add precomputed mathematical constants and specialize the second division residual

0-source/Just.PreciseMath/DoubleDouble.Arithmetic.cs

@@ -101,9 +101,13 @@ public readonly partial struct DoubleDouble :
         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;
-        return FromComponents(quotient, correction) + finalCorrection;
+        double finalRemainder = PreciseMathHelper.SubtractDivisionCorrectionHigh(remainder, right * correction);
+        double finalCorrection = finalRemainder / right._high;
+        // The entry guard gives |quotient| <= 2^901. Conservatively bounding
+        // the first product and subtraction gives |remainder.High| <= 2^457
+        // at the first correction, hence |correction| <= 2^908 and a finite sum
+        // below 2^909 in magnitude. Retain normalization before the final add.
+        return PreciseMathHelper.NormalizeFinite(quotient, correction) + finalCorrection;
     }
 
     /// <summary>Applies the expansion operation without discarding the low component.</summary>

0-source/Just.PreciseMath/DoubleDouble.cs

@@ -98,10 +98,15 @@ public readonly partial struct DoubleDouble :
     /// Represents a zero value.
     /// </summary>
     public static DoubleDouble Zero => new();
+
+    // Mathematical constants store normalized, precomputed binary64 pairs: the
+    // nearest high and then the nearest residual of the high-precision value.
+    // No DD arithmetic, parsing, normalization or heap allocation occurs on access.
+    // Reproduce with the tests' ReferenceData/generate_constants.py.
     /// <summary>
     /// Represents the ratio of the circumference of a circle to its diameter, specified by the constant, π.
     /// </summary>
-    public static DoubleDouble PI => new(3.141592653589793, 1.2246467991473532e-16);
+    public static DoubleDouble Pi => new(3.141592653589793, 1.2246467991473532e-16);
     /// <summary>
     /// Represents the natural logarithmic base, specified by the constant, e.
     /// </summary>
@@ -109,7 +114,85 @@ public readonly partial struct DoubleDouble :
     /// <summary>
     /// Represents the natural logarithm of value 2.
     /// </summary>
-    public static DoubleDouble LN2 => new(0.6931471805599453, 2.3190468138462996e-17);
+    public static DoubleDouble Ln2 => new(0.6931471805599453, 2.3190468138462996e-17);
+
+    /// <summary>Gets τ = 2π, the angle of one full turn in radians.</summary>
+    public static DoubleDouble Tau => new(6.283185307179586, 2.4492935982947064e-16);
+
+    /// <summary>Gets π/2, the angle of 90 degrees in radians.</summary>
+    public static DoubleDouble PiOver2 => new(1.5707963267948966, 6.123233995736766e-17);
+
+    /// <summary>Gets π/3, the angle of 60 degrees in radians.</summary>
+    public static DoubleDouble PiOver3 => new(1.0471975511965979, -1.072081766451091e-16);
+
+    /// <summary>Gets π/4, the angle of 45 degrees in radians.</summary>
+    public static DoubleDouble PiOver4 => new(0.7853981633974483, 3.061616997868383e-17);
+
+    /// <summary>Gets π/6, the angle of 30 degrees in radians.</summary>
+    public static DoubleDouble PiOver6 => new(0.5235987755982989, -5.360408832255455e-17);
+
+    /// <summary>Gets 1/π.</summary>
+    public static DoubleDouble InvPi => new(0.3183098861837907, -1.9678676675182486e-17);
+
+    /// <summary>Gets 1/(2π), the factor for converting radians to turns.</summary>
+    public static DoubleDouble InvTau => new(0.15915494309189535, -9.839338337591243e-18);
+
+    /// <summary>Gets π/180. Multiply an angle in degrees by this value to obtain radians.</summary>
+    public static DoubleDouble DegToRad => new(0.017453292519943295, 2.9486522708701687e-19);
+
+    /// <summary>Gets 180/π. Multiply an angle in radians by this value to obtain degrees.</summary>
+    public static DoubleDouble RadToDeg => new(57.29577951308232, -1.9878495670576283e-15);
+
+    /// <summary>Gets 1/e = exp(-1).</summary>
+    public static DoubleDouble InvE => new(0.36787944117144233, -1.2428753672788363e-17);
+
+    /// <summary>Gets ln(10), the factor for converting base-10 logarithms to natural logarithms.</summary>
+    public static DoubleDouble Ln10 => new(2.302585092994046, -2.1707562233822494e-16);
+
+    /// <summary>Gets log₂(e) = 1/ln(2), the factor for converting natural logarithms to base 2.</summary>
+    public static DoubleDouble Log2E => new(1.4426950408889634, 2.0355273740931033e-17);
+
+    /// <summary>Gets log₁₀(e) = 1/ln(10), the factor for converting natural logarithms to base 10.</summary>
+    public static DoubleDouble Log10E => new(0.4342944819032518, 1.098319650216765e-17);
+
+    /// <summary>Gets log₂(10), the factor for converting base-10 logarithms to base 2.</summary>
+    public static DoubleDouble Log2Of10 => new(3.321928094887362, 1.661617516973592e-16);
+
+    /// <summary>Gets log₁₀(2), the factor for converting base-2 logarithms to base 10.</summary>
+    public static DoubleDouble Log10Of2 => new(0.3010299956639812, -2.8037281277851704e-18);
+
+    /// <summary>Gets √2.</summary>
+    public static DoubleDouble Sqrt2 => new(1.4142135623730951, -9.667293313452913e-17);
+
+    /// <summary>Gets √3.</summary>
+    public static DoubleDouble Sqrt3 => new(1.7320508075688772, 1.0035084221806903e-16);
+
+    /// <summary>Gets √5.</summary>
+    public static DoubleDouble Sqrt5 => new(2.23606797749979, -1.0864230407365012e-16);
+
+    /// <summary>Gets 1/√2, also the sine and cosine of π/4.</summary>
+    public static DoubleDouble InvSqrt2 => new(0.7071067811865476, -4.833646656726457e-17);
+
+    /// <summary>Gets 1/√3, also the tangent of π/6.</summary>
+    public static DoubleDouble InvSqrt3 => new(0.5773502691896257, 3.3450280739356345e-17);
+
+    /// <summary>Gets √π, the Gaussian integral over the real line for exp(-x²).</summary>
+    public static DoubleDouble SqrtPi => new(1.772453850905516, -7.666586499825799e-17);
+
+    /// <summary>Gets 1/√π, a Gaussian normalization factor.</summary>
+    public static DoubleDouble InvSqrtPi => new(0.5641895835477563, 7.66772980658294e-18);
+
+    /// <summary>Gets 2/√π, the normalization factor in the error-function integral.</summary>
+    public static DoubleDouble TwoInvSqrtPi => new(1.1283791670955126, 1.533545961316588e-17);
+
+    /// <summary>Gets √(2π), used in Gaussian integrals and Stirling's approximation.</summary>
+    public static DoubleDouble SqrtTau => new(2.5066282746310007, -1.8328579980459167e-16);
+
+    /// <summary>Gets 1/√(2π), the standard normal probability density's normalization factor.</summary>
+    public static DoubleDouble InvSqrtTau => new(0.3989422804014327, -2.49232720227773e-17);
+
+    /// <summary>Gets the golden ratio φ = (1 + √5)/2.</summary>
+    public static DoubleDouble GoldenRatio => new(1.618033988749895, -5.432115203682506e-17);
     #endregion
 
     /// <summary>

0-source/Just.PreciseMath/PreciseMathHelper.cs

@@ -104,10 +104,35 @@ internal static class PreciseMathHelper
         return NormalizeFinite(sum, sumError + (middleError + lowError));
     }
 
+    // Only for DD division's second remainder: normalized finite inputs with highs
+    // bounded by 2^462, including the correction product's exact fallback results.
+    // Same sign and normal binade imply Sterbenz-exact high subtraction. Its error
+    // is +0; with canonical input lows, TwoAdd(+0, lowSum) is (lowSum, +0).
+    // Eliminate those two transforms, not the low-sum error or final normalization.
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    internal static double SubtractDivisionCorrectionHigh(DoubleDouble remainder, DoubleDouble product)
+    {
+        ulong remainderBits = BitConverter.DoubleToUInt64Bits(remainder._high);
+        ulong productBits = BitConverter.DoubleToUInt64Bits(product._high);
+        if ((remainderBits & 0x7ff0_0000_0000_0000UL) == 0
+            || ((remainderBits ^ productBits) & 0xfff0_0000_0000_0000UL) != 0)
+        {
+            return (remainder - product)._high;
+        }
+
+        double high = remainder._high - product._high;
+        (double low, double lowError) = TwoAdd(remainder._low, -product._low);
+        (double sum, double sumError) = TwoAdd(high, low);
+        double error = sumError + lowError;
+        // Preserve NormalizeFinite's zero-low shortcut, including the high zero's
+        // sign. The normalized low output is unused by the final quotient correction.
+        return error == 0.0 ? sum : sum + error;
+    }
+
     // Both components and their rounded sum must be finite. Unlike QuickTwoSum,
     // this entry point permits either magnitude order, including cancellation.
-    // AddFinite establishes these bounds; this helper does not validate them or
-    // canonicalize NaN/infinity. Use DoubleDouble.FromComponents for arbitrary pairs.
+    // AddFinite and DD division establish these bounds; this helper does not validate
+    // them or canonicalize NaN/infinity. Use DoubleDouble.FromComponents for arbitrary pairs.
     // Retain the high zero's sign when low is zero, as FromComponents does.
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
     internal static DoubleDouble NormalizeFinite(double high, double low)

1-tests/Just.PreciseMath.Tests/ArithmeticRangeTests.cs

@@ -80,6 +80,7 @@ public class ArithmeticRangeTests
     [InlineData("+")]
     [InlineData("-")]
     [InlineData("*")]
+    [InlineData("/")]
     public void FiniteKernelsPreservePreviousComponentBits(string operation)
     {
         // Differential characterization, not an independent accuracy oracle.
@@ -113,7 +114,39 @@ public class ArithmeticRangeTests
             {
                 DoubleDouble expected;
                 DoubleDouble actual;
-                if (operation == "*")
+                if (operation == "/")
+                {
+                    if (left.High == 0.0 || right.High == 0.0
+                        || Math.Abs(Math.ILogB(left.High)) > 450 || Math.Abs(Math.ILogB(right.High)) > 450)
+                    {
+                        continue;
+                    }
+                    // Retain the original division expression, including its public
+                    // normalization. Sparse corrections may use MultiplyBoundary.
+                    double quotient = left.High / right.High;
+                    DoubleDouble remainder = left - (right * quotient);
+                    double correction = remainder.High / right.High;
+                    double.IsFinite(quotient).ShouldBeTrue();
+                    double.IsFinite(correction).ShouldBeTrue();
+                    double.IsFinite(quotient + correction).ShouldBeTrue();
+                    // Conservative bounds from the entry domain and the first
+                    // scalar product/four-TwoSum remainder, not an accuracy claim.
+                    (Math.Abs(quotient) <= Math.ScaleB(1.0, 901)).ShouldBeTrue();
+                    (Math.Abs(remainder.High) <= Math.ScaleB(1.0, 457)).ShouldBeTrue();
+                    (Math.Abs(correction) <= Math.ScaleB(1.0, 908)).ShouldBeTrue();
+                    (Math.Abs(quotient + correction) < Math.ScaleB(1.0, 909)).ShouldBeTrue();
+                    DoubleDouble correctionProduct = right * correction;
+                    AssertDivisionResidualCancellation(remainder, correctionProduct);
+                    remainder -= correctionProduct;
+                    double finalCorrection = remainder.High / right.High;
+                    DoubleDouble normalized = DoubleDouble.FromComponents(quotient, correction);
+                    DoubleDouble finiteNormalized = PreciseMathHelper.NormalizeFinite(quotient, correction);
+                    BitConverter.DoubleToInt64Bits(finiteNormalized.High).ShouldBe(BitConverter.DoubleToInt64Bits(normalized.High));
+                    BitConverter.DoubleToInt64Bits(finiteNormalized.Low).ShouldBe(BitConverter.DoubleToInt64Bits(normalized.Low));
+                    expected = normalized + finalCorrection;
+                    actual = left / right;
+                }
+                else if (operation == "*")
                 {
                     if (left.High == 0.0 || right.High == 0.0)
                     {
@@ -161,6 +194,62 @@ public class ArithmeticRangeTests
         }
     }
 
+    [Fact]
+    public void DivisionResidualCancellationPreservesHighBitsAtBinadeEdges()
+    {
+        // Include both sides of normal/subnormal and binade transitions, with
+        // canonical zero, dense and sparse lows. Cross-binade/sign/zero cases
+        // are intentionally ineligible and must retain the general subtraction.
+        List<DoubleDouble> values = [new(0.0), new(-0.0)];
+        foreach (int exponent in new[] { -1074, -1022, -900, -54, 0, 1, 457, 461 })
+        {
+            foreach (double significand in new[] { 1.0, Math.BitIncrement(1.0), Math.BitDecrement(2.0) })
+            {
+                foreach (double sign in new[] { -1.0, 1.0 })
+                {
+                    double high = sign * Math.ScaleB(significand, exponent);
+                    foreach (double low in new[] { 0.0, Math.ScaleB(high, -54), -Math.ScaleB(high, -54),
+                        double.Epsilon, -double.Epsilon })
+                    {
+                        values.Add(DoubleDouble.FromComponents(high, low));
+                    }
+                }
+            }
+        }
+        foreach (DoubleDouble remainder in values)
+        {
+            foreach (DoubleDouble product in values)
+            {
+                AssertDivisionResidualCancellation(remainder, product);
+            }
+        }
+    }
+
+    private static void AssertDivisionResidualCancellation(DoubleDouble remainder, DoubleDouble product)
+    {
+        DoubleDouble expected = remainder - product;
+        double actual = PreciseMathHelper.SubtractDivisionCorrectionHigh(remainder, product);
+        BitConverter.DoubleToInt64Bits(actual).ShouldBe(BitConverter.DoubleToInt64Bits(expected.High));
+        // Independent BCL classification, rather than the proposed exponent-bit
+        // guard. Within one normal binade, same-sign subtraction is exact.
+        if (!double.IsNormal(remainder.High) || !double.IsNormal(product.High)
+            || Math.Sign(remainder.High) != Math.Sign(product.High)
+            || Math.ILogB(remainder.High) != Math.ILogB(product.High))
+        {
+            return;
+        }
+        (double high, double highError) = PreciseMathHelper.TwoAdd(remainder.High, -product.High);
+        BitConverter.DoubleToInt64Bits(highError).ShouldBe(0L);
+        (double low, double lowError) = PreciseMathHelper.TwoAdd(remainder.Low, -product.Low);
+        (double middle, double middleError) = PreciseMathHelper.TwoAdd(highError, low);
+        BitConverter.DoubleToInt64Bits(middle).ShouldBe(BitConverter.DoubleToInt64Bits(low));
+        BitConverter.DoubleToInt64Bits(middleError).ShouldBe(0L);
+        (double sum, double sumError) = PreciseMathHelper.TwoAdd(high, low);
+        double error = sumError + lowError;
+        double simplifiedHigh = error == 0.0 ? sum : sum + error;
+        BitConverter.DoubleToInt64Bits(simplifiedHigh).ShouldBe(BitConverter.DoubleToInt64Bits(expected.High));
+    }
+
     private static void AssertMultiplicationRange(double left, double right)
     {
         int exponent = Math.ILogB(left) + Math.ILogB(right);

1-tests/Just.PreciseMath.Tests/DoubleDoubleArithmeticTests.cs

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

1-tests/Just.PreciseMath.Tests/DoubleDoubleFormattingTests.cs

@@ -58,12 +58,12 @@ public class DoubleDoubleFormattingTests
     public void FormattingDoesNotRouteThroughDecimalOrRestrictExponentRange()
     {
         // Review-1 §6 verified the former decimal-based formatter threw for 1e100,
-        // printed "0" for 1e-100, threw for NaN/infinity, and emitted for PI digits
+        // printed "0" for 1e-100, threw for NaN/infinity, and emitted for pi digits
         // already wrong at binary64 precision. Expected digits are exact references:
-        // the PI pair rounded to 32 significant digits, ties to even
+        // the Pi pair rounded to 32 significant digits, ties to even
         // (Python Fraction/Decimal at precision 120), and the exact binary64 values of
         // the powers of ten.
-        DoubleDouble.PI.ToString("G32", CultureInfo.InvariantCulture).ShouldBe("3.1415926535897932384626433832795");
+        DoubleDouble.Pi.ToString("G32", CultureInfo.InvariantCulture).ShouldBe("3.1415926535897932384626433832795");
         new DoubleDouble(1e100).ToString("G", CultureInfo.InvariantCulture).ShouldBe("1.0000000000000000159028911097599E+100");
         new DoubleDouble(1e-100).ToString("G", CultureInfo.InvariantCulture).ShouldBe("1.0000000000000000199918998026029E-100");
         // The binary64 values are exactly representable as sums with a zero residual,

1-tests/Just.PreciseMath.Tests/DoubleDoubleSpanFormattingTests.cs

@@ -38,7 +38,7 @@ public class DoubleDoubleSpanFormattingTests
     public void ShortDestinationIsUnchangedAndReportsZero()
     {
         char[] buffer = ['!', '!'];
-        DoubleDouble.PI.TryFormat(buffer, out int written, "G32", CultureInfo.InvariantCulture).ShouldBeFalse();
+        DoubleDouble.Pi.TryFormat(buffer, out int written, "G32", CultureInfo.InvariantCulture).ShouldBeFalse();
         written.ShouldBe(0);
         new string(buffer).ShouldBe("!!");
         DoubleDouble.One.TryFormat(Span<char>.Empty, out written, provider: CultureInfo.InvariantCulture).ShouldBeFalse();

1-tests/Just.PreciseMath.Tests/DoubleDoubleTests.cs

@@ -6,25 +6,81 @@ namespace Just.PreciseMath.Tests;
 public class DoubleDoubleTests
 {
     [Theory]
-    [InlineData("PI", 3.141592653589793, 1.2246467991473532e-16)]
+    [InlineData("Pi", 3.141592653589793, 1.2246467991473532e-16)]
     [InlineData("E", 2.718281828459045, 1.4456468917292502e-16)]
-    [InlineData("LN2", 0.6931471805599453, 2.3190468138462996e-17)]
+    [InlineData("Ln2", 0.6931471805599453, 2.3190468138462996e-17)]
+    [InlineData("Tau", 6.283185307179586, 2.4492935982947064e-16)]
+    [InlineData("PiOver2", 1.5707963267948966, 6.123233995736766e-17)]
+    [InlineData("PiOver3", 1.0471975511965979, -1.072081766451091e-16)]
+    [InlineData("PiOver4", 0.7853981633974483, 3.061616997868383e-17)]
+    [InlineData("PiOver6", 0.5235987755982989, -5.360408832255455e-17)]
+    [InlineData("InvPi", 0.3183098861837907, -1.9678676675182486e-17)]
+    [InlineData("InvTau", 0.15915494309189535, -9.839338337591243e-18)]
+    [InlineData("DegToRad", 0.017453292519943295, 2.9486522708701687e-19)]
+    [InlineData("RadToDeg", 57.29577951308232, -1.9878495670576283e-15)]
+    [InlineData("InvE", 0.36787944117144233, -1.2428753672788363e-17)]
+    [InlineData("Ln10", 2.302585092994046, -2.1707562233822494e-16)]
+    [InlineData("Log2E", 1.4426950408889634, 2.0355273740931033e-17)]
+    [InlineData("Log10E", 0.4342944819032518, 1.098319650216765e-17)]
+    [InlineData("Log2Of10", 3.321928094887362, 1.661617516973592e-16)]
+    [InlineData("Log10Of2", 0.3010299956639812, -2.8037281277851704e-18)]
+    [InlineData("Sqrt2", 1.4142135623730951, -9.667293313452913e-17)]
+    [InlineData("Sqrt3", 1.7320508075688772, 1.0035084221806903e-16)]
+    [InlineData("Sqrt5", 2.23606797749979, -1.0864230407365012e-16)]
+    [InlineData("InvSqrt2", 0.7071067811865476, -4.833646656726457e-17)]
+    [InlineData("InvSqrt3", 0.5773502691896257, 3.3450280739356345e-17)]
+    [InlineData("SqrtPi", 1.772453850905516, -7.666586499825799e-17)]
+    [InlineData("InvSqrtPi", 0.5641895835477563, 7.66772980658294e-18)]
+    [InlineData("TwoInvSqrtPi", 1.1283791670955126, 1.533545961316588e-17)]
+    [InlineData("SqrtTau", 2.5066282746310007, -1.8328579980459167e-16)]
+    [InlineData("InvSqrtTau", 0.3989422804014327, -2.49232720227773e-17)]
+    [InlineData("GoldenRatio", 1.618033988749895, -5.432115203682506e-17)]
     public void ConstantsHaveNearestBinary64Residuals(string name, double high, double low)
     {
         // Each residual is round_binary64(constant - exact_binary64(high)).
-        // Reproduced with Python decimal at precision 90: e = Decimal(1).exp(),
-        // ln(2) = Decimal(2).ln(), pi = 16*atan(1/5) - 4*atan(1/239), using
-        // atan(x) = sum((-1)^k*x^(2k+1)/(2k+1)) until |term| < 1e-95.
-        // low = float(reference - Decimal.from_float(float(reference))).
+        // Reproduce with ReferenceData/generate_constants.py: Python Decimal at
+        // 160 and 240 digits, Machin's pi, exp/ln/sqrt and exact Fraction splitting.
+        // The generator records every formula and checks normalized components.
         DoubleDouble value = name switch
         {
-            "PI" => DoubleDouble.PI,
+            "Pi" => DoubleDouble.Pi,
             "E" => DoubleDouble.E,
-            "LN2" => DoubleDouble.LN2,
+            "Ln2" => DoubleDouble.Ln2,
+            "Tau" => DoubleDouble.Tau,
+            "PiOver2" => DoubleDouble.PiOver2,
+            "PiOver3" => DoubleDouble.PiOver3,
+            "PiOver4" => DoubleDouble.PiOver4,
+            "PiOver6" => DoubleDouble.PiOver6,
+            "InvPi" => DoubleDouble.InvPi,
+            "InvTau" => DoubleDouble.InvTau,
+            "DegToRad" => DoubleDouble.DegToRad,
+            "RadToDeg" => DoubleDouble.RadToDeg,
+            "InvE" => DoubleDouble.InvE,
+            "Ln10" => DoubleDouble.Ln10,
+            "Log2E" => DoubleDouble.Log2E,
+            "Log10E" => DoubleDouble.Log10E,
+            "Log2Of10" => DoubleDouble.Log2Of10,
+            "Log10Of2" => DoubleDouble.Log10Of2,
+            "Sqrt2" => DoubleDouble.Sqrt2,
+            "Sqrt3" => DoubleDouble.Sqrt3,
+            "Sqrt5" => DoubleDouble.Sqrt5,
+            "InvSqrt2" => DoubleDouble.InvSqrt2,
+            "InvSqrt3" => DoubleDouble.InvSqrt3,
+            "SqrtPi" => DoubleDouble.SqrtPi,
+            "InvSqrtPi" => DoubleDouble.InvSqrtPi,
+            "TwoInvSqrtPi" => DoubleDouble.TwoInvSqrtPi,
+            "SqrtTau" => DoubleDouble.SqrtTau,
+            "InvSqrtTau" => DoubleDouble.InvSqrtTau,
+            "GoldenRatio" => DoubleDouble.GoldenRatio,
             _ => throw new ArgumentOutOfRangeException(nameof(name)),
         };
         value.High.ShouldBe(high);
         value.Low.ShouldBe(low);
+        BitConverter.DoubleToInt64Bits(value.High).ShouldBe(BitConverter.DoubleToInt64Bits(high));
+        BitConverter.DoubleToInt64Bits(value.Low).ShouldBe(BitConverter.DoubleToInt64Bits(low));
+        DoubleDouble.IsCanonical(value).ShouldBeTrue();
+        DoubleDouble.IsFinite(value).ShouldBeTrue();
+        value.Low.ShouldNotBe(0.0);
     }
 
     [Fact]
@@ -46,7 +102,7 @@ public class DoubleDoubleTests
     [InlineData(3.141592653589793, 1.2246467991473532e-16)]
     [InlineData(2.718281828459045, 1.4456468917292502e-16)]
     [InlineData(0.6931471805599453, 2.3190468138462996e-17)]
-    public void AdditiveIdentity(double high, double low)
+    public void AdditiveIdentityAdd(double high, double low)
     {
         DoubleDouble value = new(high, low);
 
@@ -70,7 +126,32 @@ public class DoubleDoubleTests
     [InlineData(3.141592653589793, 1.2246467991473532e-16)]
     [InlineData(2.718281828459045, 1.4456468917292502e-16)]
     [InlineData(0.6931471805599453, 2.3190468138462996e-17)]
-    public void MultiplicativeIdentity(double high, double low)
+    public void AdditiveIdentitySubtract(double high, double low)
+    {
+        DoubleDouble value = new(high, low);
+        DoubleDouble negativeValue = -value;
+
+        DoubleDouble result = value - DoubleDouble.AdditiveIdentity;
+        DoubleDouble resultInversedOrder = DoubleDouble.AdditiveIdentity - value;
+
+        result.High.ShouldBe(high);
+        result.Low.ShouldBe(low);
+
+        resultInversedOrder.High.ShouldBe(negativeValue.High);
+        resultInversedOrder.Low.ShouldBe(negativeValue.Low);
+    }
+
+    [Theory]
+    [InlineData(1.0, 0.0)]
+    [InlineData(10.0, 0.0)]
+    [InlineData(100.0, 0.0)]
+    [InlineData(-1.0, 0.0)]
+    [InlineData(-10.0, 0.0)]
+    [InlineData(-100.0, 0.0)]
+    [InlineData(3.141592653589793, 1.2246467991473532e-16)]
+    [InlineData(2.718281828459045, 1.4456468917292502e-16)]
+    [InlineData(0.6931471805599453, 2.3190468138462996e-17)]
+    public void MultiplicativeIdentityMultiply(double high, double low)
     {
         DoubleDouble value = new(high, low);
 
@@ -83,4 +164,24 @@ public class DoubleDoubleTests
         resultInversedOrder.High.ShouldBe(high);
         resultInversedOrder.Low.ShouldBe(low);
     }
+
+    [Theory]
+    [InlineData(1.0, 0.0)]
+    [InlineData(10.0, 0.0)]
+    [InlineData(100.0, 0.0)]
+    [InlineData(-1.0, 0.0)]
+    [InlineData(-10.0, 0.0)]
+    [InlineData(-100.0, 0.0)]
+    [InlineData(3.141592653589793, 1.2246467991473532e-16)]
+    [InlineData(2.718281828459045, 1.4456468917292502e-16)]
+    [InlineData(0.6931471805599453, 2.3190468138462996e-17)]
+    public void MultiplicativeIdentityDivide(double high, double low)
+    {
+        DoubleDouble value = new(high, low);
+
+        DoubleDouble result = value / DoubleDouble.MultiplicativeIdentity;
+
+        result.High.ShouldBe(high);
+        result.Low.ShouldBe(low);
+    }
 }

1-tests/Just.PreciseMath.Tests/ReferenceData/generate_constants.py

@@ -0,0 +1,76 @@
+"""Reproduce DoubleDouble's precomputed mathematical constants.
+
+Run: python3 -B 1-tests/Just.PreciseMath.Tests/ReferenceData/generate_constants.py
+Standard library only; no dependency on the implementation under test.
+Decimal evaluates formulas at 160 and 240 digits; Fraction-based splitting
+rounds the high and then the exact residual to nearest-even binary64.
+"""
+
+from decimal import Decimal, localcontext
+from fractions import Fraction
+
+from generate_irrational_arithmetic import arctan_inverse, split
+
+
+def constants(precision: int) -> dict[str, tuple[float, float]]:
+    with localcontext() as context:
+        context.prec = precision
+        one = Decimal(1)
+        pi = 16 * arctan_inverse(5) - 4 * arctan_inverse(239)
+        e = one.exp()
+        ln2 = Decimal(2).ln()
+        ln10 = Decimal(10).ln()
+        sqrt2 = Decimal(2).sqrt()
+        sqrt3 = Decimal(3).sqrt()
+        sqrt5 = Decimal(5).sqrt()
+        sqrt_pi = pi.sqrt()
+        sqrt_tau = (2 * pi).sqrt()
+        values = {
+            "Pi": pi,
+            "E": e,
+            "Ln2": ln2,
+            "Tau": 2 * pi,
+            "PiOver2": pi / 2,
+            "PiOver3": pi / 3,
+            "PiOver4": pi / 4,
+            "PiOver6": pi / 6,
+            "InvPi": one / pi,
+            "InvTau": one / (2 * pi),
+            "DegToRad": pi / 180,
+            "RadToDeg": 180 / pi,
+            "InvE": one / e,
+            "Ln10": ln10,
+            "Log2E": one / ln2,
+            "Log10E": one / ln10,
+            "Log2Of10": ln10 / ln2,
+            "Log10Of2": ln2 / ln10,
+            "Sqrt2": sqrt2,
+            "Sqrt3": sqrt3,
+            "Sqrt5": sqrt5,
+            "InvSqrt2": one / sqrt2,
+            "InvSqrt3": one / sqrt3,
+            "SqrtPi": sqrt_pi,
+            "InvSqrtPi": one / sqrt_pi,
+            "TwoInvSqrtPi": 2 / sqrt_pi,
+            "SqrtTau": sqrt_tau,
+            "InvSqrtTau": one / sqrt_tau,
+            "GoldenRatio": (one + sqrt5) / 2,
+        }
+        pairs = {name: split(Fraction(value)) for name, value in values.items()}
+        for name, value in values.items():
+            high, low = pairs[name]
+            assert high + low == high, f"Unnormalized constant: {name}"
+            assert low != 0.0, f"Lost residual: {name}"
+            assert abs(Fraction(high) + Fraction(low) - Fraction(value)) <= abs(Fraction(value)) / (1 << 105)
+        return pairs
+
+
+if __name__ == "__main__":
+    pairs = constants(160)
+    assert pairs == constants(240), "Increase precision: binary64 constants did not stabilize"
+    print("// Expected test components")
+    for name, (high, low) in pairs.items():
+        print(f'    [InlineData("{name}", {high!r}, {low!r})]')
+    print("\n// Precomputed properties")
+    for name, (high, low) in pairs.items():
+        print(f"    public static DoubleDouble {name} => new({high!r}, {low!r});")

1-tests/Just.PreciseMath.Tests/ReferenceData/generate_irrational_arithmetic.py

@@ -0,0 +1,77 @@
+"""Print BasicArithmeticMatchesPrecomputedIrrationalResults InlineData.
+
+Run with Python 3.11+; standard library only. Decimal supplies irrational
+constants; Fraction supplies exact arithmetic on their stored binary64 pairs.
+The test suite consumes the literals, not this script or Python at runtime.
+"""
+
+from decimal import Decimal, localcontext
+from fractions import Fraction
+
+
+def arctan_inverse(inverse: int) -> Decimal:
+    """Evaluate atan(1/inverse) using its alternating power series."""
+    x = Decimal(1) / inverse
+    power = x
+    total = x
+    index = 1
+    while True:
+        power *= -(x * x)
+        updated = total + power / (2 * index + 1)
+        if updated == total:
+            return total
+        total = updated
+        index += 1
+
+
+def split(value: Fraction) -> tuple[float, float]:
+    """Round high, then the exact residual, to nearest-even binary64."""
+    high = float(value)
+    return high, float(value - Fraction(high))
+
+
+def fixtures(precision: int) -> list[tuple[str, list[float]]]:
+    with localcontext() as context:
+        context.prec = precision
+        constants = {
+            "pi": 16 * arctan_inverse(5) - 4 * arctan_inverse(239),
+            "e": Decimal(1).exp(),
+            "sqrt2": Decimal(2).sqrt(),
+            "sqrt3": Decimal(3).sqrt(),
+            "ln2": Decimal(2).ln(),
+            "phi": (1 + Decimal(5).sqrt()) / 2,
+        }
+        pairs = {name: split(Fraction(value)) for name, value in constants.items()}
+
+    rows = []
+    for left_name, right_name in [("pi", "e"), ("sqrt2", "sqrt3"), ("ln2", "phi")]:
+        for overload in ["DD/DD", "DD/double", "double/DD"]:
+            left_high, left_low = pairs[left_name]
+            right_high, right_low = pairs[right_name]
+            if overload == "DD/double":
+                right_low = 0.0
+            if overload == "double/DD":
+                left_low = 0.0
+            left = Fraction(left_high) + Fraction(left_low)
+            right = Fraction(right_high) + Fraction(right_low)
+            values = [left_high, left_low, right_high, right_low]
+            for expected in [left + right, left - right, left * right, left / right]:
+                high, low = split(expected)
+                # These precomputed references are much closer than the test's
+                # 2^-100 bound; this does not assert library correct rounding.
+                assert abs(Fraction(high) + Fraction(low) - expected) <= abs(expected) / (1 << 105)
+                values.extend([high, low])
+            rows.append((f"{left_name}, {right_name}: {overload}", values))
+    return rows
+
+
+if __name__ == "__main__":
+    rows = fixtures(160)
+    assert rows == fixtures(240), "Increase precision: binary64 fixtures did not stabilize"
+    for label, values in rows:
+        print(f"    // {label}")
+        overload = label.split(": ")[1]
+        print(f'    [InlineData("{overload}",')
+        for index in range(0, len(values), 2):
+            suffix = ")]" if index == len(values) - 2 else ","
+            print(f"        {values[index]!r}, {values[index + 1]!r}{suffix}")

README.md

@@ -16,8 +16,9 @@ for a single `double`, or `DoubleDouble.FromComponents(high, low)` for arbitrary
 components. The factory normalizes finite sums and canonicalizes NaN/infinity
 with a positive-zero low component. The two-component constructor is internal
 and performs no normalization or validation; it is reserved for trusted,
-already-normalized results. The constants `PI`, `E`, and `LN2` include binary64
-residuals checked against independently computed high-precision values.
+already-normalized results. Mathematical constants use precomputed high/low pairs
+checked against independently computed high-precision values; accessing them does
+not perform double-double arithmetic or allocate on the heap.
 
 - Arithmetic: unary `+`/`-`, binary `+`, `-`, `*`, `/`, and both operand orders with
   a `double`. Addition retains residuals under cancellation; multiplication uses
@@ -44,6 +45,39 @@ with exact component checks for selected representable cases. Near underflow,
 extended precision necessarily decreases; overflow produces infinity. Performance
 of the allocating exponent-boundary path is not covered by the basic benchmarks.
 
+## Predefined mathematical constants
+
+All constants below are static `DoubleDouble` properties. Each stores the nearest
+binary64 high component followed by the nearest binary64 residual, rather than
+calculating a ratio, root, or logarithm on access. Names use PascalCase, including
+`Pi`, `E`, and `Ln2`.
+
+| Group | Properties and values |
+|---|---|
+| Circle and common angles | `Pi` (π), `Tau` (2π), `PiOver2`, `PiOver3`, `PiOver4`, `PiOver6` |
+| Angular conversion | `DegToRad` (π/180), `RadToDeg` (180/π), `InvPi` (1/π), `InvTau` (1/(2π), radians to turns) |
+| Exponential and logarithmic | `E`, `InvE` (1/e), `Ln2` (ln 2), `Ln10` (ln 10) |
+| Log-base conversion | `Log2E` (1/ln 2), `Log10E` (1/ln 10), `Log2Of10` (ln 10/ln 2), `Log10Of2` (ln 2/ln 10) |
+| Roots and geometry | `Sqrt2`, `Sqrt3`, `Sqrt5`, `InvSqrt2`, `InvSqrt3`, `GoldenRatio` ((1+√5)/2) |
+| Gaussian and error-function factors | `SqrtPi`, `InvSqrtPi`, `TwoInvSqrtPi` (2/√π), `SqrtTau` (√(2π)), `InvSqrtTau` (1/√(2π)) |
+
+Multiply by conversion factors instead of recomputing them:
+
+```csharp
+using Just.PreciseMath;
+
+DoubleDouble degrees = new(180.0);
+DoubleDouble radians = degrees * DoubleDouble.DegToRad;
+DoubleDouble convertedDegrees = radians * DoubleDouble.RadToDeg;
+DoubleDouble quarterTurn = DoubleDouble.PiOver2;
+```
+
+The factors avoid deriving constants at runtime; the multiplication itself remains
+approximate DD arithmetic, so conversions are not guaranteed exact round trips.
+The list is mathematical and dimensionless, not a table of unit-dependent physical
+constants. Precomputed roots/logarithms do not imply general `Sqrt`/`Log` functions
+are implemented.
+
 ## Conversions and formatting
 
 - Explicit conversions support `double`, `float`, `int`, `long`, and `decimal`