Round exact quotients of approximations to the requested digits [patch]

Divide returns a terminating quotient exactly. Asin(±1), Atan2(y, 0),
DegreesToRadians and RootN(x, -n) divided a value already rounded to a
wider working precision, so when the quotient terminated they returned
the guard digits too, and those digits were wrong: Asin(1, 5) gave
1.570796326794895 and RootN(1.0486, -2, 4) gave 0.9765625.

Add DivideApproximation, which divides at working precision and then
rounds to significantDigits, and use it on those paths and on the other
approx / approx paths (Tan, TanPi, AsinPi, AcosPi, AtanPi,
RadiansToDegrees). RootN(x, -n) now takes the inner root with guard
digits.

Fixes #121

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_014jNTPLxThrQnZyafw1CZ1p

Author: claude

PreciseNumber.Test/PreciseNumberRootTests.cs

@@ -299,4 +299,12 @@ public void TestRootsOfLongSignificandsStayCorrect()
 		Assert.AreEqual(200, root.SignificantDigits, "A 200 digit input produced a shorter root");
 		AssertAgreesTo(value, root.Squared(), 195, "The root of a 200 digit value does not square back");
 	}
+
+	[TestMethod]
+	public void TestNegativeDegreeRootRoundsTheReciprocalOfAnUnroundedRoot()
+	{
+		// The square root of 1.0486 is 1.02401…, which rounds to 1.024 at four digits, and 1/1.024 is
+		// exactly 0.9765625. The true reciprocal is 0.976551…, so it must come from a wider root.
+		Assert.AreEqual(Parse("0.9766"), PreciseNumber.RootN(Parse("1.0486"), -2, 4), "RootN(1.0486, -2) is wrong");
+	}
 }

PreciseNumber.Test/PreciseNumberTrigonometryTests.cs

@@ -352,4 +352,17 @@ public void TestInverseHalfTurnFamilyReturnsHalfTurns()
 		Assert.AreEqual(PreciseNumber.One, PreciseNumber.AcosPi(PreciseNumber.NegativeOne, 50));
 		AssertAgreesTo(Parse("0.25"), PreciseNumber.AtanPi(PreciseNumber.One, 50), 49, "AtanPi(1) is a quarter turn");
 	}
+
+	[TestMethod]
+	public void TestExactQuotientsOfApproximatePiStillRoundToTheRequestedDigits()
+	{
+		// π at working precision halves, or divides by 180, without a remainder. That exact quotient
+		// must still be rounded to the digits asked for, not handed back with the guard digits on it.
+		Assert.AreEqual(Parse("1.5708"), PreciseNumber.Asin(PreciseNumber.One, 5), "Asin(1) is wrong");
+		Assert.AreEqual(Parse("-1.5708"), PreciseNumber.Asin(-PreciseNumber.One, 5), "Asin(-1) is wrong");
+		Assert.AreEqual(Parse("1.5708"), PreciseNumber.Atan2(PreciseNumber.One, PreciseNumber.Zero, 5), "Atan2(1, 0) is wrong");
+		Assert.AreEqual(Parse("-1.5708"), PreciseNumber.Atan2(-PreciseNumber.One, PreciseNumber.Zero, 5), "Atan2(-1, 0) is wrong");
+		Assert.AreEqual(Parse("3.1416"), PreciseNumber.DegreesToRadians(Parse("180"), 5), "DegreesToRadians(180) is wrong");
+		Assert.AreEqual(Parse("1.5708"), PreciseNumber.DegreesToRadians(Parse("90"), 5), "DegreesToRadians(90) is wrong");
+	}
 }

PreciseNumber/PreciseNumber.Roots.cs

@@ -158,7 +158,7 @@ public static PreciseNumber RootN(PreciseNumber x, int n, int significantDigits)
 		{
 			return x.Significand.IsZero
 				? throw new DivideByZeroException()
-				: Divide(One, RootN(x, -n, significantDigits), significantDigits);
+				: DivideApproximation(One, RootN(x, -n, significantDigits + RootGuardDigits), significantDigits + RootGuardDigits, significantDigits);
 		}
 
 		if (x.Significand.IsZero || n == 1)

PreciseNumber/PreciseNumber.Trigonometry.cs

@@ -236,7 +236,7 @@ public static PreciseNumber Tan(PreciseNumber x, int significantDigits)
 		RequireSignificantDigits(significantDigits);
 		int working = significantDigits + TrigonometricGuardDigits;
 		(PreciseNumber sin, PreciseNumber cos) = SinCos(x, working);
-		return Divide(sin, cos, significantDigits);
+		return DivideApproximation(sin, cos, working, significantDigits);
 	}
 
 	/// <summary>
@@ -281,7 +281,7 @@ public static PreciseNumber Asin(PreciseNumber x, int significantDigits)
 
 		if (magnitude == One)
 		{
-			PreciseNumber halfPi = Divide(PiTo(working), Two, significantDigits);
+			PreciseNumber halfPi = DivideApproximation(PiTo(working), Two, working, significantDigits);
 			return x.Significand.Sign > 0 ? halfPi : -halfPi;
 		}
 
@@ -431,7 +431,7 @@ public static PreciseNumber Atan2(PreciseNumber y, PreciseNumber x, int signific
 				return Zero;
 			}
 
-			PreciseNumber halfPi = Divide(PiTo(working), Two, significantDigits);
+			PreciseNumber halfPi = DivideApproximation(PiTo(working), Two, working, significantDigits);
 			return y.Significand.Sign > 0 ? halfPi : -halfPi;
 		}
 
@@ -562,7 +562,7 @@ public static PreciseNumber TanPi(PreciseNumber x, int significantDigits)
 		RequireSignificantDigits(significantDigits);
 		int working = significantDigits + TrigonometricGuardDigits;
 		(PreciseNumber sin, PreciseNumber cos) = SinCosPi(x, working);
-		return Divide(sin, cos, significantDigits);
+		return DivideApproximation(sin, cos, working, significantDigits);
 	}
 
 	/// <summary>
@@ -599,7 +599,7 @@ public static PreciseNumber AsinPi(PreciseNumber x, int significantDigits)
 		}
 
 		int working = significantDigits + TrigonometricGuardDigits;
-		return Divide(Asin(x, working), PiTo(working), significantDigits);
+		return DivideApproximation(Asin(x, working), PiTo(working), working, significantDigits);
 	}
 
 	/// <summary>
@@ -636,7 +636,7 @@ public static PreciseNumber AcosPi(PreciseNumber x, int significantDigits)
 		}
 
 		int working = significantDigits + TrigonometricGuardDigits;
-		return Divide(Acos(x, working), PiTo(working), significantDigits);
+		return DivideApproximation(Acos(x, working), PiTo(working), working, significantDigits);
 	}
 
 	/// <summary>
@@ -664,7 +664,7 @@ public static PreciseNumber AtanPi(PreciseNumber x, int significantDigits)
 		}
 
 		int working = significantDigits + TrigonometricGuardDigits;
-		return Divide(Atan(x, working), PiTo(working), significantDigits);
+		return DivideApproximation(Atan(x, working), PiTo(working), working, significantDigits);
 	}
 
 	/// <summary>
@@ -692,7 +692,7 @@ public static PreciseNumber DegreesToRadians(PreciseNumber degrees, int signific
 	{
 		RequireSignificantDigits(significantDigits);
 		int working = significantDigits + TrigonometricGuardDigits;
-		return Divide(Multiply(degrees, PiTo(working)), OneEighty, significantDigits);
+		return DivideApproximation(Multiply(degrees, PiTo(working)), OneEighty, working, significantDigits);
 	}
 
 	/// <summary>
@@ -720,7 +720,7 @@ public static PreciseNumber RadiansToDegrees(PreciseNumber radians, int signific
 	{
 		RequireSignificantDigits(significantDigits);
 		int working = significantDigits + TrigonometricGuardDigits;
-		return Divide(Multiply(radians, OneEighty), PiTo(working), significantDigits);
+		return DivideApproximation(Multiply(radians, OneEighty), PiTo(working), working, significantDigits);
 	}
 
 	/// <summary>
@@ -853,6 +853,23 @@ private static (PreciseNumber Sin, PreciseNumber Cos) SinCosSeries(PreciseNumber
 			$"The trigonometric series did not converge to {workingDigits.ToString(InvariantCulture)} significant digits.");
 	}
 
+	/// <summary>
+	/// Divides a value that is already an approximation, then rounds to the digits the caller asked for.
+	/// </summary>
+	/// <param name="numerator">The dividend, carried at <paramref name="workingDigits"/>.</param>
+	/// <param name="denominator">The divisor.</param>
+	/// <param name="workingDigits">The significant digits the operands were computed to.</param>
+	/// <param name="significantDigits">The number of significant digits to produce.</param>
+	/// <returns>The quotient, rounded to <paramref name="significantDigits"/>.</returns>
+	/// <remarks>
+	/// <see cref="Divide(PreciseNumber, PreciseNumber, int)"/> returns a terminating quotient exactly,
+	/// without rounding it. When an operand is a rounded approximation, such as π at working precision,
+	/// that exact quotient still carries the operand's guard digits, and they are wrong. Rounding
+	/// afterwards keeps the result to the digits requested whether or not the quotient terminates.
+	/// </remarks>
+	private static PreciseNumber DivideApproximation(PreciseNumber numerator, PreciseNumber denominator, int workingDigits, int significantDigits) =>
+		Divide(numerator, denominator, workingDigits).ReduceSignificance(significantDigits);
+
 	/// <summary>
 	/// Sums the arc tangent series for a small argument.
 	/// </summary>