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 = NegativeZero();
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 = NegativeZero();
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 = NegativeZero();
276     } else {
277       result.value = NotANumber();
278     }
279   } else if (IsInfinite()) {
280     // SQRT(+Inf) == +Inf
281     result.value = Infinity(false);
282   } else if (IsZero()) {
283     result.value = PositiveZero();
284   } else {
285     int expo{UnbiasedExponent()};
286     if (expo < -1 || expo > 1) {
287       // Reduce the range to [0.5 .. 4.0) by dividing by an integral power
288       // of four to avoid trouble with very large and very small values
289       // (esp. truncation of subnormals).
290       // SQRT(2**(2a) * x) = SQRT(2**(2a)) * SQRT(x) = 2**a * SQRT(x)
291       Real scaled;
292       int adjust{expo / 2};
293       scaled.Normalize(false, expo - 2 * adjust + exponentBias, GetFraction());
294       result = scaled.SQRT(rounding);
295       result.value.Normalize(false,
296           result.value.UnbiasedExponent() + adjust + exponentBias,
297           result.value.GetFraction());
298       return result;
299     }
300     // Compute the square root of the reduced value with the slow but
301     // reliable bit-at-a-time method.  Start with a clear significand and
302     // half of the unbiased exponent, and then try to set significand bits
303     // in descending order of magnitude without exceeding the exact result.
304     expo = expo / 2 + exponentBias;
305     result.value.Normalize(false, expo, Fraction::MASKL(1));
306     Real initialSq{result.value.Multiply(result.value).value};
307     if (Compare(initialSq) == Relation::Less) {
308       // Initial estimate is too large; this can happen for values just
309       // under 1.0.
310       --expo;
311       result.value.Normalize(false, expo, Fraction::MASKL(1));
312     }
313     for (int bit{significandBits - 1}; bit >= 0; --bit) {
314       Word word{result.value.word_};
315       result.value.word_ = word.IBSET(bit);
316       auto squared{result.value.Multiply(result.value, rounding)};
317       if (squared.flags.test(RealFlag::Overflow) ||
318           squared.flags.test(RealFlag::Underflow) ||
319           Compare(squared.value) == Relation::Less) {
320         result.value.word_ = word;
321       }
322     }
323     // The computed square root has a square that's not greater than the
324     // original argument.  Check this square against the square of the next
325     // larger Real and return that one if its square is closer in magnitude to
326     // the original argument.
327     Real resultSq{result.value.Multiply(result.value).value};
328     Real diff{Subtract(resultSq).value.ABS()};
329     if (diff.IsZero()) {
330       return result; // exact
331     }
332     Real ulp;
333     ulp.Normalize(false, expo, Fraction::MASKR(1));
334     Real nextAfter{result.value.Add(ulp).value};
335     auto nextAfterSq{nextAfter.Multiply(nextAfter)};
336     if (!nextAfterSq.flags.test(RealFlag::Overflow) &&
337         !nextAfterSq.flags.test(RealFlag::Underflow)) {
338       Real nextAfterDiff{Subtract(nextAfterSq.value).value.ABS()};
339       if (nextAfterDiff.Compare(diff) == Relation::Less) {
340         result.value = nextAfter;
341         if (nextAfterDiff.IsZero()) {
342           return result; // exact
343         }
344       }
345     }
346     result.flags.set(RealFlag::Inexact);
347   }
348   return result;
349 }
350 
351 template <typename W, int P>
352 ValueWithRealFlags<Real<W, P>> Real<W, P>::NEAREST(bool upward) const {
353   ValueWithRealFlags<Real> result;
354   if (IsFinite()) {
355     Fraction fraction{GetFraction()};
356     int expo{Exponent()};
357     Fraction one{1};
358     Fraction nearest;
359     bool isNegative{IsNegative()};
360     if (upward != isNegative) { // upward in magnitude
361       auto next{fraction.AddUnsigned(one)};
362       if (next.carry) {
363         ++expo;
364         nearest = Fraction::Least(); // MSB only
365       } else {
366         nearest = next.value;
367       }
368     } else { // downward in magnitude
369       if (IsZero()) {
370         nearest = 1; // smallest magnitude negative subnormal
371         isNegative = !isNegative;
372       } else {
373         auto sub1{fraction.SubtractSigned(one)};
374         if (sub1.overflow) {
375           nearest = Fraction{0}.NOT();
376           --expo;
377         } else {
378           nearest = sub1.value;
379         }
380       }
381     }
382     result.flags = result.value.Normalize(isNegative, expo, nearest);
383   } else {
384     result.flags.set(RealFlag::InvalidArgument);
385     result.value = *this;
386   }
387   return result;
388 }
389 
390 // HYPOT(x,y) = SQRT(x**2 + y**2) by definition, but those squared intermediate
391 // values are susceptible to over/underflow when computed naively.
392 // Assuming that x>=y, calculate instead:
393 //   HYPOT(x,y) = SQRT(x**2 * (1+(y/x)**2))
394 //              = ABS(x) * SQRT(1+(y/x)**2)
395 template <typename W, int P>
396 ValueWithRealFlags<Real<W, P>> Real<W, P>::HYPOT(
397     const Real &y, Rounding rounding) const {
398   ValueWithRealFlags<Real> result;
399   if (IsNotANumber() || y.IsNotANumber()) {
400     result.flags.set(RealFlag::InvalidArgument);
401     result.value = NotANumber();
402   } else if (ABS().Compare(y.ABS()) == Relation::Less) {
403     return y.HYPOT(*this);
404   } else if (IsZero()) {
405     return result; // x==y==0
406   } else {
407     auto yOverX{y.Divide(*this, rounding)}; // y/x
408     bool inexact{yOverX.flags.test(RealFlag::Inexact)};
409     auto squared{yOverX.value.Multiply(yOverX.value, rounding)}; // (y/x)**2
410     inexact |= squared.flags.test(RealFlag::Inexact);
411     Real one;
412     one.Normalize(false, exponentBias, Fraction::MASKL(1)); // 1.0
413     auto sum{squared.value.Add(one, rounding)}; // 1.0 + (y/x)**2
414     inexact |= sum.flags.test(RealFlag::Inexact);
415     auto sqrt{sum.value.SQRT()};
416     inexact |= sqrt.flags.test(RealFlag::Inexact);
417     result = sqrt.value.Multiply(ABS(), rounding);
418     if (inexact) {
419       result.flags.set(RealFlag::Inexact);
420     }
421   }
422   return result;
423 }
424 
425 // MOD(x,y) = x - AINT(x/y)*y
426 template <typename W, int P>
427 ValueWithRealFlags<Real<W, P>> Real<W, P>::MOD(
428     const Real &y, Rounding rounding) const {
429   ValueWithRealFlags<Real> result;
430   Real quotient{Divide(y, rounding).AccumulateFlags(result.flags)};
431   Real toInt{quotient.ToWholeNumber(common::RoundingMode::ToZero)
432                  .AccumulateFlags(result.flags)};
433   Real product{toInt.Multiply(y, rounding).AccumulateFlags(result.flags)};
434   result.value = Subtract(product, rounding).AccumulateFlags(result.flags);
435   return result;
436 }
437 
438 // MODULO(x,y) = x - FLOOR(x/y)*y
439 template <typename W, int P>
440 ValueWithRealFlags<Real<W, P>> Real<W, P>::MODULO(
441     const Real &y, Rounding rounding) const {
442   ValueWithRealFlags<Real> result;
443   Real quotient{Divide(y, rounding).AccumulateFlags(result.flags)};
444   Real toInt{quotient.ToWholeNumber(common::RoundingMode::Down)
445                  .AccumulateFlags(result.flags)};
446   Real product{toInt.Multiply(y, rounding).AccumulateFlags(result.flags)};
447   result.value = Subtract(product, rounding).AccumulateFlags(result.flags);
448   return result;
449 }
450 
451 template <typename W, int P>
452 ValueWithRealFlags<Real<W, P>> Real<W, P>::DIM(
453     const Real &y, Rounding rounding) const {
454   ValueWithRealFlags<Real> result;
455   if (IsNotANumber() || y.IsNotANumber()) {
456     result.flags.set(RealFlag::InvalidArgument);
457     result.value = NotANumber();
458   } else if (Compare(y) == Relation::Greater) {
459     result = Subtract(y, rounding);
460   } else {
461     // result is already zero
462   }
463   return result;
464 }
465 
466 template <typename W, int P>
467 ValueWithRealFlags<Real<W, P>> Real<W, P>::ToWholeNumber(
468     common::RoundingMode mode) const {
469   ValueWithRealFlags<Real> result{*this};
470   if (IsNotANumber()) {
471     result.flags.set(RealFlag::InvalidArgument);
472     result.value = NotANumber();
473   } else if (IsInfinite()) {
474     result.flags.set(RealFlag::Overflow);
475   } else {
476     constexpr int noClipExponent{exponentBias + binaryPrecision - 1};
477     if (Exponent() < noClipExponent) {
478       Real adjust; // ABS(EPSILON(adjust)) == 0.5
479       adjust.Normalize(IsSignBitSet(), noClipExponent, Fraction::MASKL(1));
480       // Compute ival=(*this + adjust), losing any fractional bits; keep flags
481       result = Add(adjust, Rounding{mode});
482       result.flags.reset(RealFlag::Inexact); // result *is* exact
483       // Return (ival-adjust) with original sign in case we've generated a zero.
484       result.value =
485           result.value.Subtract(adjust, Rounding{common::RoundingMode::ToZero})
486               .value.SIGN(*this);
487     }
488   }
489   return result;
490 }
491 
492 template <typename W, int P>
493 RealFlags Real<W, P>::Normalize(bool negative, int exponent,
494     const Fraction &fraction, Rounding rounding, RoundingBits *roundingBits) {
495   int lshift{fraction.LEADZ()};
496   if (lshift == fraction.bits /* fraction is zero */ &&
497       (!roundingBits || roundingBits->empty())) {
498     // No fraction, no rounding bits -> +/-0.0
499     exponent = lshift = 0;
500   } else if (lshift < exponent) {
501     exponent -= lshift;
502   } else if (exponent > 0) {
503     lshift = exponent - 1;
504     exponent = 0;
505   } else if (lshift == 0) {
506     exponent = 1;
507   } else {
508     lshift = 0;
509   }
510   if (exponent >= maxExponent) {
511     // Infinity or overflow
512     if (rounding.mode == common::RoundingMode::TiesToEven ||
513         rounding.mode == common::RoundingMode::TiesAwayFromZero ||
514         (rounding.mode == common::RoundingMode::Up && !negative) ||
515         (rounding.mode == common::RoundingMode::Down && negative)) {
516       word_ = Word{maxExponent}.SHIFTL(significandBits); // Inf
517     } else {
518       // directed rounding: round to largest finite value rather than infinity
519       // (x86 does this, not sure whether it's standard behavior)
520       word_ = Word{word_.MASKR(word_.bits - 1)}.IBCLR(significandBits);
521     }
522     if (negative) {
523       word_ = word_.IBSET(bits - 1);
524     }
525     RealFlags flags{RealFlag::Overflow};
526     if (!fraction.IsZero()) {
527       flags.set(RealFlag::Inexact);
528     }
529     return flags;
530   }
531   word_ = Word::ConvertUnsigned(fraction).value;
532   if (lshift > 0) {
533     word_ = word_.SHIFTL(lshift);
534     if (roundingBits) {
535       for (; lshift > 0; --lshift) {
536         if (roundingBits->ShiftLeft()) {
537           word_ = word_.IBSET(lshift - 1);
538         }
539       }
540     }
541   }
542   if constexpr (isImplicitMSB) {
543     word_ = word_.IBCLR(significandBits);
544   }
545   word_ = word_.IOR(Word{exponent}.SHIFTL(significandBits));
546   if (negative) {
547     word_ = word_.IBSET(bits - 1);
548   }
549   return {};
550 }
551 
552 template <typename W, int P>
553 RealFlags Real<W, P>::Round(
554     Rounding rounding, const RoundingBits &bits, bool multiply) {
555   int origExponent{Exponent()};
556   RealFlags flags;
557   bool inexact{!bits.empty()};
558   if (inexact) {
559     flags.set(RealFlag::Inexact);
560   }
561   if (origExponent < maxExponent &&
562       bits.MustRound(rounding, IsNegative(), word_.BTEST(0) /* is odd */)) {
563     typename Fraction::ValueWithCarry sum{
564         GetFraction().AddUnsigned(Fraction{}, true)};
565     int newExponent{origExponent};
566     if (sum.carry) {
567       // The fraction was all ones before rounding; sum.value is now zero
568       sum.value = sum.value.IBSET(binaryPrecision - 1);
569       if (++newExponent >= maxExponent) {
570         flags.set(RealFlag::Overflow); // rounded away to an infinity
571       }
572     }
573     flags |= Normalize(IsNegative(), newExponent, sum.value);
574   }
575   if (inexact && origExponent == 0) {
576     // inexact subnormal input: signal Underflow unless in an x86-specific
577     // edge case
578     if (rounding.x86CompatibleBehavior && Exponent() != 0 && multiply &&
579         bits.sticky() &&
580         (bits.guard() ||
581             (rounding.mode != common::RoundingMode::Up &&
582                 rounding.mode != common::RoundingMode::Down))) {
583       // x86 edge case in which Underflow fails to signal when a subnormal
584       // inexact multiplication product rounds to a normal result when
585       // the guard bit is set or we're not using directed rounding
586     } else {
587       flags.set(RealFlag::Underflow);
588     }
589   }
590   return flags;
591 }
592 
593 template <typename W, int P>
594 void Real<W, P>::NormalizeAndRound(ValueWithRealFlags<Real> &result,
595     bool isNegative, int exponent, const Fraction &fraction, Rounding rounding,
596     RoundingBits roundingBits, bool multiply) {
597   result.flags |= result.value.Normalize(
598       isNegative, exponent, fraction, rounding, &roundingBits);
599   result.flags |= result.value.Round(rounding, roundingBits, multiply);
600 }
601 
602 inline enum decimal::FortranRounding MapRoundingMode(
603     common::RoundingMode rounding) {
604   switch (rounding) {
605   case common::RoundingMode::TiesToEven:
606     break;
607   case common::RoundingMode::ToZero:
608     return decimal::RoundToZero;
609   case common::RoundingMode::Down:
610     return decimal::RoundDown;
611   case common::RoundingMode::Up:
612     return decimal::RoundUp;
613   case common::RoundingMode::TiesAwayFromZero:
614     return decimal::RoundCompatible;
615   }
616   return decimal::RoundNearest; // dodge gcc warning about lack of result
617 }
618 
619 inline RealFlags MapFlags(decimal::ConversionResultFlags flags) {
620   RealFlags result;
621   if (flags & decimal::Overflow) {
622     result.set(RealFlag::Overflow);
623   }
624   if (flags & decimal::Inexact) {
625     result.set(RealFlag::Inexact);
626   }
627   if (flags & decimal::Invalid) {
628     result.set(RealFlag::InvalidArgument);
629   }
630   return result;
631 }
632 
633 template <typename W, int P>
634 ValueWithRealFlags<Real<W, P>> Real<W, P>::Read(
635     const char *&p, Rounding rounding) {
636   auto converted{
637       decimal::ConvertToBinary<P>(p, MapRoundingMode(rounding.mode))};
638   const auto *value{reinterpret_cast<Real<W, P> *>(&converted.binary)};
639   return {*value, MapFlags(converted.flags)};
640 }
641 
642 template <typename W, int P> std::string Real<W, P>::DumpHexadecimal() const {
643   if (IsNotANumber()) {
644     return "NaN0x"s + word_.Hexadecimal();
645   } else if (IsNegative()) {
646     return "-"s + Negate().DumpHexadecimal();
647   } else if (IsInfinite()) {
648     return "Inf"s;
649   } else if (IsZero()) {
650     return "0.0"s;
651   } else {
652     Fraction frac{GetFraction()};
653     std::string result{"0x"};
654     char intPart = '0' + frac.BTEST(frac.bits - 1);
655     result += intPart;
656     result += '.';
657     int trailz{frac.TRAILZ()};
658     if (trailz >= frac.bits - 1) {
659       result += '0';
660     } else {
661       int remainingBits{frac.bits - 1 - trailz};
662       int wholeNybbles{remainingBits / 4};
663       int lostBits{remainingBits - 4 * wholeNybbles};
664       if (wholeNybbles > 0) {
665         std::string fracHex{frac.SHIFTR(trailz + lostBits)
666                                 .IAND(frac.MASKR(4 * wholeNybbles))
667                                 .Hexadecimal()};
668         std::size_t field = wholeNybbles;
669         if (fracHex.size() < field) {
670           result += std::string(field - fracHex.size(), '0');
671         }
672         result += fracHex;
673       }
674       if (lostBits > 0) {
675         result += frac.SHIFTR(trailz)
676                       .IAND(frac.MASKR(lostBits))
677                       .SHIFTL(4 - lostBits)
678                       .Hexadecimal();
679       }
680     }
681     result += 'p';
682     int exponent = Exponent() - exponentBias;
683     if (intPart == '0') {
684       exponent += 1;
685     }
686     result += Integer<32>{exponent}.SignedDecimal();
687     return result;
688   }
689 }
690 
691 template <typename W, int P>
692 llvm::raw_ostream &Real<W, P>::AsFortran(
693     llvm::raw_ostream &o, int kind, bool minimal) const {
694   if (IsNotANumber()) {
695     o << "(0._" << kind << "/0.)";
696   } else if (IsInfinite()) {
697     if (IsNegative()) {
698       o << "(-1._" << kind << "/0.)";
699     } else {
700       o << "(1._" << kind << "/0.)";
701     }
702   } else {
703     using B = decimal::BinaryFloatingPointNumber<P>;
704     B value{word_.template ToUInt<typename B::RawType>()};
705     char buffer[common::MaxDecimalConversionDigits(P) +
706         EXTRA_DECIMAL_CONVERSION_SPACE];
707     decimal::DecimalConversionFlags flags{}; // default: exact representation
708     if (minimal) {
709       flags = decimal::Minimize;
710     }
711     auto result{decimal::ConvertToDecimal<P>(buffer, sizeof buffer, flags,
712         static_cast<int>(sizeof buffer), decimal::RoundNearest, value)};
713     const char *p{result.str};
714     if (DEREF(p) == '-' || *p == '+') {
715       o << *p++;
716     }
717     int expo{result.decimalExponent};
718     if (*p != '0') {
719       --expo;
720     }
721     o << *p << '.' << (p + 1);
722     if (expo != 0) {
723       o << 'e' << expo;
724     }
725     o << '_' << kind;
726   }
727   return o;
728 }
729 
730 // 16.9.180
731 template <typename W, int P> Real<W, P> Real<W, P>::RRSPACING() const {
732   if (IsNotANumber()) {
733     return *this;
734   } else if (IsInfinite()) {
735     return NotANumber();
736   } else {
737     Real result;
738     result.Normalize(false, binaryPrecision + exponentBias - 1, GetFraction());
739     return result;
740   }
741 }
742 
743 // 16.9.180
744 template <typename W, int P> Real<W, P> Real<W, P>::SPACING() const {
745   if (IsNotANumber()) {
746     return *this;
747   } else if (IsInfinite()) {
748     return NotANumber();
749   } else if (IsZero()) {
750     return TINY();
751   } else {
752     Real result;
753     result.Normalize(
754         false, Exponent() - binaryPrecision + 1, Fraction::MASKL(1));
755     return result;
756   }
757 }
758 
759 template class Real<Integer<16>, 11>;
760 template class Real<Integer<16>, 8>;
761 template class Real<Integer<32>, 24>;
762 template class Real<Integer<64>, 53>;
763 template class Real<Integer<80>, 64>;
764 template class Real<Integer<128>, 113>;
765 } // namespace Fortran::evaluate::value
766