Apply approximation to Asinh(x) and Acosh(x)

src/Numerics/Trigonometry.cs

@@ -713,7 +713,13 @@ namespace MathNet.Numerics
         /// <returns>The hyperbolic angle, i.e. the area of its hyperbolic sector.</returns>
         public static double Asinh(double value)
         {
-            return Math.Log(value + Math.Sqrt((value * value) + 1), Math.E);
+            // asinh(x) = Sign(x) * ln(|x| + sqrt(x*x + 1))
+            // if |x| > huge, asinh(x) ~= Sign(x) * ln(2|x|)
+
+            if (Math.Abs(value) >= 268435456.0) // 2^28, taken from freeBSD
+                return Math.Sign(value) * (Math.Log(Math.Abs(value)) + Math.Log(2.0));
+
+            return Math.Sign(value) * Math.Log(Math.Abs(value) + Math.Sqrt((value * value) + 1));
         }
 
         /// <summary>
@@ -733,6 +739,12 @@ namespace MathNet.Numerics
         /// <returns>The hyperbolic angle, i.e. the area of its hyperbolic sector.</returns>
         public static double Acosh(double value)
         {
+            // acosh(x) = ln(x + sqrt(x*x - 1))
+            // if |x| >= 2^28, acosh(x) ~ ln(x) + ln(2)
+
+            if (Math.Abs(value) >= 268435456.0) // 2^28, taken from freeBSD
+                return Math.Log(value) + Math.Log(2.0);
+
             return Math.Log(value + (Math.Sqrt(value - 1) * Math.Sqrt(value + 1)), Math.E);
         }
 

src/UnitTests/TrigonometryTest.cs

@@ -338,6 +338,7 @@ namespace MathNet.Numerics.UnitTests
         /// <param name="expected">Expected value.</param>
         [TestCase(1.0, 0.0)]
         [TestCase(8388608, 16.635532333438682)]
+        [TestCase(1.7976931348623157E+308, 710.47586007394394203711)]
         public void CanComputeInverseHyperbolicCosine(double value, double expected)
         {
             var actual = Trig.Acosh(value);
@@ -383,6 +384,8 @@ namespace MathNet.Numerics.UnitTests
         [TestCase(-8388608, -16.63553233343869)]
         [TestCase(1.19209289550780998537e-7, 1.1920928955078072e-7)]
         [TestCase(-1.19209289550780998537e-7, -1.1920928955078072e-7)]
+        [TestCase(1.7976931348623157E+308, 710.47586007394394203711)]
+        [TestCase(-1.7976931348623157E+308, -710.47586007394394203711)]
         public void CanComputeInverseHyperbolicSine(double value, double expected)
         {
             var actual = Trig.Asinh(value);