GetFraction());
292       result = scaled.SQRT(rounding);
293       result.value.Normalize(false,
294           result.value.UnbiasedExponent() + adjust + exponentBias,
295           result.value.GetFraction());
296       return result;
297     }
298     // Compute the square root of the reduced value with the slow but
299     // reliable bit-at-a-time method.  Start with a clear significand and
300     // half of the unbiased exponent, and then try to set significand bits
301     // in descending order of magnitude without exceeding the exact result.
302     expo = expo / 2 + exponentBias;
303     result.value.Normalize(false, expo, Fraction::MASKL(1));
304     Real initialSq{result.value.Multiply(result.value).value};
305     if (Compare(initialSq) == Relation::Less) {
306       // Initial estimate is too large; this can happen for values just
307       // under 1.0.
308       --expo;
309       result.value.Normalize(false, expo, Fraction::MASKL(1));
310     }
311     for (int bit{significandBits - 1}; bit >= 0; --bit) {
312       Word word{result.value.word_};
313       result.value.word_ = word.IBSET(bit);
314       auto squared{result.value.Multiply(result.value, rounding)};
315       if (squared.flags.test(RealFlag::Overflow) ||
316           squared.flags.test(RealFlag::Underflow) ||
317           Compare(squared.value) == Relation::Less) {
318         result.value.word_ = word;
319       }
320     }
321     // The computed square root has a square that's not greater than the
322     // original argument.  Check this square against the square of the next
323     // larger Real and return that one if its square is closer in magnitude to
324     // the original argument.
325     Real resultSq{result.value.Multiply(result.value).value};
326     Real diff{Subtract(resultSq).value.ABS()};
327     if (diff.IsZero()) {
328       return result; // exact
329     }
330     Real ulp;
331     ulp.Normalize(false, expo, Fraction::MASKR(1));
332     Real nextAfter{result.value.Add(ulp).value};
333     auto nextAfterSq{nextAfter.Multiply(nextAfter)};
334     if (!nextAfterSq.flags.test(RealFlag::Overflow) &&
335         !nextAfterSq.flags.test(RealFlag::Underflow)) {
336       Real nextAfterDiff{Subtract(nextAfterSq.value).value.ABS()};
337       if (nextAfterDiff.Compare(diff) == Relation::Less) {
338         result.value = nextAfter;
339         if (nextAfterDiff.IsZero()) {
340           return result; // exact
341         }
342       }
343     }
344     result.flags.set(RealFlag::Inexact);
345   }
346   return result;
347 }
348 
349 template <typename W, int P>
350 ValueWithRealFlags<Real<W, P>> Real<W, P>::NEAREST(bool upward) const {
351   ValueWithRealFlags<Real> result;
352   if (IsFinite()) {
353     Fraction fraction{GetFraction()};
354     int expo{Exponent()};
355     Fraction one{1};
356     Fraction nearest;
357     bool isNegative{IsNegative()};
358     if (upward != isNegative) { // upward in magnitude
359       auto next{fraction.AddUnsigned(one)};
360       if (next.carry) {
361         ++expo;
362         nearest = Fraction::Least(); // MSB only
363       } else {
364         nearest = next.value;
365       }
366     } else { // downward in magnitude
367       if (IsZero()) {
368         nearest = 1; // smallest magnitude negative subnormal
369         isNegative = !isNegative;
370       } else {
371         auto sub1{fraction.SubtractSigned(one)};
372         if (sub1.overflow) {
373           nearest = Fraction{0}.NOT();
374           --expo;
375         } else {
376           nearest = sub1.value;
377         }
378       }
379     }
380     result.flags = result.value.Normalize(isNegative, expo, nearest);
381   } else {
382     result.flags.set(RealFlag::InvalidArgument);
383     result.value = *this;
384   }
385   return result;
386 }
387 
388 // HYPOT(x,y) = SQRT(x**2 + y**2) by definition, but those squared intermediate
389 // values are susceptible to over/underflow when computed naively.
390 // Assuming that x>=y, calculate instead:
391 //   HYPOT(x,y) = SQRT(x**2 * (1+(y/x)**2))
392 //              = ABS(x) * SQRT(1+(y/x)**2)
393 template <typename W, int P>
394 ValueWithRealFlags<Real<W, P>> Real<W, P>::HYPOT(
395     const Real &y, Rounding rounding) const {
396   ValueWithRealFlags<Real> result;
397   if (IsNotANumber() || y.IsNotANumber()) {
398     result.flags.set(RealFlag::InvalidArgument);
399     result.value = NotANumber();
400   } else if (ABS().Compare(y.ABS()) == Relation::Less) {
401     return y.HYPOT(*this);
402   } else if (IsZero()) {
403     return result; // x==y==0
404   } else {
405     auto yOverX{y.Divide(*this, rounding)}; // y/x
406     bool inexact{yOverX.flags.test(RealFlag::Inexact)};
407     auto squared{yOverX.value.Multiply(yOverX.value, rounding)}; // (y/x)**2
408     inexact |= squared.flags.test(RealFlag::Inexact);
409     Real one;
410     one.Normalize(false, exponentBias, Fraction::MASKL(1)); // 1.0
411     auto sum{squared.value.Add(one, rounding)}; // 1.0 + (y/x)**2
412     inexact |= sum.flags.test(RealFlag::Inexact);
413     auto sqrt{sum.value.SQRT()};
414     inexact |= sqrt.flags.test(RealFlag::Inexact);
415     result = sqrt.value.Multiply(ABS(), rounding);
416     if (inexact) {
417       result.flags.set(RealFlag::Inexact);
418     }
419   }
420   return result;
421 }
422 
423 template <typename W, int P>
424 ValueWithRealFlags<Real<W, P>> Real<W, P>::ToWholeNumber(
425     common::RoundingMode mode) const {
426   ValueWithRealFlags<Real> result{*this};
427   if (IsNotANumber()) {
428     result.flags.set(RealFlag::InvalidArgument);
429     result.value = NotANumber();
430   } else if (IsInfinite()) {
431     result.flags.set(RealFlag::Overflow);
432   } else {
433     constexpr int noClipExponent{exponentBias + binaryPrecision - 1};
434     if (Exponent() < noClipExponent) {
435       Real adjust; // ABS(EPSILON(adjust)) == 0.5
436       adjust.Normalize(IsSignBitSet(), noClipExponent, Fraction::MASKL(1));
437       // Compute ival=(*this + adjust), losing any fractional bits; keep flags
438       result = Add(adjust, Rounding{mode});
439       result.flags.reset(RealFlag::Inexact); // result *is* exact
440       // Return (ival-adjust) with original sign in case we've generated a zero.
441       result.value =
442           result.value.Subtract(adjust, Rounding{common::RoundingMode::ToZero})
443               .value.SIGN(*this);
444     }
445   }
446   return result;
447 }
448 
449 template <typename W, int P>
450 RealFlags Real<W, P>::Normalize(bool negative, int exponent,
451     const Fraction &fraction, Rounding rounding, RoundingBits *roundingBits) {
452   int lshift{fraction.LEADZ()};
453   if (lshift == fraction.bits /* fraction is zero */ &&
454       (!roundingBits || roundingBits->empty())) {
455     // No fraction, no rounding bits -> +/-0.0
456     exponent = lshift = 0;
457   } else if (lshift < exponent) {
458     exponent -= lshift;
459   } else if (exponent > 0) {
460     lshift = exponent - 1;
461     exponent = 0;
462   } else if (lshift == 0) {
463     exponent = 1;
464   } else {
465     lshift = 0;
466   }
467   if (exponent >= maxExponent) {
468     // Infinity or overflow
469     if (rounding.mode == common::RoundingMode::TiesToEven ||
470         rounding.mode == common::RoundingMode::TiesAwayFromZero ||
471         (rounding.mode == common::RoundingMode::Up && !negative) ||
472         (rounding.mode == common::RoundingMode::Down && negative)) {
473       word_ = Word{maxExponent}.SHIFTL(significandBits); // Inf
474     } else {
475       // directed rounding: round to largest finite value rather than infinity
476       // (x86 does this, not sure whether it's standard behavior)
477       word_ = Word{word_.MASKR(word_.bits - 1)}.IBCLR(significandBits);
478     }
479     if (negative) {
480       word_ = word_.IBSET(bits - 1);
481     }
482     RealFlags flags{RealFlag::Overflow};
483     if (!fraction.IsZero()) {
484       flags.set(RealFlag::Inexact);
485     }
486     return flags;
487   }
488   word_ = Word::ConvertUnsigned(fraction).value;
489   if (lshift > 0) {
490     word_ = word_.SHIFTL(lshift);
491     if (roundingBits) {
492       for (; lshift > 0; --lshift) {
493         if (roundingBits->ShiftLeft()) {
494           word_ = word_.IBSET(lshift - 1);
495         }
496       }
497     }
498   }
499   if constexpr (isImplicitMSB) {
500     word_ = word_.IBCLR(significandBits);
501   }
502   word_ = word_.IOR(Word{exponent}.SHIFTL(significandBits));
503   if (negative) {
504     word_ = word_.IBSET(bits - 1);
505   }
506   return {};
507 }
508 
509 template <typename W, int P>
510 RealFlags Real<W, P>::Round(
511     Rounding rounding, const RoundingBits &bits, bool multiply) {
512   int origExponent{Exponent()};
513   RealFlags flags;
514   bool inexact{!bits.empty()};
515   if (inexact) {
516     flags.set(RealFlag::Inexact);
517   }
518   if (origExponent < maxExponent &&
519       bits.MustRound(rounding, IsNegative(), word_.BTEST(0) /* is odd */)) {
520     typename Fraction::ValueWithCarry sum{
521         GetFraction().AddUnsigned(Fraction{}, true)};
522     int newExponent{origExponent};
523     if (sum.carry) {
524       // The fraction was all ones before rounding; sum.value is now zero
525       sum.value = sum.value.IBSET(binaryPrecision - 1);
526       if (++newExponent >= maxExponent) {
527         flags.set(RealFlag::Overflow); // rounded away to an infinity
528       }
529     }
530     flags |= Normalize(IsNegative(), newExponent, sum.value);
531   }
532   if (inexact && origExponent == 0) {
533     // inexact subnormal input: signal Underflow unless in an x86-specific
534     // edge case
535     if (rounding.x86CompatibleBehavior && Exponent() != 0 && multiply &&
536         bits.sticky() &&
537         (bits.guard() ||
538             (rounding.mode != common::RoundingMode::Up &&
539                 rounding.mode != common::RoundingMode::Down))) {
540       // x86 edge case in which Underflow fails to signal when a subnormal
541       // inexact multiplication product rounds to a normal result when
542       // the guard bit is set or we're not using directed rounding
543     } else {
544       flags.set(RealFlag::Underflow);
545     }
546   }
547   return flags;
548 }
549 
550 template <typename W, int P>
551 void Real<W, P>::NormalizeAndRound(ValueWithRealFlags<Real> &result,
552     bool isNegative, int exponent, const Fraction &fraction, Rounding rounding,
553     RoundingBits roundingBits, bool multiply) {
554   result.flags |= result.value.Normalize(
555       isNegative, exponent, fraction, rounding, &roundingBits);
556   result.flags |= result.value.Round(rounding, roundingBits, multiply);
557 }
558 
559 inline enum decimal::FortranRounding MapRoundingMode(
560     common::RoundingMode rounding) {
561   switch (rounding) {
562   case common::RoundingMode::TiesToEven:
563     break;
564   case common::RoundingMode::ToZero:
565     return decimal::RoundToZero;
566   case common::RoundingMode::Down:
567     return decimal::RoundDown;
568   case common::RoundingMode::Up:
569     return decimal::RoundUp;
570   case common::RoundingMode::TiesAwayFromZero:
571     return decimal::RoundCompatible;
572   }
573   return decimal::RoundNearest; // dodge gcc warning about lack of result
574 }
575 
576 inline RealFlags MapFlags(decimal::ConversionResultFlags flags) {
577   RealFlags result;
578   if (flags & decimal::Overflow) {
579     result.set(RealFlag::Overflow);
580   }
581   if (flags & decimal::Inexact) {
582     result.set(RealFlag::Inexact);
583   }
584   if (flags & decimal::Invalid) {
585     result.set(RealFlag::InvalidArgument);
586   }
587   return result;
588 }
589 
590 template <typename W, int P>
591 ValueWithRealFlags<Real<W, P>> Real<W, P>::Read(
592     const char *&p, Rounding rounding) {
593   auto converted{
594       decimal::ConvertToBinary<P>(p, MapRoundingMode(rounding.mode))};
595   const auto *value{reinterpret_cast<Real<W, P> *>(&converted.binary)};
596   return {*value, MapFlags(converted.flags)};
597 }
598 
599 template <typename W, int P> std::string Real<W, P>::DumpHexadecimal() const {
600   if (IsNotANumber()) {
601     return "NaN0x"s + word_.Hexadecimal();
602   } else if (IsNegative()) {
603     return "-"s + Negate().DumpHexadecimal();
604   } else if (IsInfinite()) {
605     return "Inf"s;
606   } else if (IsZero()) {
607     return "0.0"s;
608   } else {
609     Fraction frac{GetFraction()};
610     std::string result{"0x"};
611     char intPart = '0' + frac.BTEST(frac.bits - 1);
612     result += intPart;
613     result += '.';
614     int trailz{frac.TRAILZ()};
615     if (trailz >= frac.bits - 1) {
616       result += '0';
617     } else {
618       int remainingBits{frac.bits - 1 - trailz};
619       int wholeNybbles{remainingBits / 4};
620       int lostBits{remainingBits - 4 * wholeNybbles};
621       if (wholeNybbles > 0) {
622         std::string fracHex{frac.SHIFTR(trailz + lostBits)
623                                 .IAND(frac.MASKR(4 * wholeNybbles))
624                                 .Hexadecimal()};
625         std::size_t field = wholeNybbles;
626         if (fracHex.size() < field) {
627           result += std::string(field - fracHex.size(), '0');
628         }
629         result += fracHex;
630       }
631       if (lostBits > 0) {
632         result += frac.SHIFTR(trailz)
633                       .IAND(frac.MASKR(lostBits))
634                       .SHIFTL(4 - lostBits)
635                       .Hexadecimal();
636       }
637     }
638     result += 'p';
639     int exponent = Exponent() - exponentBias;
640     result += Integer<32>{exponent}.SignedDecimal();
641     return result;
642   }
643 }
644 
645 template <typename W, int P>
646 llvm::raw_ostream &Real<W, P>::AsFortran(
647     llvm::raw_ostream &o, int kind, bool minimal) const {
648   if (IsNotANumber()) {
649     o << "(0._" << kind << "/0.)";
650   } else if (IsInfinite()) {
651     if (IsNegative()) {
652       o << "(-1._" << kind << "/0.)";
653     } else {
654       o << "(1._" << kind << "/0.)";
655     }
656   } else {
657     using B = decimal::BinaryFloatingPointNumber<P>;
658     B value{word_.template ToUInt<typename B::RawType>()};
659     char buffer[common::MaxDecimalConversionDigits(P) +
660         EXTRA_DECIMAL_CONVERSION_SPACE];
661     decimal::DecimalConversionFlags flags{}; // default: exact representation
662     if (minimal) {
663       flags = decimal::Minimize;
664     }
665     auto result{decimal::ConvertToDecimal<P>(buffer, sizeof buffer, flags,
666         static_cast<int>(sizeof buffer), decimal::RoundNearest, value)};
667     const char *p{result.str};
668     if (DEREF(p) == '-' || *p == '+') {
669       o << *p++;
670     }
671     int expo{result.decimalExponent};
672     if (*p != '0') {
673       --expo;
674     }
675     o << *p << '.' << (p + 1);
676     if (expo != 0) {
677       o << 'e' << expo;
678     }
679     o << '_' << kind;
680   }
681   return o;
682 }
683 
684 template class Real<Integer<16>, 11>;
685 template class Real<Integer<16>, 8>;
686 template class Real<Integer<32>, 24>;
687 template class Real<Integer<64>, 53>;
688 template class Real<Integer<80>, 64>;
689 template class Real<Integer<128>, 113>;
690 } // namespace Fortran::evaluate::value
691