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 // HYPOT(x,y) = SQRT(x**2 + y**2) by definition, but those squared intermediate
350 // values are susceptible to over/underflow when computed naively.
351 // Assuming that x>=y, calculate instead:
352 //   HYPOT(x,y) = SQRT(x**2 * (1+(y/x)**2))
353 //              = ABS(x) * SQRT(1+(y/x)**2)
354 template <typename W, int P>
355 ValueWithRealFlags<Real<W, P>> Real<W, P>::HYPOT(
356     const Real &y, Rounding rounding) const {
357   ValueWithRealFlags<Real> result;
358   if (IsNotANumber() || y.IsNotANumber()) {
359     result.flags.set(RealFlag::InvalidArgument);
360     result.value = NotANumber();
361   } else if (ABS().Compare(y.ABS()) == Relation::Less) {
362     return y.HYPOT(*this);
363   } else if (IsZero()) {
364     return result; // x==y==0
365   } else {
366     auto yOverX{y.Divide(*this, rounding)}; // y/x
367     bool inexact{yOverX.flags.test(RealFlag::Inexact)};
368     auto squared{yOverX.value.Multiply(yOverX.value, rounding)}; // (y/x)**2
369     inexact |= squared.flags.test(RealFlag::Inexact);
370     Real one;
371     one.Normalize(false, exponentBias, Fraction::MASKL(1)); // 1.0
372     auto sum{squared.value.Add(one, rounding)}; // 1.0 + (y/x)**2
373     inexact |= sum.flags.test(RealFlag::Inexact);
374     auto sqrt{sum.value.SQRT()};
375     inexact |= sqrt.flags.test(RealFlag::Inexact);
376     result = sqrt.value.Multiply(ABS(), rounding);
377     if (inexact) {
378       result.flags.set(RealFlag::Inexact);
379     }
380   }
381   return result;
382 }
383 
384 template <typename W, int P>
385 ValueWithRealFlags<Real<W, P>> Real<W, P>::ToWholeNumber(
386     common::RoundingMode mode) const {
387   ValueWithRealFlags<Real> result{*this};
388   if (IsNotANumber()) {
389     result.flags.set(RealFlag::InvalidArgument);
390     result.value = NotANumber();
391   } else if (IsInfinite()) {
392     result.flags.set(RealFlag::Overflow);
393   } else {
394     constexpr int noClipExponent{exponentBias + binaryPrecision - 1};
395     if (Exponent() < noClipExponent) {
396       Real adjust; // ABS(EPSILON(adjust)) == 0.5
397       adjust.Normalize(IsSignBitSet(), noClipExponent, Fraction::MASKL(1));
398       // Compute ival=(*this + adjust), losing any fractional bits; keep flags
399       result = Add(adjust, Rounding{mode});
400       result.flags.reset(RealFlag::Inexact); // result *is* exact
401       // Return (ival-adjust) with original sign in case we've generated a zero.
402       result.value =
403           result.value.Subtract(adjust, Rounding{common::RoundingMode::ToZero})
404               .value.SIGN(*this);
405     }
406   }
407   return result;
408 }
409 
410 template <typename W, int P>
411 RealFlags Real<W, P>::Normalize(bool negative, int exponent,
412     const Fraction &fraction, Rounding rounding, RoundingBits *roundingBits) {
413   int lshift{fraction.LEADZ()};
414   if (lshift == fraction.bits /* fraction is zero */ &&
415       (!roundingBits || roundingBits->empty())) {
416     // No fraction, no rounding bits -> +/-0.0
417     exponent = lshift = 0;
418   } else if (lshift < exponent) {
419     exponent -= lshift;
420   } else if (exponent > 0) {
421     lshift = exponent - 1;
422     exponent = 0;
423   } else if (lshift == 0) {
424     exponent = 1;
425   } else {
426     lshift = 0;
427   }
428   if (exponent >= maxExponent) {
429     // Infinity or overflow
430     if (rounding.mode == common::RoundingMode::TiesToEven ||
431         rounding.mode == common::RoundingMode::TiesAwayFromZero ||
432         (rounding.mode == common::RoundingMode::Up && !negative) ||
433         (rounding.mode == common::RoundingMode::Down && negative)) {
434       word_ = Word{maxExponent}.SHIFTL(significandBits); // Inf
435     } else {
436       // directed rounding: round to largest finite value rather than infinity
437       // (x86 does this, not sure whether it's standard behavior)
438       word_ = Word{word_.MASKR(word_.bits - 1)}.IBCLR(significandBits);
439     }
440     if (negative) {
441       word_ = word_.IBSET(bits - 1);
442     }
443     RealFlags flags{RealFlag::Overflow};
444     if (!fraction.IsZero()) {
445       flags.set(RealFlag::Inexact);
446     }
447     return flags;
448   }
449   word_ = Word::ConvertUnsigned(fraction).value;
450   if (lshift > 0) {
451     word_ = word_.SHIFTL(lshift);
452     if (roundingBits) {
453       for (; lshift > 0; --lshift) {
454         if (roundingBits->ShiftLeft()) {
455           word_ = word_.IBSET(lshift - 1);
456         }
457       }
458     }
459   }
460   if constexpr (isImplicitMSB) {
461     word_ = word_.IBCLR(significandBits);
462   }
463   word_ = word_.IOR(Word{exponent}.SHIFTL(significandBits));
464   if (negative) {
465     word_ = word_.IBSET(bits - 1);
466   }
467   return {};
468 }
469 
470 template <typename W, int P>
471 RealFlags Real<W, P>::Round(
472     Rounding rounding, const RoundingBits &bits, bool multiply) {
473   int origExponent{Exponent()};
474   RealFlags flags;
475   bool inexact{!bits.empty()};
476   if (inexact) {
477     flags.set(RealFlag::Inexact);
478   }
479   if (origExponent < maxExponent &&
480       bits.MustRound(rounding, IsNegative(), word_.BTEST(0) /* is odd */)) {
481     typename Fraction::ValueWithCarry sum{
482         GetFraction().AddUnsigned(Fraction{}, true)};
483     int newExponent{origExponent};
484     if (sum.carry) {
485       // The fraction was all ones before rounding; sum.value is now zero
486       sum.value = sum.value.IBSET(binaryPrecision - 1);
487       if (++newExponent >= maxExponent) {
488         flags.set(RealFlag::Overflow); // rounded away to an infinity
489       }
490     }
491     flags |= Normalize(IsNegative(), newExponent, sum.value);
492   }
493   if (inexact && origExponent == 0) {
494     // inexact subnormal input: signal Underflow unless in an x86-specific
495     // edge case
496     if (rounding.x86CompatibleBehavior && Exponent() != 0 && multiply &&
497         bits.sticky() &&
498         (bits.guard() ||
499             (rounding.mode != common::RoundingMode::Up &&
500                 rounding.mode != common::RoundingMode::Down))) {
501       // x86 edge case in which Underflow fails to signal when a subnormal
502       // inexact multiplication product rounds to a normal result when
503       // the guard bit is set or we're not using directed rounding
504     } else {
505       flags.set(RealFlag::Underflow);
506     }
507   }
508   return flags;
509 }
510 
511 template <typename W, int P>
512 void Real<W, P>::NormalizeAndRound(ValueWithRealFlags<Real> &result,
513     bool isNegative, int exponent, const Fraction &fraction, Rounding rounding,
514     RoundingBits roundingBits, bool multiply) {
515   result.flags |= result.value.Normalize(
516       isNegative, exponent, fraction, rounding, &roundingBits);
517   result.flags |= result.value.Round(rounding, roundingBits, multiply);
518 }
519 
520 inline enum decimal::FortranRounding MapRoundingMode(
521     common::RoundingMode rounding) {
522   switch (rounding) {
523   case common::RoundingMode::TiesToEven:
524     break;
525   case common::RoundingMode::ToZero:
526     return decimal::RoundToZero;
527   case common::RoundingMode::Down:
528     return decimal::RoundDown;
529   case common::RoundingMode::Up:
530     return decimal::RoundUp;
531   case common::RoundingMode::TiesAwayFromZero:
532     return decimal::RoundCompatible;
533   }
534   return decimal::RoundNearest; // dodge gcc warning about lack of result
535 }
536 
537 inline RealFlags MapFlags(decimal::ConversionResultFlags flags) {
538   RealFlags result;
539   if (flags & decimal::Overflow) {
540     result.set(RealFlag::Overflow);
541   }
542   if (flags & decimal::Inexact) {
543     result.set(RealFlag::Inexact);
544   }
545   if (flags & decimal::Invalid) {
546     result.set(RealFlag::InvalidArgument);
547   }
548   return result;
549 }
550 
551 template <typename W, int P>
552 ValueWithRealFlags<Real<W, P>> Real<W, P>::Read(
553     const char *&p, Rounding rounding) {
554   auto converted{
555       decimal::ConvertToBinary<P>(p, MapRoundingMode(rounding.mode))};
556   const auto *value{reinterpret_cast<Real<W, P> *>(&converted.binary)};
557   return {*value, MapFlags(converted.flags)};
558 }
559 
560 template <typename W, int P> std::string Real<W, P>::DumpHexadecimal() const {
561   if (IsNotANumber()) {
562     return "NaN0x"s + word_.Hexadecimal();
563   } else if (IsNegative()) {
564     return "-"s + Negate().DumpHexadecimal();
565   } else if (IsInfinite()) {
566     return "Inf"s;
567   } else if (IsZero()) {
568     return "0.0"s;
569   } else {
570     Fraction frac{GetFraction()};
571     std::string result{"0x"};
572     char intPart = '0' + frac.BTEST(frac.bits - 1);
573     result += intPart;
574     result += '.';
575     int trailz{frac.TRAILZ()};
576     if (trailz >= frac.bits - 1) {
577       result += '0';
578     } else {
579       int remainingBits{frac.bits - 1 - trailz};
580       int wholeNybbles{remainingBits / 4};
581       int lostBits{remainingBits - 4 * wholeNybbles};
582       if (wholeNybbles > 0) {
583         std::string fracHex{frac.SHIFTR(trailz + lostBits)
584                                 .IAND(frac.MASKR(4 * wholeNybbles))
585                                 .Hexadecimal()};
586         std::size_t field = wholeNybbles;
587         if (fracHex.size() < field) {
588           result += std::string(field - fracHex.size(), '0');
589         }
590         result += fracHex;
591       }
592       if (lostBits > 0) {
593         result += frac.SHIFTR(trailz)
594                       .IAND(frac.MASKR(lostBits))
595                       .SHIFTL(4 - lostBits)
596                       .Hexadecimal();
597       }
598     }
599     result += 'p';
600     int exponent = Exponent() - exponentBias;
601     result += Integer<32>{exponent}.SignedDecimal();
602     return result;
603   }
604 }
605 
606 template <typename W, int P>
607 llvm::raw_ostream &Real<W, P>::AsFortran(
608     llvm::raw_ostream &o, int kind, bool minimal) const {
609   if (IsNotANumber()) {
610     o << "(0._" << kind << "/0.)";
611   } else if (IsInfinite()) {
612     if (IsNegative()) {
613       o << "(-1._" << kind << "/0.)";
614     } else {
615       o << "(1._" << kind << "/0.)";
616     }
617   } else {
618     using B = decimal::BinaryFloatingPointNumber<P>;
619     B value{word_.template ToUInt<typename B::RawType>()};
620     char buffer[common::MaxDecimalConversionDigits(P) +
621         EXTRA_DECIMAL_CONVERSION_SPACE];
622     decimal::DecimalConversionFlags flags{}; // default: exact representation
623     if (minimal) {
624       flags = decimal::Minimize;
625     }
626     auto result{decimal::ConvertToDecimal<P>(buffer, sizeof buffer, flags,
627         static_cast<int>(sizeof buffer), decimal::RoundNearest, value)};
628     const char *p{result.str};
629     if (DEREF(p) == '-' || *p == '+') {
630       o << *p++;
631     }
632     int expo{result.decimalExponent};
633     if (*p != '0') {
634       --expo;
635     }
636     o << *p << '.' << (p + 1);
637     if (expo != 0) {
638       o << 'e' << expo;
639     }
640     o << '_' << kind;
641   }
642   return o;
643 }
644 
645 template class Real<Integer<16>, 11>;
646 template class Real<Integer<16>, 8>;
647 template class Real<Integer<32>, 24>;
648 template class Real<Integer<64>, 53>;
649 template class Real<Integer<80>, 64>;
650 template class Real<Integer<128>, 113>;
651 } // namespace Fortran::evaluate::value
652