Data update
This commit is contained in:
Ingy döt Net
parent
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commit
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10248 changed files with 63654 additions and 6775 deletions
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@ -1,57 +1,55 @@
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parse version version; say version; glob. = ''
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say 'Gamma function in arbitrary precision'
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say
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include Settings
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say version; say 'Gamma'; say
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arg n; if n = '' then n = 100; numeric digits n
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say '(Half)integers formulas'
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w = '-99.5 -10.5 -5.5 -2.5 -1.5 -0.5 0.5 1 1.5 2 2.5 5 5.5 10 10.5 99 99.5'
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numeric digits 100
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do i = 1 to words(w)
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x = word(w,i); call time('r'); r = Gamma(x); e = format(time('e'),,3)
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say 'Formulas' format(x,4,1) r '('e 'seconds)'
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numeric digits n
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do i = 1 to Words(w)
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x = Word(w,i); call Time('r'); r = Gamma(x); e = Format(Time('e'),,3)
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say 'Formulas' Format(x,4,1) r '('e 'seconds)'
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end
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say
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say 'Lanczos (max 60 digits) vs Spouge (no limit) vs Stirling (no limit) approximation'
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say 'Lanczos (max 60 decimals) vs Spouge (no limit) vs Stirling (no limit) approximation'
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w = '-12.8 -6.4 -3.2 -1.6 -0.8 -0.4 -0.2 -0.1 0.1 0.2 0.4 0.8 1.6 3.2 6.4 12.8'
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do i = 1 to words(w)
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x = word(w,i)
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numeric digits 60
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call time('r'); r = Gamma(x); e = format(time('e'),,3)
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say 'Lanczos ' format(x,4,1) r '('e 'seconds)'
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numeric digits 61
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call time('r'); r = Gamma(x); e = format(time('e'),,3)
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say 'Spouge ' format(x,4,1) r '('e 'seconds)'
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do i = 1 to Words(w)
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x = Word(w,i)
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numeric digits Min(60,n)
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call Time('r'); r = Gamma(x); e = Format(Time('e'),,3)
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say 'Lanczos ' Format(x,4,1) r '('e 'seconds)'
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numeric digits n
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call Time('r'); r = Gamma(x); e = Format(Time('e'),,3)
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say 'Spouge ' Format(x,4,1) r '('e 'seconds)'
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if x > 0 then do
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call time('r'); r = Stirling(x); e = format(time('e'),,3)
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say 'Stirling' format(x,4,1) r '('e 'seconds)'
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call Time('r'); r = Stirling(x); e = Format(Time('e'),,3)
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say 'Stirling' Format(x,4,1) r '('e 'seconds)'
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end
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end
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say
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say 'Same for a bigger number'
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w = '-99.9 99.9'
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do i = 1 to words(w)
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x = word(w,i)
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numeric digits 60
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call time('r'); r = Gamma(x); e = format(time('e'),,3)
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say 'Lanczos ' format(x,4,1) r '('e 'seconds)'
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numeric digits 100
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call time('r'); r = Gamma(x); e = format(time('e'),,3)
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say 'Spouge ' format(x,4,1) r '('e 'seconds)'
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do i = 1 to Words(w)
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x = Word(w,i)
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numeric digits Min(60,n)
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call Time('r'); r = Gamma(x); e = Format(Time('e'),,3)
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say 'Lanczos ' Format(x,4,1) r '('e 'seconds)'
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numeric digits n
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call Time('r'); r = Gamma(x); e = Format(Time('e'),,3)
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say 'Spouge ' Format(x,4,1) r '('e 'seconds)'
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if x > 0 then do
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call time('r'); r = Stirling(x); e = format(time('e'),,3)
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say 'Stirling' format(x,4,1) r '('e 'seconds)'
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call Time('r'); r = Stirling(x); e = Format(Time('e'),,3)
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say 'Stirling' Format(x,4,1) r '('e 'seconds)'
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end
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end
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exit
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Gamma:
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/* Gamma */
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procedure expose glob.
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procedure expose glob. fact.
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arg x
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/* Validity */
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if x < 1 & Whole(x) then
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return 'X'
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/* Formulas for negative and positive (half)integers */
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if x < 0 then do
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if Half(x) then do
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if IsHalf(x) then do
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numeric digits Digits()+2
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i = Abs(Floor(x)); y = (-1)**i*2**(2*i)*Fact(i)*Sqrt(Pi())/Fact(2*i)
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numeric digits Digits()-2
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@ -59,9 +57,9 @@ if x < 0 then do
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end
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end
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if x > 0 then do
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if Whole(x) then
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if IsWhole(x) then
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return Fact(x-1)
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if Half(x) then do
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if IsHalf(x) then do
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numeric digits Digits()+2
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i = Floor(x); y = Fact(2*i)*Sqrt(Pi())/(2**(2*i)*Fact(i))
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numeric digits Digits()-2
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@ -167,351 +165,11 @@ return y+0
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Stirling:
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/* Sterling */
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procedure expose glob.
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procedure expose glob. fact.
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arg x
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return sqrt(2*pi()/x) * power(x/e(),x)
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return Sqrt(2*Pi()/x) * Power(x/e(),x)
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E:
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/* Euler number */
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procedure expose glob.
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p = Digits()
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/* In memory? */
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if glob.e.p <> '' then
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return glob.e.p
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if p < 101 then
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/* Fast value */
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glob.e.p = 2.71828182845904523536028747135266249775724709369995957496696762772407663035354759457138217852516643+0
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else do
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numeric digits Digits()+2
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/* Taylor series */
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y = 2; t = 1; v = y
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do n = 2
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t = t/n; y = y+t
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if y = v then
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leave
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v = y
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end
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numeric digits Digits()-2
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glob.e.p = y+0
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end
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return glob.e.p
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Exp:
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/* Exponential e^x */
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procedure expose glob.
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arg x
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numeric digits Digits()+2
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/* Fast values */
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if Whole(x) then
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return E()**x
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/* Argument reduction */
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i = x%1
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if Abs(x-i) > 0.5 then
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i = i+Sign(x)
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/* Taylor series */
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x = x-i; y = 1; t = 1; v = y
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do n = 1
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t = (t*x)/n; y = y+t
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if y = v then
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leave
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v = y
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end
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/* Inverse reduction */
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y = y*e()**i
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numeric digits Digits()-2
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return y+0
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Fact:
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/* Factorial n! */
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procedure expose glob.
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arg x
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/* Validity */
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if \ Whole(x) then
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return 'X'
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if x < 0 then
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return 'X'
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/* Current in memory? */
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if glob.fact.x <> '' then
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return glob.fact.x
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w = x-1
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/* Previous in memory? */
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if glob.fact.w = '' then do
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/* Loop cf definition */
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y = 1
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do n = 2 to x
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y = y*n
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end
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glob.fact.x = y
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end
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else
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/* Multiply */
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glob.fact.x = glob.fact.w*x
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return glob.fact.x
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Floor:
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/* Floor */
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procedure expose glob.
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arg x
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/* Formula */
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if Whole(x) then
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return x
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else
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return Trunc(x)-(x<0)
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Frac:
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/* Fractional part */
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procedure expose glob.
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arg x
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/* Formula */
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return x-x%1
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Half:
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/* Is a number half integer? */
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procedure expose glob.
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arg x
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/* Formula */
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return (Frac(Abs(x))=0.5)
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Ln:
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/* Natural logarithm base e */
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procedure expose glob.
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arg x
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/* Validity */
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if x <= 0 then
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return 'X'
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/* Fast values */
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if x = 1 then
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return 0
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p = Digits()
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/* In memory? */
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if glob.ln.x.p <> '' then
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return glob.ln.x.p
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/* Precalculated values */
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if x = 2 & p < 101 then do
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glob.ln.x.p = Ln2()
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return glob.ln.x.p
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end
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if x = 4 & p < 101 then do
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glob.ln.x.p = Ln4()
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return glob.ln.x.p
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end
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if x = 8 & p < 101 then do
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glob.ln.x.p = Ln8()
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return glob.ln.x.p
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end
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if x = 10 & p < 101 then do
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glob.ln.x.p = Ln10()
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return glob.ln.x.p
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end
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numeric digits p+2
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/* Argument reduction */
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z = x; i = 0; e = 1/E()
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if z < 0.5 then do
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y = 1/z
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do while y > 1.5
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y = y*e; i = i-1
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end
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z = 1/y
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end
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if z > 1.5 then do
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do while z > 1.5
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z = z*e; i = i+1
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end
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end
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/* Taylor series */
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q = (z-1)/(z+1); f = q; y = q; v = q; q = q*q
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do n = 3 by 2
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f = f*q; y = y+f/n
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if y = v then
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leave
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v = y
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end
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numeric digits p
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/* Inverse reduction */
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glob.ln.x.p = 2*y+i
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return glob.ln.x.p
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Power:
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/* Power function x^y */
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procedure expose glob.
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arg x,y
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/* Validity */
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if x < 0 then
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return 'X'
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/* Fast values */
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if x = 0 then
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return 0
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if y = 0 then
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return 1
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/* Fast formula */
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if Whole(y) then
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return x**y
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/* Formulas */
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if Abs(y//1) = 0.5 then
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return Sqrt(x)**Sign(y)*x**(y%1)
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else
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return Exp(y*Ln(x))
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Sin:
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/* Sine */
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procedure expose glob.
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arg x
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numeric digits Digits()+2
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/* Argument reduction */
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u = Pi(); x = x//(2*u)
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if Abs(x) > u then
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x = x-Sign(x)*2*u
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/* Taylor series */
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t = x; y = x; x = x*x; v = y
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do n = 2 by 2
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t = -t*x/(n*(n+1)); y = y+t
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if y = v then
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leave
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v = y
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end
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numeric digits Digits()-2
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return y+0
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Sqrt:
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/* Square root x^(1/2) */
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procedure expose glob.
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arg x
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/* Validity */
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if x < 0 then
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return 'X'
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/* Fast values */
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if x = 0 then
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return 0
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if x = 1 then
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return 1
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p = Digits()
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/* Predefined values */
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if x = 2 & p < 101 then
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return Sqrt2()
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if x = 3 & p < 101 then
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return Sqrt3()
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if x = 5 & p < 101 then
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return Sqrt5()
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numeric digits p+2
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/* Argument reduction to [0,100) */
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i = Xpon(x); i = (i-(i<0))%2; x = x/100**i
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/* First guess 1 digit accurate */
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t = '2.5 6.5 12.5 20.5 30.5 42.5 56.5 72.5 90.5 100'
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do y = 1 until word(t,y) > x
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end
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/* Dynamic precision */
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d = Digits()
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do n = 1 while d > 2
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d.n = d; d = d%2+1
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end
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d.n = 2
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/* Method Heron */
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do k = n to 1 by -1
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numeric digits d.k
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y = (y+x/y)*0.5
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end
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numeric digits p
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return y*10**i
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Whole:
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/* Is a number integer? */
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procedure expose glob.
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arg x
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/* Formula */
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return Datatype(x,'w')
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Xpon:
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/* Exponent */
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procedure expose glob.
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arg x
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/* Formula */
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if x = 0 then
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return 0
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else
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return Right(x*1E+99999,6)-99999
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Ln2:
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/* Natural log of 2 */
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procedure expose glob.
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/* Fast value */
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y = 0.6931471805599453094172321214581765680755001343602552541206800094933936219696947156058633269964186875
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return y+0
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Ln4:
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/* Natural log of 4 */
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procedure expose glob.
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/* Fast value */
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y = 1.386294361119890618834464242916353136151000268720510508241360018986787243939389431211726653992837375
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return y+0
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Ln8:
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/* Natural log of 8 */
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procedure expose glob.
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/* Fast value */
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y = 2.079441541679835928251696364374529704226500403080765762362040028480180865909084146817589980989256063
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return y+0
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Ln10:
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/* Natural log of 10 */
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procedure expose glob.
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/* Fast value */
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y = 2.30258509299404568401799145468436420760110148862877297603332790096757260967735248023599720508959830
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return y+0
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Pi:
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/* Pi */
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procedure expose glob.
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p = Digits()
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/* In memory? */
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if glob.pi.p <> '' then
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return glob.pi.p
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if p < 101 then
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/* Fast value */
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glob.pi.p = 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211707+0
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else do
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numeric digits Digits()+2
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if p < 201 then do
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/* Method Chudnovsky series */
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y = 0
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do n = 0
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v = y; y = y + Fact(6*n)*(13591409+545140134*n)/(Fact(3*n)*Fact(n)**3*-640320**(3*n))
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if y = v then
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leave
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end
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y = 4270934400/(Sqrt(10005)*y)
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end
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else do
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/* Method Agmean series */
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y = 0.25; a = 1; g = Sqrt(0.5); n = 1
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do until a = v
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v = a
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x = (a+g)*0.5; g = Sqrt(a*g)
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y = y-n*(x-a)**2; n = n+n; a = x
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end
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y = a*a/y
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end
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numeric digits Digits()-2
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glob.pi.p = y+0
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end
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return glob.pi.p
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Sqrt2:
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/* Square root of 2 */
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procedure expose glob.
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/* Fast value */
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y = 1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
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return y+0
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Sqrt3:
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/* Square root of 3 */
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procedure expose glob.
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/* Fast value */
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y = 1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576
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return y+0
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Sqrt5:
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/* Square root of 5 */
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procedure expose glob.
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/* Fast value */
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y = 2.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275
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return y+0
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include Constants
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include Functions
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include Numbers
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include Abend
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