flang/lib/Evaluate/real.cpp

   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 && expo > 1) {
 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 in the standard; unfortunately, this definition
 405 // can be pretty inaccurate when x is much larger than y in magnitude due to
 406 // cancellation.  Implement instead with (essentially) arbitrary precision
 407 // long division, discarding the quotient and returning the remainder.
 408 // See runtime/numeric.cpp for more details.
 409 template <typename W, int P>
 410 ValueWithRealFlags<Real<W, P>> Real<W, P>::MOD(
 411     const Real &p, Rounding rounding) const {
 412   ValueWithRealFlags<Real> result;
 413   if (IsNotANumber() || p.IsNotANumber() || IsInfinite()) {
 414     result.flags.set(RealFlag::InvalidArgument);
 415     result.value = NotANumber();
 416   } else if (p.IsZero()) {
 417     result.flags.set(RealFlag::DivideByZero);
 418     result.value = NotANumber();
 419   } else if (p.IsInfinite()) {
 420     result.value = *this;
 421   } else {
 422     result.value = ABS();
 423     auto pAbs{p.ABS()};
 424     Real half, adj;
 425     half.Normalize(false, exponentBias - 1, Fraction::MASKL(1)); // 0.5
 426     for (adj.Normalize(false, Exponent(), pAbs.GetFraction());
 427          result.value.Compare(pAbs) != Relation::Less;
 428          adj = adj.Multiply(half).value) {
 429       if (result.value.Compare(adj) != Relation::Less) {
 430         result.value =
 431             result.value.Subtract(adj, rounding).AccumulateFlags(result.flags);
 432         if (result.value.IsZero()) {
 433           break;
 434         }
 435       }
 436     }
 437     if (IsNegative()) {
 438       result.value = result.value.Negate();
 439     }
 440   }
 441   return result;
 442 }
 443
 444 // MODULO(x,y) = x - FLOOR(x/y)*y in the standard; here, it is defined
 445 // in terms of MOD() with adjustment of the result.
 446 template <typename W, int P>
 447 ValueWithRealFlags<Real<W, P>> Real<W, P>::MODULO(
 448     const Real &p, Rounding rounding) const {
 449   ValueWithRealFlags<Real> result{MOD(p, rounding)};
 450   if (IsNegative() != p.IsNegative()) {
 451     if (result.value.IsZero()) {
 452       result.value = result.value.Negate();
 453     } else {
 454       result.value =
 455           result.value.Add(p, rounding).AccumulateFlags(result.flags);
 456     }
 457   }
 458   return result;
 459 }
 460
 461 template <typename W, int P>
 462 ValueWithRealFlags<Real<W, P>> Real<W, P>::DIM(
 463     const Real &y, Rounding rounding) const {
 464   ValueWithRealFlags<Real> result;
 465   if (IsNotANumber() || y.IsNotANumber()) {
 466     result.flags.set(RealFlag::InvalidArgument);
 467     result.value = NotANumber();
 468   } else if (Compare(y) == Relation::Greater) {
 469     result = Subtract(y, rounding);
 470   } else {
 471     // result is already zero
 472   }
 473   return result;
 474 }
 475
 476 template <typename W, int P>
 477 ValueWithRealFlags<Real<W, P>> Real<W, P>::ToWholeNumber(
 478     common::RoundingMode mode) const {
 479   ValueWithRealFlags<Real> result{*this};
 480   if (IsNotANumber()) {
 481     result.flags.set(RealFlag::InvalidArgument);
 482     result.value = NotANumber();
 483   } else if (IsInfinite()) {
 484     result.flags.set(RealFlag::Overflow);
 485   } else {
 486     constexpr int noClipExponent{exponentBias + binaryPrecision - 1};
 487     if (Exponent() < noClipExponent) {
 488       Real adjust; // ABS(EPSILON(adjust)) == 0.5
 489       adjust.Normalize(IsSignBitSet(), noClipExponent, Fraction::MASKL(1));
 490       // Compute ival=(*this + adjust), losing any fractional bits; keep flags
 491       result = Add(adjust, Rounding{mode});
 492       result.flags.reset(RealFlag::Inexact); // result *is* exact
 493       // Return (ival-adjust) with original sign in case we've generated a zero.
 494       result.value =
 495           result.value.Subtract(adjust, Rounding{common::RoundingMode::ToZero})
 496               .value.SIGN(*this);
 497     }
 498   }
 499   return result;
 500 }
 501
 502 template <typename W, int P>
 503 RealFlags Real<W, P>::Normalize(bool negative, int exponent,
 504     const Fraction &fraction, Rounding rounding, RoundingBits *roundingBits) {
 505   int lshift{fraction.LEADZ()};
 506   if (lshift == fraction.bits /* fraction is zero */ &&
 507       (!roundingBits || roundingBits->empty())) {
 508     // No fraction, no rounding bits -> +/-0.0
 509     exponent = lshift = 0;
 510   } else if (lshift < exponent) {
 511     exponent -= lshift;
 512   } else if (exponent > 0) {
 513     lshift = exponent - 1;
 514     exponent = 0;
 515   } else if (lshift == 0) {
 516     exponent = 1;
 517   } else {
 518     lshift = 0;
 519   }
 520   if (exponent >= maxExponent) {
 521     // Infinity or overflow
 522     if (rounding.mode == common::RoundingMode::TiesToEven ||
 523         rounding.mode == common::RoundingMode::TiesAwayFromZero ||
 524         (rounding.mode == common::RoundingMode::Up && !negative) ||
 525         (rounding.mode == common::RoundingMode::Down && negative)) {
 526       word_ = Word{maxExponent}.SHIFTL(significandBits); // Inf
 527     } else {
 528       // directed rounding: round to largest finite value rather than infinity
 529       // (x86 does this, not sure whether it's standard behavior)
 530       word_ = Word{word_.MASKR(word_.bits - 1)}.IBCLR(significandBits);
 531     }
 532     if (negative) {
 533       word_ = word_.IBSET(bits - 1);
 534     }
 535     RealFlags flags{RealFlag::Overflow};
 536     if (!fraction.IsZero()) {
 537       flags.set(RealFlag::Inexact);
 538     }
 539     return flags;
 540   }
 541   word_ = Word::ConvertUnsigned(fraction).value;
 542   if (lshift > 0) {
 543     word_ = word_.SHIFTL(lshift);
 544     if (roundingBits) {
 545       for (; lshift > 0; --lshift) {
 546         if (roundingBits->ShiftLeft()) {
 547           word_ = word_.IBSET(lshift - 1);
 548         }
 549       }
 550     }
 551   }
 552   if constexpr (isImplicitMSB) {
 553     word_ = word_.IBCLR(significandBits);
 554   }
 555   word_ = word_.IOR(Word{exponent}.SHIFTL(significandBits));
 556   if (negative) {
 557     word_ = word_.IBSET(bits - 1);
 558   }
 559   return {};
 560 }
 561
 562 template <typename W, int P>
 563 RealFlags Real<W, P>::Round(
 564     Rounding rounding, const RoundingBits &bits, bool multiply) {
 565   int origExponent{Exponent()};
 566   RealFlags flags;
 567   bool inexact{!bits.empty()};
 568   if (inexact) {
 569     flags.set(RealFlag::Inexact);
 570   }
 571   if (origExponent < maxExponent &&
 572       bits.MustRound(rounding, IsNegative(), word_.BTEST(0) /* is odd */)) {
 573     typename Fraction::ValueWithCarry sum{
 574         GetFraction().AddUnsigned(Fraction{}, true)};
 575     int newExponent{origExponent};
 576     if (sum.carry) {
 577       // The fraction was all ones before rounding; sum.value is now zero
 578       sum.value = sum.value.IBSET(binaryPrecision - 1);
 579       if (++newExponent >= maxExponent) {
 580         flags.set(RealFlag::Overflow); // rounded away to an infinity
 581       }
 582     }
 583     flags |= Normalize(IsNegative(), newExponent, sum.value);
 584   }
 585   if (inexact && origExponent == 0) {
 586     // inexact subnormal input: signal Underflow unless in an x86-specific
 587     // edge case
 588     if (rounding.x86CompatibleBehavior && Exponent() != 0 && multiply &&
 589         bits.sticky() &&
 590         (bits.guard() ||
 591             (rounding.mode != common::RoundingMode::Up &&
 592                 rounding.mode != common::RoundingMode::Down))) {
 593       // x86 edge case in which Underflow fails to signal when a subnormal
 594       // inexact multiplication product rounds to a normal result when
 595       // the guard bit is set or we're not using directed rounding
 596     } else {
 597       flags.set(RealFlag::Underflow);
 598     }
 599   }
 600   return flags;
 601 }
 602
 603 template <typename W, int P>
 604 void Real<W, P>::NormalizeAndRound(ValueWithRealFlags<Real> &result,
 605     bool isNegative, int exponent, const Fraction &fraction, Rounding rounding,
 606     RoundingBits roundingBits, bool multiply) {
 607   result.flags |= result.value.Normalize(
 608       isNegative, exponent, fraction, rounding, &roundingBits);
 609   result.flags |= result.value.Round(rounding, roundingBits, multiply);
 610 }
 611
 612 inline enum decimal::FortranRounding MapRoundingMode(
 613     common::RoundingMode rounding) {
 614   switch (rounding) {
 615   case common::RoundingMode::TiesToEven:
 616     break;
 617   case common::RoundingMode::ToZero:
 618     return decimal::RoundToZero;
 619   case common::RoundingMode::Down:
 620     return decimal::RoundDown;
 621   case common::RoundingMode::Up:
 622     return decimal::RoundUp;
 623   case common::RoundingMode::TiesAwayFromZero:
 624     return decimal::RoundCompatible;
 625   }
 626   return decimal::RoundNearest; // dodge gcc warning about lack of result
 627 }
 628
 629 inline RealFlags MapFlags(decimal::ConversionResultFlags flags) {
 630   RealFlags result;
 631   if (flags & decimal::Overflow) {
 632     result.set(RealFlag::Overflow);
 633   }
 634   if (flags & decimal::Inexact) {
 635     result.set(RealFlag::Inexact);
 636   }
 637   if (flags & decimal::Invalid) {
 638     result.set(RealFlag::InvalidArgument);
 639   }
 640   return result;
 641 }
 642
 643 template <typename W, int P>
 644 ValueWithRealFlags<Real<W, P>> Real<W, P>::Read(
 645     const char *&p, Rounding rounding) {
 646   auto converted{
 647       decimal::ConvertToBinary<P>(p, MapRoundingMode(rounding.mode))};
 648   const auto *value{reinterpret_cast<Real<W, P> *>(&converted.binary)};
 649   return {*value, MapFlags(converted.flags)};
 650 }
 651
 652 template <typename W, int P> std::string Real<W, P>::DumpHexadecimal() const {
 653   if (IsNotANumber()) {
 654     return "NaN0x"s + word_.Hexadecimal();
 655   } else if (IsNegative()) {
 656     return "-"s + Negate().DumpHexadecimal();
 657   } else if (IsInfinite()) {
 658     return "Inf"s;
 659   } else if (IsZero()) {
 660     return "0.0"s;
 661   } else {
 662     Fraction frac{GetFraction()};
 663     std::string result{"0x"};
 664     char intPart = '0' + frac.BTEST(frac.bits - 1);
 665     result += intPart;
 666     result += '.';
 667     int trailz{frac.TRAILZ()};
 668     if (trailz >= frac.bits - 1) {
 669       result += '0';
 670     } else {
 671       int remainingBits{frac.bits - 1 - trailz};
 672       int wholeNybbles{remainingBits / 4};
 673       int lostBits{remainingBits - 4 * wholeNybbles};
 674       if (wholeNybbles > 0) {
 675         std::string fracHex{frac.SHIFTR(trailz + lostBits)
 676                                 .IAND(frac.MASKR(4 * wholeNybbles))
 677                                 .Hexadecimal()};
 678         std::size_t field = wholeNybbles;
 679         if (fracHex.size() < field) {
 680           result += std::string(field - fracHex.size(), '0');
 681         }
 682         result += fracHex;
 683       }
 684       if (lostBits > 0) {
 685         result += frac.SHIFTR(trailz)
 686                       .IAND(frac.MASKR(lostBits))
 687                       .SHIFTL(4 - lostBits)
 688                       .Hexadecimal();
 689       }
 690     }
 691     result += 'p';
 692     int exponent = Exponent() - exponentBias;
 693     if (intPart == '0') {
 694       exponent += 1;
 695     }
 696     result += Integer<32>{exponent}.SignedDecimal();
 697     return result;
 698   }
 699 }
 700
 701 template <typename W, int P>
 702 llvm::raw_ostream &Real<W, P>::AsFortran(
 703     llvm::raw_ostream &o, int kind, bool minimal) const {
 704   if (IsNotANumber()) {
 705     o << "(0._" << kind << "/0.)";
 706   } else if (IsInfinite()) {
 707     if (IsNegative()) {
 708       o << "(-1._" << kind << "/0.)";
 709     } else {
 710       o << "(1._" << kind << "/0.)";
 711     }
 712   } else {
 713     using B = decimal::BinaryFloatingPointNumber<P>;
 714     B value{word_.template ToUInt<typename B::RawType>()};
 715     char buffer[common::MaxDecimalConversionDigits(P) +
 716         EXTRA_DECIMAL_CONVERSION_SPACE];
 717     decimal::DecimalConversionFlags flags{}; // default: exact representation
 718     if (minimal) {
 719       flags = decimal::Minimize;
 720     }
 721     auto result{decimal::ConvertToDecimal<P>(buffer, sizeof buffer, flags,
 722         static_cast<int>(sizeof buffer), decimal::RoundNearest, value)};
 723     const char *p{result.str};
 724     if (DEREF(p) == '-' || *p == '+') {
 725       o << *p++;
 726     }
 727     int expo{result.decimalExponent};
 728     if (*p != '0') {
 729       --expo;
 730     }
 731     o << *p << '.' << (p + 1);
 732     if (expo != 0) {
 733       o << 'e' << expo;
 734     }
 735     o << '_' << kind;
 736   }
 737   return o;
 738 }
 739
 740 // 16.9.180
 741 template <typename W, int P> Real<W, P> Real<W, P>::RRSPACING() const {
 742   if (IsNotANumber()) {
 743     return *this;
 744   } else if (IsInfinite()) {
 745     return NotANumber();
 746   } else {
 747     Real result;
 748     result.Normalize(false, binaryPrecision + exponentBias - 1, GetFraction());
 749     return result;
 750   }
 751 }
 752
 753 // 16.9.180
 754 template <typename W, int P> Real<W, P> Real<W, P>::SPACING() const {
 755   if (IsNotANumber()) {
 756     return *this;
 757   } else if (IsInfinite()) {
 758     return NotANumber();
 759   } else if (IsZero() || IsSubnormal()) {
 760     return TINY(); // mandated by standard
 761   } else {
 762     Real result;
 763     result.Normalize(false, Exponent(), Fraction::MASKR(1));
 764     return result.IsZero() ? TINY() : result;
 765   }
 766 }
 767
 768 // 16.9.171
 769 template <typename W, int P>
 770 Real<W, P> Real<W, P>::SET_EXPONENT(std::int64_t expo) const {
 771   if (IsNotANumber()) {
 772     return *this;
 773   } else if (IsInfinite()) {
 774     return NotANumber();
 775   } else if (IsZero()) {
 776     return *this;
 777   } else {
 778     return SCALE(Integer<64>(expo - UnbiasedExponent() - 1)).value;
 779   }
 780 }
 781
 782 // 16.9.171
 783 template <typename W, int P> Real<W, P> Real<W, P>::FRACTION() const {
 784   return SET_EXPONENT(0);
 785 }
 786
 787 template class Real<Integer<16>, 11>;
 788 template class Real<Integer<16>, 8>;
 789 template class Real<Integer<32>, 24>;
 790 template class Real<Integer<64>, 53>;
 791 template class Real<Integer<80>, 64>;
 792 template class Real<Integer<128>, 113>;
 793 } // namespace Fortran::evaluate::value