# picomath

## Scheme

### gamma.scm

```; Visit http://www.johndcook.com/stand_alone_code.html for the source of this code and more like it.

; Note that the functions Gamma and LogGamma are mutually dependent.

(define (gamma x)

(if (<= x 0)
(error "Invalid input"))

; Split the function domain into three intervals:
; (0, 0.001), [0.001, 12), and (12, infinity)

(cond

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; First interval: (0, 0.001)
;
; For small x, 1/Gamma(x) has power series x + gamma x^2  - ...
; So in this range, 1/Gamma(x) = x + gamma x^2 with error on the order of x^3.
; The relative error over this interval is less than 6e-7.

((< x 0.001)
(let ((gamma 0.577215664901532860606512090)) ; Euler's gamma constant
(/ 1.0 (* x (+ 1.0 (* gamma x))))))

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Second interval: [0.001, 12)

((< x 12.0)
; The algorithm directly approximates gamma over (1,2) and uses
; reduction identities to reduce other arguments to this interval.

(letrec ((gamma-iter (lambda (z num den ps qs)
(if (null? ps)
(list num den)
(let ((new-num (* (+ num (car ps)) z))
(new-den (+ (* den z) (car qs))))
(gamma-iter z new-num new-den (cdr ps) (cdr qs))))))
(gamma-z-n (lambda (result y n)
(if (= n 0)
result
(gamma-z-n (* result y) (+ y 1) (- n 1))))))

(let* ((arg-was-less-than-one (< x 1.0))
; Add or subtract integers as necessary to bring y into (1,2)
; Will correct for this below
(n (if arg-was-less-than-one 0 (- (truncate x) 1)))
(y (if arg-was-less-than-one (+ x 1.0) (- x n)))

; numerator coefficients for approximation over the interval (1,2)
(p '(-1.71618513886549492533811E+0
2.47656508055759199108314E+1
-3.79804256470945635097577E+2
6.29331155312818442661052E+2
8.66966202790413211295064E+2
-3.14512729688483675254357E+4
-3.61444134186911729807069E+4
6.64561438202405440627855E+4))

; denominator coefficients for approximation over the interval (1,2)
(q '(-3.08402300119738975254353E+1
3.15350626979604161529144E+2
-1.01515636749021914166146E+3
-3.10777167157231109440444E+3
2.25381184209801510330112E+4
4.75584627752788110767815E+3
-1.34659959864969306392456E+5
-1.15132259675553483497211E+5))

(z (- y 1))
(a (gamma-iter z 0.0 1.0 p q))
(num (car a))
(result (+ (/ num den) 1.0)))

; Apply correction if argument was not initially in (1,2)
(if arg-was-less-than-one
(/ result (- y 1.0))
(gamma-z-n result y n)))))

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Third interval: [12, infinity)

((<= x 171.624)
(exp (log-gamma x)))

(else
; Correct answer too large to display.
(inf))))

(define (log-gamma x)
(if (<= x 0)
(error "Invalid input"))

(if (< x 12.0)
(log (abs (gamma x)))

; Abramowitz and Stegun 6.1.41
; Asymptotic series should be good to at least 11 or 12 figures
; For error analysis, see Whittiker and Watson
; A Course in Modern Analysis (1927), page 252

(letrec ((log-gamma-iter (lambda (z sum c)
(if (null? c)
sum
(let ((s (+ (* sum z) (car c))))
(log-gamma-iter z s (cdr c)))))))

(let* ((c (reverse (list (/  1.0 12.0)
(/ -1.0 360.0)
(/  1.0 1260.0)
(/ -1.0 1680.0)
(/  1.0 1188.0)
(/ -691.0 360360.0)
(/  1.0 156.0)
(/ -3617.0 122400.0))))
(z (/ 1.0 (* x x)))
(sum (log-gamma-iter z 0 c))
(series (/ sum x))
(half-log-two-pi 0.91893853320467274178032973640562))
(+ (* (- x 0.5) (log x)) (- x) half-log-two-pi series)))))
```