root/qd-gamma.lisp @ 19dd5231c42c6eec40a7876bd9ffb2661fb05012

;;;; -*- Mode: lisp -*-
;;;;
;;;; Copyright (c) 2011 Raymond Toy
;;;; Permission is hereby granted, free of charge, to any person
;;;; obtaining a copy of this software and associated documentation
;;;; files (the "Software"), to deal in the Software without
;;;; restriction, including without limitation the rights to use,
;;;; copy, modify, merge, publish, distribute, sublicense, and/or sell
;;;; copies of the Software, and to permit persons to whom the
;;;; Software is furnished to do so, subject to the following
;;;; conditions:
;;;;
;;;; The above copyright notice and this permission notice shall be
;;;; included in all copies or substantial portions of the Software.
;;;;
;;;; THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
;;;; EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
;;;; OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
;;;; NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
;;;; HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
;;;; WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
;;;; FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
;;;; OTHER DEALINGS IN THE SOFTWARE.

(in-package #:oct)

(eval-when (:compile-toplevel :load-toplevel :execute)
  (setf *readtable* *oct-readtable*))

;; For log-gamma we use the asymptotic formula
;;
;; log(gamma(z)) ~ (z - 1/2)*log(z) + log(2*%pi)/2
;;                   + sum(bern(2*k)/(2*k)/(2*k-1)/z^(2k-1), k, 1, inf)
;;
;;               = (z - 1/2)*log(z) + log(2*%pi)/2
;;                  + 1/12/z*(1 - 1/30/z^2 + 1/105/z^4 + 1/140/z^6 + ...
;;                              + 174611/10450/z^18 + ...)
;;
;; For double-floats, let's stop the series at the power z^18.  The
;; next term is 77683/483/z^20.  This means that for |z| > 8.09438,
;; the series has double-float precision.
;;
;; For quad-doubles, let's stop the series at the power z^62.  The
;; next term is about 6.364d37/z^64.  So for |z| > 38.71, the series
;; has quad-double precision.
(defparameter *log-gamma-asymp-coef*
  #(-1/30 1/105 -1/140 1/99 -691/30030 1/13 -3617/10200 43867/20349 
    -174611/10450 77683/483 -236364091/125580 657931/25 -3392780147/7830 
    1723168255201/207669 -7709321041217/42160 151628697551/33 
    -26315271553053477373/201514950 154210205991661/37 
    -261082718496449122051/1758900 1520097643918070802691/259161 
    -2530297234481911294093/9890 25932657025822267968607/2115 
    -5609403368997817686249127547/8725080 19802288209643185928499101/539 
    -61628132164268458257532691681/27030 29149963634884862421418123812691/190323 
    -354198989901889536240773677094747/31900 
    2913228046513104891794716413587449/3363 
    -1215233140483755572040304994079820246041491/16752085350 
    396793078518930920708162576045270521/61 
    -106783830147866529886385444979142647942017/171360 
    133872729284212332186510857141084758385627191/2103465
    ))

#+nil
(defun log-gamma-asymp-series (z nterms)
  ;; Sum the asymptotic formula for n terms
  ;;
  ;; 1 + sum(c[k]/z^(2*k+2), k, 0, nterms)
  (let ((z2 (* z z))
        (sum 1)
        (term 1))
    (dotimes (k nterms)
      (setf term (* term z2))
      (incf sum (/ (aref *log-gamma-asymp-coef* k) term)))
    sum))

(defun log-gamma-asymp-series (z nterms)
  (loop with y = (* z z)
     for k from 1 to nterms
     for x = 0 then
       (setf x (/ (+ x (aref *log-gamma-asymp-coef* (- nterms k)))
                  y))
     finally (return (+ 1 x))))
       

(defun log-gamma-asymp-principal (z nterms log2pi/2)
  (+ (- (* (- z 1/2)
           (log z))
        z)
     log2pi/2))

(defun log-gamma-asymp (z nterms log2pi/2)
  (+ (log-gamma-asymp-principal z nterms log2pi/2)
     (* 1/12 (/ (log-gamma-asymp-series z nterms) z))))

(defun log2pi/2 (precision)
  (ecase precision
    (single-float
     (coerce (/ (log (* 2 pi)) 2) 'single-float))
    (double-float
     (coerce (/ (log (* 2 pi)) 2) 'double-float))
    (qd-real
     (/ (log +2pi+) 2))))

(defun log-gamma-aux (z limit nterms)
  (let ((precision (float-contagion z)))
    (cond ((minusp (realpart z))
           ;; Use reflection formula if realpart(z) < 0
           ;;   log(gamma(-z)) = log(pi)-log(-z)-log(sin(pi*z))-log(gamma(z))
           ;; Or
           ;;   log(gamma(z)) = log(pi)-log(-z)-log(sin(pi*z))-log(gamma(-z))
           (- (apply-contagion (log pi) precision)
              (log (- z))
              (apply-contagion (log (sin (* pi z))) precision)
              (log-gamma (- z))))
          (t
           (let ((absz (abs z)))
             (cond ((>= absz limit)
                    ;; Can use the asymptotic formula directly with 9 terms
                    (log-gamma-asymp z nterms (log2pi/2 precision)))
                   (t
                    ;; |z| is too small.  Use the formula
                    ;; log(gamma(z)) = log(gamma(z+1)) - log(z)
                    (- (log-gamma (+ z 1))
                       (log z)))))))))

(defmethod log-gamma ((z cl:number))
  (log-gamma-aux z 9 9))

(defmethod log-gamma ((z qd-real))
  (log-gamma-aux z 26 26))

(defmethod log-gamma ((z qd-complex))
  (log-gamma-aux z 26 26))

(defun gamma-aux (z limit nterms)
  (let ((precision (float-contagion z)))
    (cond ((<= (realpart z) 0)
           ;; Use reflection formula if realpart(z) < 0:
           ;;  gamma(-z) = -pi*csc(pi*z)/gamma(z+1)
           ;; or
           ;;  gamma(z) = pi*csc(pi*z)/gamma(1-z)
           (if (and (realp z)
                    (= (truncate z) z))
               ;; Gamma of a negative integer is infinity.  Signal an error
               (error "Gamma of non-positive integer ~S" z)
               (/ (float-pi z)
                  (sin (* (float-pi z) z))
                  (gamma-aux (- 1 z) limit nterms))))
          ((and (zerop (imagpart z))
                (= z (truncate z)))
           ;; We have gamma(n) where an integer value n and is small
           ;; enough.  In this case, just compute the product
           ;; directly.  We do this because our current implementation
           ;; has some round-off for these values, and that's annoying
           ;; and unexpected.
           (let ((n (truncate z)))
             (loop
                for prod = (apply-contagion 1 precision) then (* prod k)
                for k from 2 below n
                finally (return (apply-contagion prod precision)))))
          (t
           (let ((absz (abs z)))
             (cond ((>= absz limit)
                    ;; Use log gamma directly:
                    ;;  log(gamma(z)) = principal part + 1/12/z*(series part)
                    ;; so
                    ;;  gamma(z) = exp(principal part)*exp(1/12/z*series)
                    (exp (log-gamma z))
                    #+nil
                    (* (exp (log-gamma-asymp-principal z nterms
                                                       (log2pi/2 precision)))
                       (exp (* 1/12 (/ (log-gamma-asymp-series z nterms) z)))))
                   (t
                    ;; 1 <= |z| <= limit
                    ;; gamma(z) = gamma(z+1)/z
                    (/ (gamma-aux (+ 1 z) limit nterms) z))))))))
                 
(defmethod gamma ((z cl:number))
  (gamma-aux z 9 9))

(defmethod gamma ((z qd-real))
  (gamma-aux z 39 32))

(defmethod gamma ((z qd-complex))
  (gamma-aux z 39 32))

;; Lentz's algorithm for evaluating continued fractions.
;;
;; Let the continued fraction be:
;;
;;      a1    a2    a3
;; b0 + ----  ----  ----
;;      b1 +  b2 +  b3 +
;;

(defvar *debug-cf-eval*
  nil
  "When true, enable some debugging prints when evaluating a
  continued fraction.")

;; Max number of iterations allowed when evaluating the continued
;; fraction.  When this is reached, we assume that the continued
;; fraction did not converge.
(defvar *max-cf-iterations*
  10000
  "Max number of iterations allowed when evaluating the continued
  fraction.  When this is reached, we assume that the continued
  fraction did not converge.")

(defun lentz (bf af)
  (let ((tiny-value-count 0))
    (flet ((value-or-tiny (v)
             (if (zerop v)
                 (progn
                   (incf tiny-value-count)
                   (etypecase v
                     ((or double-float cl:complex)
                      least-positive-normalized-double-float)
                     ((or qd-real qd-complex)
                      (make-qd least-positive-normalized-double-float))))
                 v)))
      (let* ((f (value-or-tiny (funcall bf 0)))
             (c f)
             (d 0)
             (eps (epsilon f)))
        (loop
           for j from 1 upto *max-cf-iterations*
           for an = (funcall af j)
           for bn = (funcall bf j)
           do (progn
                (setf d (value-or-tiny (+ bn (* an d))))
                (setf c (value-or-tiny (+ bn (/ an c))))
                (when *debug-cf-eval*
                  (format t "~&j = ~d~%" j)
                  (format t "  an = ~s~%" an)
                  (format t "  bn = ~s~%" bn)
                  (format t "  c  = ~s~%" c)
                  (format t "  d  = ~s~%" d))
                (let ((delta (/ c d)))
                  (setf d (/ d))
                  (setf f (* f delta))
                  (when *debug-cf-eval*
                    (format t "  dl= ~S~%" delta)
                    (format t "  f = ~S~%" f))
                  (when (<= (abs (- delta 1)) eps)
                    (return-from lentz (values f j tiny-value-count)))))
           finally
             (error 'simple-error
                    :format-control "~<Continued fraction failed to converge after ~D iterations.~%    Delta = ~S~>"
                    :format-arguments (list *max-cf-iterations* (/ c d))))))))

;; Continued fraction for erf(b):
;;
;; z[n] = 1+2*n-2*z^2
;; a[n] = 4*n*z^2
;;
;; This works ok, but has problems for z > 3 where sometimes the
;; result is greater than 1.
#+nil
(defun erf (z)
  (let* ((z2 (* z z))
         (twoz2 (* 2 z2)))
    (* (/ (* 2 z)
          (sqrt (float-pi z)))
       (exp (- z2))
       (/ (lentz #'(lambda (n)
                     (- (+ 1 (* 2 n))
                        twoz2))
                 #'(lambda (n)
                     (* 4 n z2)))))))

;; Tail of the incomplete gamma function:
;; integrate(x^(a-1)*exp(-x), x, z, inf)
;;
;; The continued fraction, valid for all z except the negative real
;; axis:
;;
;; b[n] = 1+2*n+z-a
;; a[n] = n*(a-n)
;;
;; See http://functions.wolfram.com/06.06.10.0003.01
(defun cf-incomplete-gamma-tail (a z)
  (when (and (zerop (imagpart z)) (minusp (realpart z)))
    (error 'domain-error
           :function-name 'cf-incomplete-gamma-tail
           :format-arguments (list 'z z)
           :format-control "Argument ~S should not be on the negative real axis:  ~S"))
  (/ (handler-case (* (expt z a)
                      (exp (- z)))
       (arithmetic-error ()
         ;; z^a*exp(-z) can overflow prematurely.  In this case, use
         ;; the equivalent exp(a*log(z)-z).  We don't use this latter
         ;; form because it has more roundoff error than the former.
         (exp (- (* a (log z)) z))))
     (let ((z-a (- z a)))
       (lentz #'(lambda (n)
                  (+ n n 1 z-a))
              #'(lambda (n)
                  (* n (- a n)))))))

;; Incomplete gamma function:
;; integrate(x^(a-1)*exp(-x), x, 0, z)
;;
;; The continued fraction, valid for all z:
;;
;; b[n] = n - 1 + z + a
;; a[n] = -z*(a + n)
;;
;; See http://functions.wolfram.com/06.06.10.0007.01.  We modified the
;; continued fraction slightly and discarded the first quotient from
;; the fraction.
#+nil
(defun cf-incomplete-gamma (a z)
  (/ (handler-case (* (expt z a)
                      (exp (- z)))
       (arithmetic-error ()
         ;; z^a*exp(-z) can overflow prematurely.  In this case, use
         ;; the equivalent exp(a*log(z)-z).  We don't use this latter
         ;; form because it has more roundoff error than the former.
         (exp (- (* a (log z)) z))))
     (let ((za1 (+ z a 1)))
       (- a (/ (* a z)
               (lentz #'(lambda (n)
                          (+ n za1))
                      #'(lambda (n)
                          (- (* z (+ a n))))))))))

;; Incomplete gamma function:
;; integrate(x^(a-1)*exp(-x), x, 0, z)
;;
;; The continued fraction, valid for all z:
;;
;; b[n] = a + n
;; a[n] = -(a+n/2)*z if n odd
;;        n/2*z      if n even
;;
;; See http://functions.wolfram.com/06.06.10.0009.01.
;;
;; Some experiments indicate that this converges faster than the above
;; and is actually quite a bit more accurate, expecially near the
;; negative real axis.
(defun cf-incomplete-gamma (a z)
  (/ (handler-case (* (expt z a)
                      (exp (- z)))
       (arithmetic-error ()
         ;; z^a*exp(-z) can overflow prematurely.  In this case, use
         ;; the equivalent exp(a*log(z)-z).  We don't use this latter
         ;; form because it has more roundoff error than the former.
         (exp (- (* a (log z)) z))))
     (lentz #'(lambda (n)
                (+ n a))
            #'(lambda (n)
                (if (evenp n)
                    (* (ash n -1) z)
                    (- (* (+ a (ash n -1)) z)))))))

;; Series expansion for incomplete gamma.  Intended for |a|<1 and
;; |z|<1.  The series is
;;
;; g(a,z) = z^a * sum((-z)^k/k!/(a+k), k, 0, inf)
(defun s-incomplete-gamma (a z)
  (let ((-z (- z))
        (eps (epsilon z)))
    (loop for k from 0
       for term = 1 then (* term (/ -z k))
       for sum = (/ a) then (+ sum (/ term (+ a k)))
       when (< (abs term) (* (abs sum) eps))
       return (* sum (expt z a)))))

;; Tail of the incomplete gamma function.
(defun incomplete-gamma-tail (a z)
  "Tail of the incomplete gamma function defined by:

  integrate(t^(a-1)*exp(-t), t, z, inf)"
  (let* ((prec (float-contagion a z))
         (a (apply-contagion a prec))
         (z (apply-contagion z prec)))
    (if (zerop a)
        ;; incomplete_gamma_tail(0, z) = exp_integral_e(1,z)
        (exp-integral-e 1 z)
        (if (and (zerop (imagpart a))
                 (zerop (imagpart z)))
            ;; For real values, we split the result to compute either the
            ;; tail directly from the continued fraction or from gamma(a)
            ;; - incomplete-gamma.  The continued fraction doesn't
            ;; converge on the negative real axis, so we can't use that
            ;; there.  And accuracy appears to be better if z is "small".
            ;; We take this to mean |z| < |a-1|.  Note that |a-1| is the
            ;; peak of the integrand.
            (if (and (> (abs z) (abs (- a 1)))
                     (not (minusp (realpart z))))
                (cf-incomplete-gamma-tail a z)
                (- (gamma a) (cf-incomplete-gamma a z)))
            ;; If the argument is close enough to the negative real axis,
            ;; the continued fraction for the tail is not very accurate.
            ;; Use the incomplete gamma function to evaluate in this
            ;; region.  (Arbitrarily selected the region to be a sector.
            ;; But what is the correct size of this sector?)
            (if (<= (abs (phase z)) 3.1)
                (cf-incomplete-gamma-tail a z)
                (- (gamma a) (cf-incomplete-gamma a z)))))))

(defun incomplete-gamma (a z)
  "Incomplete gamma function defined by:

  integrate(t^(a-1)*exp(-t), t, 0, z)"
  (let* ((prec (float-contagion a z))
         (a (apply-contagion a prec))
         (z (apply-contagion z prec)))
    (if (and (< (abs a) 1) (< (abs z) 1))
        (s-incomplete-gamma a z)
        (if (and (realp a) (realp z))
            (if (< z (- a 1))
                (cf-incomplete-gamma a z)
                (- (gamma a) (cf-incomplete-gamma-tail a z)))
            ;; The continued fraction doesn't converge very fast if a
            ;; and z are small.  In this case, use the series
            ;; expansion instead, which converges quite rapidly.
            (if (< (abs z) (abs a))
                (cf-incomplete-gamma a z)
                (- (gamma a) (cf-incomplete-gamma-tail a z)))))))

(defun erf (z)
  "Error function:

    erf(z) = 2/sqrt(%pi)*sum((-1)^k*z^(2*k+1)/k!/(2*k+1), k, 0, inf)

  For real z, this is equivalent to

    erf(z) = 2/sqrt(%pi)*integrate(exp(-t^2), t, 0, z) for real z."
  ;;
  ;; Erf is an odd function: erf(-z) = -erf(z)
  (if (minusp (realpart z))
      (- (erf (- z)))
      (/ (incomplete-gamma 1/2 (* z z))
         (sqrt (float-pi z)))))

(defun erfc (z)
  "Complementary error function:

    erfc(z) = 1 - erf(z)"
  ;; Compute erfc(z) via 1 - erf(z) is not very accurate if erf(z) is
  ;; near 1.  Wolfram says
  ;;
  ;; erfc(z) = 1 - sqrt(z^2)/z * (1 - 1/sqrt(pi)*gamma_incomplete_tail(1/2, z^2))
  ;;
  ;; For real(z) > 0, sqrt(z^2)/z is 1 so
  ;;
  ;; erfc(z) = 1 - (1 - 1/sqrt(pi)*gamma_incomplete_tail(1/2,z^2))
  ;;         = 1/sqrt(pi)*gamma_incomplete_tail(1/2,z^2)
  ;;
  ;; For real(z) < 0, sqrt(z^2)/z is -1 so
  ;;
  ;; erfc(z) = 1 + (1 - 1/sqrt(pi)*gamma_incomplete_tail(1/2,z^2))
  ;;         = 1 + 1/sqrt(pi)*gamma_incomplete(1/2,z^2)
  (if (>= (realpart z) 0)
      (/ (incomplete-gamma-tail 1/2 (* z z))
         (sqrt (float-pi z)))
      (+ 1
         (/ (incomplete-gamma 1/2 (* z z))
            (sqrt (float-pi z))))))

(defun cf-exp-integral-e (v z)
  ;; We use the continued fraction
  ;;
  ;; E(v,z) = exp(-z)/cf(z)
  ;;
  ;; where the continued fraction cf(z) is
  ;;
  ;; a[k] = -k*(k+v-1)
  ;; b[k] = v + 2*k + z
  ;;
  ;; for k = 1, inf
  (let ((z+v (+ z v)))
    (/ (exp (- z))
       (lentz #'(lambda (k)
                  (+ z+v (* 2 k)))
              #'(lambda (k)
                  (* (- k)
                     (+ k v -1)))))))


;; For v not an integer:
;;
;; E(v,z) = gamma(1-v)*z^(v-1) - sum((-1)^k*z^k/(k-v+1)/k!, k, 0, inf)
;;
;; For v an integer:
;;
;; E(v,z) = (-z)^(v-1)/(v-1)!*(psi(v)-log(z))
;;          - sum((-1)^k*z^k/(k-v+1)/k!, k, 0, inf, k != n-1)
;;
(defun s-exp-integral-e (v z)
  ;; E(v,z) = gamma(1-v)*z^(v-1) - sum((-1)^k*z^k/(k-v+1)/k!, k, 0, inf)
  (let ((-z (- z))
        (-v (- v))
        (eps (epsilon z)))
    (if (and (realp v)
             (= v (ftruncate v)))
        ;; v is an integer
        (let* ((n (truncate v))
               (n-1 (1- n)))
          (- (* (/ (expt -z n-1)
                   (gamma v))
                (- (psi v) (log z)))
             (loop for k from 0
                   for term = 1 then (* term (/ -z k))
                   for sum = (if (= v 1) 0 (/ (- 1 v)))
                     then (+ sum (let ((denom (- k n-1)))
                                   (if (zerop denom)
                                       0
                                       (/ term denom))))
                   when (< (abs term) (* (abs sum) eps))
                     return sum)))
        (loop for k from 0
              for term = 1 then (* term (/ -z k))
              for sum = (/ (- 1 v)) then (+ sum (/ term (+ k 1 -v)))
              when (< (abs term) (* (abs sum) eps))
                return (- (* (gamma (- 1 v)) (expt z (- v 1)))
                                    sum)))))

(defun exp-integral-e (v z)
  "Exponential integral E:

   E(v,z) = integrate(exp(-t)/t^v, t, 1, inf)"
  ;; E(v,z) = z^(v-1) * integrate(t^(-v)*exp(-t), t, z, inf);
  ;;
  ;; for |arg(z)| < pi.
  ;;
  ;;
  (cond ((and (realp v) (minusp v))
         ;; E(-v, z) = z^(-v-1)*incomplete_gamma_tail(v+1,z)
         (let ((-v (- v)))
           (* (expt z (- v 1))
              (incomplete-gamma-tail (+ -v 1) z))))
        ((< (abs z) 1)
         ;; Use series for small z
         (s-exp-integral-e v z))
        ((>= (abs (phase z)) 3.1)
         ;; The continued fraction doesn't converge on the negative
         ;; real axis, and converges very slowly near the negative
         ;; real axis, so use the incomplete-gamma-tail function in
         ;; this region.  "Closeness" to the negative real axis is
         ;; teken to mean that z is in a sector near the axis.
         ;;
         ;; E(v,z) = z^(v-1)*incomplete_gamma_tail(1-v,z)
         (* (expt z (- v 1))
            (incomplete-gamma-tail (- 1 v) z)))
        (t
         ;; Use continued fraction for everything else.
         (cf-exp-integral-e v z))))

;; Series for Fresnel S
;;
;;   S(z) = z^3*sum((%pi/2)^(2*k+1)(-z^4)^k/(2*k+1)!/(4*k+3), k, 0, inf)
;;
;; Compute as
;;
;;   S(z) = z^3*sum(a(k)/(4*k+3), k, 0, inf)
;;
;; where
;;
;;   a(k+1) = -a(k) * (%pi/2)^2 * z^4 / (2*k+2) / (2*k+3)
;;
;;   a(0) = %pi/2.
(defun fresnel-s-series (z)
  (let* ((pi/2 (* 1/2 (float-pi z)))
         (factor (- (* (expt z 4) pi/2 pi/2)))
         (eps (epsilon z))
         (sum 0)
         (term pi/2))
    (loop for k2 from 0 by 2
       until (< (abs term) (* eps (abs sum)))
       do
       (incf sum (/ term (+ 3 k2 k2)))
       (setf term (/ (* term factor)
                     (* (+ k2 2)
                        (+ k2 3)))))
    (* sum (expt z 3))))
    
(defun fresnel-s (z)
  "Fresnel S:

   S(z) = integrate(sin(%pi*t^2/2), t, 0, z) "
  (let ((prec (float-contagion z))
        (sqrt-pi (sqrt (float-pi z))))
    (flet ((fs (z)
             ;; Wolfram gives
             ;;
             ;;  S(z) = (1+%i)/4*(erf(c*z) - %i*erf(conjugate(c)*z))
             ;;
             ;; where c = sqrt(%pi)/2*(1+%i).
             ;;
             ;; But for large z, we should use erfc.  Then
             ;;  S(z) = 1/2 - (1+%i)/4*(erfc(c*z) - %i*erfc(conjugate(c)*z))
             (if (and t (> (abs z) 2))
                 (- 1/2
                    (* #c(1/4 1/4)
                       (- (erfc (* #c(1/2 1/2) sqrt-pi z))
                          (* #c(0 1)
                             (erfc (* #c(1/2 -1/2) sqrt-pi z))))))
                 (* #c(1/4 1/4)
                    (- (erf (* #c(1/2 1/2) sqrt-pi z))
                       (* #c(0 1)
                          (erf (* #c(1/2 -1/2) sqrt-pi z)))))))
          (rfs (z)
            ;; When z is real, recall that erf(conjugate(z)) =
            ;; conjugate(erf(z)).  Then
            ;;
            ;;  S(z) = 1/2*(realpart(erf(c*z)) - imagpart(erf(c*z)))
            ;;
            ;; But for large z, we should use erfc.  Then
            ;;
            ;;  S(z) = 1/2 - 1/2*(realpart(erfc(c*z)) - imagpart(erf(c*z)))
            (if (> (abs z) 2)
                (let ((s (erfc (* #c(1/2 1/2) sqrt-pi z))))
                  (- 1/2
                     (* 1/2 (- (realpart s) (imagpart s)))))
                (let ((s (erf (* #c(1/2 1/2) sqrt-pi z))))
                  (* 1/2 (- (realpart s) (imagpart s)))))))
      ;; For small z, the erf terms above suffer from subtractive
      ;; cancellation.  So use the series in this case.  Some simple
      ;; tests were done to determine that for double-floats we want
      ;; to use the series for z < 1 to give max accuracy.  For
      ;; qd-real, the above formula is good enough for z > 1d-5.
      (if (< (abs z) (ecase prec
                       (single-float 1.5f0)
                       (double-float 1d0)
                       (qd-real #q1)))
          (fresnel-s-series z)
          (if (realp z)
              ;; FresnelS is real for a real argument. And it is odd.
              (if (minusp z)
                  (- (rfs (- z)))
                  (rfs z))
              (fs z))))))

(defun fresnel-c (z)
  "Fresnel C:

   C(z) = integrate(cos(%pi*t^2/2), t, 0, z) "
  (let ((sqrt-pi (sqrt (float-pi z))))
    (flet ((fs (z)
             ;; Wolfram gives
             ;;
             ;;  C(z) = (1-%i)/4*(erf(c*z) + %i*erf(conjugate(c)*z))
             ;;
             ;; where c = sqrt(%pi)/2*(1+%i).
             (* #c(1/4 -1/4)
                (+ (erf (* #c(1/2 1/2) sqrt-pi z))
                   (* #c(0 1)
                      (erf (* #c(1/2 -1/2) sqrt-pi z)))))))
      (if (realp z)
          ;; FresnelS is real for a real argument. And it is odd.
          (if (minusp z)
              (- (realpart (fs (- z))))
              (realpart (fs z)))
          (fs z)))))

(defun sin-integral (z)
  "Sin integral:

  Si(z) = integrate(sin(t)/t, t, 0, z)"
  ;; Wolfram has
  ;;
  ;; Si(z) = %i/2*(gamma_inc_tail(0, -%i*z) - gamma_inc_tail(0, %i*z) + log(-%i*z)-log(%i*z))
  ;;
  (flet ((si (z)
           (* #c(0 1/2)
              (let ((iz (* #c(0 1) z))
                    (-iz (* #c(0 -1) z)))
                (+ (- (incomplete-gamma-tail 0 -iz)
                      (incomplete-gamma-tail 0 iz))
                   (- (log -iz)
                      (log iz)))))))
    (if (realp z)
        ;; Si is odd and real for real z.  In this case, we have
        ;;
        ;; Si(x) = %i/2*(gamma_inc_tail(0, -%i*x) - gamma_inc_tail(0, %i*x) - %i*%pi)
        ;;       = %pi/2 + %i/2*(gamma_inc_tail(0, -%i*x) - gamma_inc_tail(0, %i*x))
        ;; But gamma_inc_tail(0, conjugate(z)) = conjugate(gamma_inc_tail(0, z)), so
        ;;
        ;; Si(x) = %pi/2 + imagpart(gamma_inc_tail(0, %i*x))
        (cond ((< z 0)
               (- (sin-integral (- z))))
              ((= z 0)
               (* 0 z))
              (t
               (+ (* 1/2 (float-pi z))
                  (imagpart (incomplete-gamma-tail 0 (complex 0 z))))))
        (si z))))

(defun cos-integral (z)
  "Cos integral:

    Ci(z) = integrate((cos(t) - 1)/t, t, 0, z) + log(z) + gamma

  where gamma is Euler-Mascheroni constant"
  ;; Wolfram has
  ;;
  ;; Ci(z) = log(z) - 1/2*(gamma_inc_tail(0, -%i*z) + gamma_inc_tail(0, %i*z) + log(-%i*z)+log(%i*z))
  ;;
  (flet ((ci (z)
           (- (log z)
              (* 1/2
                 (let ((iz (* #c(0 1) z))
                       (-iz (* #c(0 -1) z)))
                   (+ (+ (incomplete-gamma-tail 0 -iz)
                         (incomplete-gamma-tail 0 iz))
                      (+ (log -iz)
                         (log iz))))))))
    (if (and (realp z) (plusp z))
        (realpart (ci z))
        (ci z))))

;; Array of values of the Bernoulli numbers.  We only have enough for
;; the evaluation of the psi function.
(defconstant bern-values
  (make-array 55
              :initial-contents
              '(1
                -1/2
                1/6
                0
                -1/30
                0
                1/42
                0
                -1/30
                0
                5/66
                0
                -691/2730
                0
                7/6
                0
                -3617/510
                0
                43867/798
                0
                -174611/330
                0
                854513/138
                0
                -236364091/2730
                0
                8553103/6
                0
                -23749461029/870
                0
                8615841276005/14322
                0
                -7709321041217/510
                0
                2577687858367/6
                0
                -26315271553053477373/1919190
                0
                2929993913841559/6
                0
                -261082718496449122051/13530
                0
                1520097643918070802691/1806
                0
                -27833269579301024235023/690
                0
                596451111593912163277961/282
                0
                -5609403368997817686249127547/46410
                0
                495057205241079648212477525/66
                0
                -801165718135489957347924991853/1590
                0
                29149963634884862421418123812691/798
                )))
                
(defun bern (k)
  (aref bern-values k))

(defun psi (z)
  "Digamma function defined by

  - %gamma + sum(1/k-1/(k+z-1), k, 1, inf)

  where %gamma is Euler's constant"

  ;; A&S 6.3.7:  Reflection formula
  ;;
  ;;   psi(1-z) = psi(z) + %pi*cot(%pi*z)
  ;;
  ;; A&S 6.3.6:  Recurrence formula
  ;;
  ;;   psi(n+z) = 1/(z+n-1)+1/(z+n-2)+...+1/(z+2)+1/(1+z)+psi(1+z)
  ;;
  ;; A&S 6.3.8:  Asymptotic formula
  ;;
  ;;   psi(z) ~ log(z) - sum(bern(2*n)/(2*n*z^(2*n)), n, 1, inf)
  ;;
  ;; So use reflection formula if Re(z) < 0.  For z > 0, use the recurrence
  ;; formula to increase the argument and then apply the asymptotic formula.

  (cond ((= z 1)
         ;; psi(1) = -%gamma
         (- (float +%gamma+ (if (integerp z) 0.0 z))))
        ((minusp (realpart z))
         (let ((p (float-pi z)))
           (flet ((cot-pi (z)
                    ;; cot(%pi*z), carefully.  If z is an odd multiple
                    ;; of 1/2, cot is 0.
                    (if (and (realp z)
                             (= 1/2 (- z (ftruncate z))))
                        (float 0 z)
                        (/ (tan (* p z))))))
             (- (psi (- 1 z))
                (* p (cot-pi z))))))
        (t
         (let* ((k (* 2 (1+ (floor (* .41 (- (log (epsilon (float (realpart z))) 10)))))))
                (m 0)
                (y (expt (+ z k) 2))
                (x 0))
           (loop for i from 1 upto (floor k 2) do
             (progn
               (incf m (+ (/ (+ z i i -1))
                          (/ (+ z i i -2))))
               (setf x (/ (+ x (/ (bern (+ k 2 (* -2 i)))
                                  (- k i i -2)))
                          y))))
           (- (log (+ z k))
              (/ (* 2 (+ z k)))
              x
              m)))))