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