Make Spouge's constants from exact ratios and powers of e

Sources/AngouriMath/Functions/InternalAMExtensions.cs

@@ -1245,17 +1245,24 @@ internal static EDecimal[] GetSpougeFactorialConstants(int a)
 
                 constants[0] = EDecimal.PI(mc).Multiply(2, mc).Sqrt(mc);
 
-                var negative = false;
+                // c_k = (-1)^(k - 1) (a - k)^(k - 1)/(k - 1)! sqrt(a - k) e^(a - k), the first factor a ratio of exact
+                // integers and e^(a - k) a product of e's, with digits to spare for the a multiplications that make it.
+                // With a power of a - k to k - 1/2 and an exponential for each k, the constants for 300 digits took 20
+                // seconds, and for 600 digits 218.
+                var work = EContext.ForPrecision(a * 15 / 10 + 10);
+                var e = EDecimal.One.Exp(work);
+                var powersOfE = new EDecimal[a];
+                powersOfE[0] = EDecimal.One;
+                for (int m = 1; m < a; m++)
+                    powersOfE[m] = powersOfE[m - 1].Multiply(e, work);
                 for (int k = 1; k < a; k++)
                 {
-                    var deltaAK = EDecimal.FromInt32(a - k);
-                    var ck = deltaAK.Pow(EDecimal.FromInt32(k).Subtract(0.5m), mc);
-                    ck = deltaAK.Exp(mc).Multiply(ck, mc);
-                    ck = ck.Divide(Factorial(k - 1), mc);
-                    if (negative)
-                        ck = ck.Negate();
-                    constants[k] = ck;
-                    negative = !negative;
+                    var m = a - k;
+                    var ck = EDecimal.FromEInteger(EInteger.FromInt32(m).Pow(k - 1)).Divide(EDecimal.FromEInteger(Factorial(k - 1)), work)
+                        .Multiply(EDecimal.FromInt32(m).Sqrt(work), work)
+                        .Multiply(powersOfE[m], work)
+                        .RoundToPrecision(mc);
+                    constants[k] = k % 2 == 0 ? ck.Negate() : ck;
                 }
 
                 return constants;

Sources/Tests/UnitTests/Core/FactorialFarFromZeroTest.cs

@@ -1,4 +1,4 @@
-//
+//
 // Copyright (c) 2019-2026 Angouri.
 // AngouriMath is licensed under MIT.
 // Details: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.
@@ -78,6 +78,24 @@ public void ADivisorsModulusNearAWholeNumberIsNotRoundedToIt()
             Close("-0.002993872340987814525483818825494753213055396187438519990678101458851071", value.ImaginaryPart.EDecimal, 60);
         }
 
+        /// <summary>
+        /// At 300 digits, on the real line and off it. Spouge's 390 constants were made with a power and an exponential
+        /// each, and the first factorial at that precision took 20 seconds; they are a ratio of integers, a square root and
+        /// a product of <c>e</c>'s now.
+        /// </summary>
+        [Fact]
+        public void AtThreeHundredDigits()
+        {
+            var context = EContext.ForPrecision(300);
+            Close("0.89297951156924921121856431365822588137622979265243370031680944253013920338924793984699429634706292980638634973335742111190636152316815741999257112256948992272242994101666773944751677408238564727251848331757626146492648043525527613387077304719882217534457586134019354517131441367733861313846090028508017368",
+                Evaluate("(1/3)!", context).RealPart.EDecimal, 295);
+            var value = Evaluate("(1/3 + i)!", context);
+            Close("0.55310533708451368643420143855148996023577256508846542223376506819123717723035945992089147907370521801219397887942272059738687781730094054546525636372408182990085240891013953705988378166744479672870436455961985418938874599630727253813312731648659256268261654562175265755195218626689594794087234395264053064",
+                value.RealPart.EDecimal, 295);
+            Close("0.0097530148049768242524751667053160295250646412881078165005696874705860541880090280173509525760804027158706765197226375317213454843960898145149813764056673011748957471092198259369271410261828558753507851947578513569017247888018679994921424076865568363297838883827120115940210102202849352569014259005743340527",
+                value.ImaginaryPart.EDecimal, 293);
+        }
+
         /// <summary>The reflection formula off the real line, which the complex factorial now takes left of zero.</summary>
         [Fact]
         public void LeftOfZeroOffTheRealLine()