add precomputed mathematical constants and specialize the second division residual
.NET Test / .NET tests (push) Successful in 2m23s
.NET Test / .NET tests (push) Successful in 2m23s
This commit is contained in:
@@ -6,6 +6,303 @@ namespace Just.PreciseMath.Tests;
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public class DoubleDoubleArithmeticTests
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{
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[Theory]
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[InlineData(6.0, 2.0, 8.0, 4.0, 12.0, 3.0)]
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[InlineData(-6.0, 2.0, -4.0, -8.0, -12.0, -3.0)]
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[InlineData(6.0, -2.0, 4.0, 8.0, -12.0, -3.0)]
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[InlineData(-6.0, -2.0, -8.0, -4.0, 12.0, 3.0)]
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[InlineData(1.5, 0.5, 2.0, 1.0, 0.75, 3.0)]
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[InlineData(-1.5, 0.5, -1.0, -2.0, -0.75, -3.0)]
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[InlineData(0.75, 1.5, 2.25, -0.75, 1.125, 0.5)]
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[InlineData(0.0, 2.0, 2.0, -2.0, 0.0, 0.0)]
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[InlineData(2.0, 2.0, 4.0, 0.0, 4.0, 1.0)]
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public void BasicArithmeticHasExactDyadicResults(double left, double right,
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double sum, double difference, double product, double quotient)
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{
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// Small integers and binary fractions: each expected result is an exact
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// rational representable in binary64, so no accuracy tolerance is needed.
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DoubleDouble a = new(left);
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DoubleDouble b = new(right);
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CheckBoundary(a + b, sum, 0.0);
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CheckBoundary(a + right, sum, 0.0);
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CheckBoundary(left + b, sum, 0.0);
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CheckBoundary(a - b, difference, 0.0);
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CheckBoundary(a - right, difference, 0.0);
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CheckBoundary(left - b, difference, 0.0);
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CheckBoundary(a * b, product, 0.0);
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CheckBoundary(a * right, product, 0.0);
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CheckBoundary(left * b, product, 0.0);
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CheckBoundary(a / b, quotient, 0.0);
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CheckBoundary(a / right, quotient, 0.0);
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CheckBoundary(left / b, quotient, 0.0);
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}
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[Theory]
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[InlineData(30.0, 0.5235987755982989, -5.360408832255455e-17)]
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[InlineData(45.0, 0.7853981633974483, 3.061616997868383e-17)]
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[InlineData(60.0, 1.0471975511965979, -1.072081766451091e-16)]
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[InlineData(90.0, 1.5707963267948966, 6.123233995736766e-17)]
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[InlineData(180.0, 3.141592653589793, 1.2246467991473532e-16)]
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[InlineData(360.0, 6.283185307179586, 2.4492935982947064e-16)]
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public void AngleConversionConstantsRetainExtendedPrecision(double degrees, double radiansHigh, double radiansLow)
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{
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// Independent precomputed pi fractions from ReferenceData/generate_constants.py.
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// Check each direction against its own reference, not a computed round trip.
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foreach (double sign in new[] { -1.0, 1.0 })
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{
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double signedDegrees = sign * degrees;
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BigInteger radiansUnits = Units(sign * radiansHigh) + Units(sign * radiansLow);
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AssertRelative(signedDegrees * DoubleDouble.DegToRad, radiansUnits, BigInteger.One);
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AssertRelative(DoubleDouble.DegToRad * signedDegrees, radiansUnits, BigInteger.One);
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AssertRelative(new DoubleDouble(signedDegrees) * DoubleDouble.DegToRad, radiansUnits, BigInteger.One);
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DoubleDouble radians = DoubleDouble.FromComponents(sign * radiansHigh, sign * radiansLow);
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AssertRelative(radians * DoubleDouble.RadToDeg, Units(signedDegrees), BigInteger.One);
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AssertRelative(DoubleDouble.RadToDeg * radians, Units(signedDegrees), BigInteger.One);
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}
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}
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// Generated by: python3 1-tests/Just.PreciseMath.Tests/ReferenceData/generate_irrational_arithmetic.py
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// Constants: Python Decimal at 160 digits, checked again at 240 digits;
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// pi uses Machin's formula, e/roots/ln use Decimal's exp/sqrt/ln, phi=(1+sqrt(5))/2.
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// Expected operations use exact Fraction sums of the stored input components,
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// then round high and residual separately to binary64. A scalar input has no low.
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// Column pairs: left, right, sum, difference, product, quotient.
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[Theory]
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// pi, e: DD/DD
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[InlineData("DD/DD",
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3.141592653589793, 1.2246467991473532e-16,
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2.718281828459045, 1.4456468917292502e-16,
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5.859874482048839, -1.7705984076240228e-16,
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0.423310825130748, -2.2100009258189695e-17,
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8.539734222673568, -6.773815290502424e-16,
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1.1557273497909217, -1.3998972600526045e-17)]
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// pi, e: DD/double
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[InlineData("DD/double",
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3.141592653589793, 1.2246467991473532e-16,
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2.718281828459045, 0.0,
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5.859874482048839, -3.2162452993532727e-16,
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0.42331082513074814, 1.1442377452219667e-17,
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8.539734222673566, 6.44811944875855e-16,
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1.1557273497909217, 4.746535510161172e-17)]
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// pi, e: double/DD
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[InlineData("double/DD",
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3.141592653589793, 0.0,
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2.718281828459045, 1.4456468917292502e-16,
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5.859874482048839, -2.995245206771376e-16,
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0.42331082513074786, 2.1968764520848465e-17,
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8.539734222673566, 7.660817963097297e-16,
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1.1557273497909217, -5.905121061079843e-17)]
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// sqrt2, sqrt3: DD/DD
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[InlineData("DD/DD",
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1.4142135623730951, -9.667293313452913e-17,
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1.7320508075688772, 1.0035084221806903e-16,
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3.1462643699419726, -2.1836669584149143e-16,
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-0.31783724519578227, 2.5020829572433146e-17,
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2.449489742783178, 2.168616518103246e-16,
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0.816496580927726, -1.7276510382355668e-18)]
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// sqrt2, sqrt3: DD/double
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[InlineData("DD/double",
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1.4142135623730951, -9.667293313452913e-17,
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1.7320508075688772, 0.0,
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3.146264369941972, 1.2537167179050217e-16,
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-0.31783724519578216, 1.434936932798652e-17,
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2.449489742783178, 7.494412974996883e-17,
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0.816496580927726, 4.5578189648549696e-17)]
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// sqrt2, sqrt3: double/DD
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[InlineData("double/DD",
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1.4142135623730951, 0.0,
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1.7320508075688772, 1.0035084221806903e-16,
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3.1462643699419726, -1.2169376270696227e-16,
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-0.31783724519578216, 1.0671460244446626e-17,
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2.4494897427831783, -5.978512613402474e-17,
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0.816496580927726, 5.408649293033552e-17)]
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// ln2, phi: DD/DD
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[InlineData("DD/DD",
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0.6931471805599453, 2.3190468138462996e-17,
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1.618033988749895, -5.432115203682506e-17,
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2.3111811693098403, -1.421529863608777e-16,
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-0.9248868081899495, -3.35106822872276e-17,
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1.121535697352152, -9.053143373999594e-17,
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0.4283885167922066, -2.699599415943276e-18)]
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// ln2, phi: DD/double
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[InlineData("DD/double",
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0.6931471805599453, 2.3190468138462996e-17,
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1.618033988749895, 0.0,
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2.3111811693098403, -8.783183432405266e-17,
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-0.9248868081899496, 2.3190468138462996e-17,
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1.121535697352152, -5.2878880360902515e-17,
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0.4283885167922066, -1.708159504353726e-17)]
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// ln2, phi: double/DD
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[InlineData("double/DD",
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0.6931471805599453, 0.0,
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1.618033988749895, -5.432115203682506e-17,
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2.3111811693098403, -1.653434544993407e-16,
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-0.9248868081899496, 5.432115203682506e-17,
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1.1215356973521518, 9.399020552198074e-17,
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0.4283885167922066, -1.703209694053491e-17)]
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public void BasicArithmeticMatchesPrecomputedIrrationalResults(string overload,
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double leftHigh, double leftLow, double rightHigh, double rightLow,
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double sumHigh, double sumLow, double differenceHigh, double differenceLow,
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double productHigh, double productLow, double quotientHigh, double quotientLow)
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{
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foreach (double sign in new[] { -1.0, 1.0 })
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{
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DoubleDouble left = DoubleDouble.FromComponents(sign * leftHigh, sign * leftLow);
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DoubleDouble right = DoubleDouble.FromComponents(sign * rightHigh, sign * rightLow);
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(DoubleDouble sum, DoubleDouble difference, DoubleDouble product, DoubleDouble quotient) = overload switch
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{
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"DD/DD" => (left + right, left - right, left * right, left / right),
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"DD/double" => (left + right.High, left - right.High, left * right.High, left / right.High),
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"double/DD" => (left.High + right, left.High - right, left.High * right, left.High / right),
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_ => throw new ArgumentOutOfRangeException(nameof(overload))
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};
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// Negating both inputs negates sum/difference but not product/quotient.
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// Compare complete expansions at the established 2^-100 relative + epsilon bound,
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// not individual component equality: correct rounding is not guaranteed.
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// The generator checks reference-rounding error is below 2^-105 relative.
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AssertRelative(sum, Units(sign * sumHigh) + Units(sign * sumLow), BigInteger.One);
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AssertRelative(difference, Units(sign * differenceHigh) + Units(sign * differenceLow), BigInteger.One);
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AssertRelative(product, Units(productHigh) + Units(productLow), BigInteger.One);
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AssertRelative(quotient, Units(quotientHigh) + Units(quotientLow), BigInteger.One);
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}
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}
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[Theory]
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[InlineData("+", "DD/DD")]
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[InlineData("-", "DD/DD")]
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[InlineData("*", "DD/DD")]
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[InlineData("/", "DD/DD")]
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[InlineData("+", "DD/double")]
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[InlineData("-", "DD/double")]
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[InlineData("*", "DD/double")]
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[InlineData("/", "DD/double")]
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[InlineData("+", "double/DD")]
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[InlineData("-", "double/DD")]
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[InlineData("*", "double/DD")]
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[InlineData("/", "double/DD")]
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public void GeneralArithmeticMeetsExactRationalBound(string operation, string overload)
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{
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// Fixed seed, with dense, sparse and zero residuals independent of the
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// high sign. Include ordinary exponents as well as dispatch/range edges.
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// Every pair is checked in both orders; overflow cases are asserted, not skipped.
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Random random = new(65537);
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int[] exponents = [-1074, -1022, -969, -451, -450, -1, 0, 1, 450, 451, 900, 1020, 1021, 1023];
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foreach (int leftExponent in exponents)
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{
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foreach (int rightExponent in exponents)
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{
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for (int sample = 0; sample < 4; ++sample)
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{
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DoubleDouble left = GeneralArithmeticSample(random, leftExponent, sample);
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DoubleDouble right = GeneralArithmeticSample(random, rightExponent, (sample + 1) % 4);
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AssertGeneralArithmetic(left, right, operation, overload);
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AssertGeneralArithmetic(right, left, operation, overload);
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}
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}
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}
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for (int sample = 0; sample < 512; ++sample)
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{
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int exponent = random.Next(-450, 451);
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DoubleDouble left = GeneralArithmeticSample(random, exponent, sample % 4);
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DoubleDouble right = GeneralArithmeticSample(random, random.Next(-450, 451), (sample + 1) % 4);
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AssertGeneralArithmetic(left, right, operation, overload);
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AssertGeneralArithmetic(right, left, operation, overload);
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// Correlate highs to force cancellation rather than hoping random
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// independent values happen to exercise it. Keep different low terms.
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DoubleDouble neighbor = DoubleDouble.FromComponents(left.High, -left.Low);
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AssertGeneralArithmetic(left, neighbor, operation, overload);
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AssertGeneralArithmetic(left, -neighbor, operation, overload);
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}
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}
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private static DoubleDouble GeneralArithmeticSample(Random random, int exponent, int residualKind)
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{
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double sign = random.Next(2) == 0 ? -1.0 : 1.0;
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double high = Math.ScaleB(sign * (1.0 + (0.75 * random.NextDouble())), exponent);
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double low = residualKind switch
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{
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0 => 0.0,
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1 => Math.ScaleB(random.NextDouble() - 0.5, exponent - 53),
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2 => Math.ScaleB(random.NextDouble() - 0.5, exponent - 106),
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3 => random.Next(2) == 0 ? -double.Epsilon : double.Epsilon,
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_ => throw new ArgumentOutOfRangeException(nameof(residualKind))
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};
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// At the subnormal floor a residual can cancel the high completely.
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// Use a scalar there so this finite-input matrix never divides by zero;
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// the separate special-value matrix covers zero denominators and NaNs.
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if (exponent < -1022)
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{
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low = 0.0;
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}
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DoubleDouble result = DoubleDouble.FromComponents(high, low);
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double.IsFinite(result.High).ShouldBeTrue();
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result.High.ShouldNotBe(0.0);
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return result;
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}
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private static void AssertGeneralArithmetic(DoubleDouble left, DoubleDouble right, string operation, string overload)
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{
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DoubleDouble actual = (operation, overload) switch
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{
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("+", "DD/DD") => left + right,
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("-", "DD/DD") => left - right,
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("*", "DD/DD") => left * right,
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("/", "DD/DD") => left / right,
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("+", "DD/double") => left + right.High,
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("-", "DD/double") => left - right.High,
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("*", "DD/double") => left * right.High,
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("/", "DD/double") => left / right.High,
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("+", "double/DD") => left.High + right,
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("-", "double/DD") => left.High - right,
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("*", "double/DD") => left.High * right,
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("/", "double/DD") => left.High / right,
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_ => throw new ArgumentOutOfRangeException(nameof(operation))
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};
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// Units decodes both IEEE-754 components independently of library arithmetic.
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// Expected numerator/denominator is measured in units of epsilon, not doubles.
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BigInteger x = overload == "double/DD" ? Units(left.High) : Units(left);
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BigInteger y = overload == "DD/double" ? Units(right.High) : Units(right);
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(BigInteger numerator, BigInteger denominator) = operation switch
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{
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"+" => (x + y, BigInteger.One),
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"-" => (x - y, BigInteger.One),
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"*" => (x * y, BigInteger.One << 1074),
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"/" => (x << 1074, y),
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_ => throw new ArgumentOutOfRangeException(nameof(operation))
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};
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if (denominator.Sign < 0)
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{
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numerator = -numerator;
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denominator = -denominator;
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}
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string context = $"{overload}: ({left.High:R}, {left.Low:R}) {operation} ({right.High:R}, {right.Low:R}); "
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+ $"actual ({actual.High:R}, {actual.Low:R})";
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// Nearest-even binary64 overflow begins at 2^1024 - 2^970.
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BigInteger overflowUnits = Units(double.MaxValue) + Units(Math.ScaleB(1.0, 970));
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if (BigInteger.Abs(numerator) >= overflowUnits * denominator)
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{
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actual.High.ShouldBe(numerator.Sign < 0 ? double.NegativeInfinity : double.PositiveInfinity, context);
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BitConverter.DoubleToInt64Bits(actual.Low).ShouldBe(0L, context);
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return;
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}
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double.IsFinite(actual.High).ShouldBeTrue(context);
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double.IsFinite(actual.Low).ShouldBeTrue(context);
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BigInteger error = BigInteger.Abs((Units(actual) * denominator) - numerator);
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// Same conservative 2^-100 relative + epsilon contract as the existing
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// suite, cross-multiplied exactly to retain all low-component information.
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(error <= (BigInteger.Abs(numerator) >> 100) + denominator).ShouldBeTrue(context);
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(actual.High + actual.Low).ShouldBe(actual.High, context);
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if (actual.Low == 0.0)
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{
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BitConverter.DoubleToInt64Bits(actual.Low).ShouldBe(0L, context);
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}
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if (actual.High == 0.0)
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{
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// Exact cancellation is +0; a nonzero underflow keeps its sign.
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BitConverter.DoubleToInt64Bits(actual.High).ShouldBe(numerator.Sign < 0 ? long.MinValue : 0L, context);
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}
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}
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[Fact]
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public void CancellationRetainsBothLowSumTerms()
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{
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@@ -232,6 +529,56 @@ public class DoubleDoubleArithmeticTests
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AssertRelative(-1.0 / value, -numerator, Units(value));
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}
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[Fact]
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public void DivisionNormalizesTheFirstCorrectionBeforeAddingTheLast()
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{
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// Exact rational: (1 + 2^-54) / (17/16 - 2^-54).
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// Independently round that rational to binary64, then round its exact
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// residual: high bits 3FEE1E1E1E1E1E1F, low bits 3C4FE3A76B2EF2C4.
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// Adding the last correction to the unnormalized pair instead loses
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// four low-component ULPs. Common power-of-two scaling preserves the ratio.
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double expectedHigh = BitConverter.UInt64BitsToDouble(0x3fee_1e1e_1e1e_1e1f);
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double expectedLow = BitConverter.UInt64BitsToDouble(0x3c4f_e3a7_6b2e_f2c4);
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foreach (int exponent in new[] { -450, 0, 450 })
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{
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foreach (double leftSign in new[] { -1.0, 1.0 })
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{
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foreach (double rightSign in new[] { -1.0, 1.0 })
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{
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DoubleDouble left = DoubleDouble.FromComponents(leftSign * Math.ScaleB(1.0, exponent),
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leftSign * Math.ScaleB(1.0, exponent - 54));
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DoubleDouble right = DoubleDouble.FromComponents(rightSign * Math.ScaleB(1.0625, exponent),
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-rightSign * Math.ScaleB(1.0, exponent - 54));
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double sign = leftSign * rightSign;
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DoubleDouble actual = left / right;
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CheckBoundary(actual, sign * expectedHigh, sign * expectedLow);
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AssertRelative(actual, Units(left) << 1074, Units(right));
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}
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}
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}
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}
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[Fact]
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public void DivisionSecondResidualPreservesBitsAndRationalAccuracy()
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{
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Random random = new(104729);
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for (int i = 0; i < 4096; ++i)
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{
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int leftExponent = random.Next(-450, 451);
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int rightExponent = random.Next(-450, 451);
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||||
double leftHigh = Math.ScaleB(1.0 + (0.75 * random.NextDouble()), leftExponent);
|
||||
double rightHigh = Math.ScaleB(1.0 + (0.75 * random.NextDouble()), rightExponent);
|
||||
double leftLow = i % 3 == 0 ? 0.0 : i % 3 == 1 ? double.Epsilon
|
||||
: Math.ScaleB(random.NextDouble() - 0.5, leftExponent - 53);
|
||||
double rightLow = i % 3 == 1 ? 0.0 : i % 3 == 2 ? -double.Epsilon
|
||||
: Math.ScaleB(random.NextDouble() - 0.5, rightExponent - 53);
|
||||
DoubleDouble left = DoubleDouble.FromComponents(i % 2 == 0 ? leftHigh : -leftHigh, leftLow);
|
||||
DoubleDouble right = DoubleDouble.FromComponents(i % 4 < 2 ? rightHigh : -rightHigh, rightLow);
|
||||
AssertDivisionMatchesPrevious(left, right);
|
||||
AssertDivisionMatchesPrevious(right, left);
|
||||
}
|
||||
}
|
||||
|
||||
[Fact]
|
||||
public void DivisionRetainsSubnormalCorrectionsWithOrdinaryHighComponents()
|
||||
{
|
||||
@@ -537,6 +884,21 @@ public class DoubleDoubleArithmeticTests
|
||||
}
|
||||
}
|
||||
|
||||
private static void AssertDivisionMatchesPrevious(DoubleDouble left, DoubleDouble right)
|
||||
{
|
||||
// Freeze the pre-specialization expression. The unchanged public operators
|
||||
// form the bitwise reference; Units supplies the independent rational oracle.
|
||||
double quotient = left.High / right.High;
|
||||
DoubleDouble remainder = left - (right * quotient);
|
||||
double correction = remainder.High / right.High;
|
||||
remainder -= right * correction;
|
||||
double finalCorrection = remainder.High / right.High;
|
||||
DoubleDouble expected = DoubleDouble.FromComponents(quotient, correction) + finalCorrection;
|
||||
DoubleDouble actual = left / right;
|
||||
CheckBoundary(actual, expected.High, expected.Low);
|
||||
AssertRelative(actual, Units(left) << 1074, Units(right));
|
||||
}
|
||||
|
||||
private static void Check(DoubleDouble value, double high, double low)
|
||||
{
|
||||
value.High.ShouldBe(high);
|
||||
|
||||
Reference in New Issue
Block a user