How useful are the different CAS languages for implementing numerical routines? Prompted by a comparison of R and C for implementing Fisher's exact test for 2x2 tables (http://fluff.info/blog/arch/00000172.htm), I thought that it would be interesting to implement this particular test in Spad, Boot, Reduce, Maxima, Common Lisp and Sage (see below). Each set of code was required to implement a univariate root finder and the hypergeometric distribution to calculate the p-value under different alternatives, together with the 95% confidence interval and the conditional maximum likelihood estimator for the odds ratio. The reference implementation is R, where the code and output would be:

As a caveat: I have little experience with these programs. Any changes or improvements to the programs would be welcomed.

To summarise, all six languages (Spad, Boot, Reduce, Maxima, Common Lisp and Sage) provide arbitrary length integers and fractions, ensuring that the hypergeometric distribution was straightforward to implement. The Lisp and R implementations were very similar, which is not surprising, given that the two languages are closely related. Nested functions in Spad required the use of #1 and #2 argument references, although this has changed recently in Fricas; I appreciated Spad's lexical scoping and facility to fall back to a symbolic analysis. An Aldor implementation would be similar to the Spad implementation. Boot's implementation was initially difficult, as I was unclear how to pass values to the nested functions with using function argument (which was not possible for the univariate root finder :-(). The use of the "$" prefix on derived variables seemed clumsy. Finally, my version of Fricas:Boot defaulted to single precision floats, which caused problems with precision. Type specification in OpenAxiom:Boot and more recent versions of Fricas would negate the need for explicit coercion to double-floats. For Reduce, the lack of nested procedures in the algebraic mode made progress slow; importantly, the switch to symbolic mode made the implementation quite straightforward, with reasonably good debugging. The Maxima and Sage versions were particularly short, given that they already provide a root solver.

This begs the question: when would one use any of these languages for mixed numerical/symbolic analysis? In my opinion, Boot is the least likely to be used, although it does play closely and well with Common Lisp (a la Maxima). One could code for numerical analysis in Boot and Common Lisp - however Boot's lack of lexical scoping may be a detraction. For an R user, Spad's type system seemed both fussy and extremely elegant; I found that debugging could be slow. Importantly, Lisp and Boot are able to evaluate Spad functions, potentially allowing the use of Lisp functions such as those translated from Fortran using f2cl (see also SandboxMLE for another example of numerical code in Spad). Axiom, Reduce, Maxima and Sage provide polished environments worthy of further consideration.

First, the Spad implementation:

)abbrev package TESTP TestPackage
R ==> Float
I ==> Integer
fisherRec ==> Record(PValue:R, CI:List R, Estimate:R)
TestPackage: with
   ridder: (R->R,R,R) -> R
   msign: (R,R) -> R
   choose:(I, I)->Fraction I 
   --chooseNew:(Integer, Integer)->Fraction Integer 
   dhyper:(I, I, I, I)->Fraction Integer
   phyper:(I, I, I, I, Boolean)->Fraction Integer
   fisherTest:(I,I,I,I, String, R, Boolean, R)-> fisherRec
   testTolerance:(R, R, R)->Boolean   
   test1: () -> Boolean
   test2: () -> Boolean
   test3: () -> Boolean
   test4: () -> Boolean
   test5: () -> Boolean
   test6: () -> Boolean
   test7: () -> Boolean
   test8: () -> Boolean
   test9: () -> Boolean
   test10: () -> Boolean
   alltests: () -> List Boolean
  == add
   import TrigonometricFunctionCategory -- for test1()
   --import OrderedCompletion(Float) for plusInfinity()$OrderedCompletion(Float)
   ridder(func, x1, x2) ==
    eps:= 1.0e-16::R
    maxit:= 30::Integer
    --verbose:= false
    fl:R := func x1
    fh:R := func x2
    xl:R := x1
    xh:R := x2
    ans:R := -1.11e30::R
    xnew:R := 0.0e0::R
    iterNum:= 0::Integer
    if fl=0.0::R then return x1
    else if fh=0.0::R then return x2
    else if (fl*fh) > 0.0::R then error "Initial points are not either side of zero."
    --if (fl*fh) < 0.0 then
    else repeat
                xm:= 0.5::R *(xl+xh)
                fm:= func xm
                ss:= sqrt((fm*fm) - (fl*fh))
                if ss =0.0::R then return ans
                xnew:= xm + (((xm - xl) * (if (fl>fh) then 1.0::R else -1.0::R) * fm) / ss)
                if abs(xnew-ans) <= eps then return ans
                ans:= xnew 
                fnew:= func ans
                if fnew=0.0::R then return ans
                if msign(fm,fnew) ~= fm then
                    xl:= xm 
                    fl:= fm 
                    xh:= ans 
                    fh:= fnew
                else if msign(fl, fnew) ~= fl then
                    xh:= ans 
                    fh:= fnew
                else if msign(fh, fnew) ~= fh then
                    xl:= ans 
                    fl:= fnew
                iterNum:=iterNum+1::Integer
                if iterNum >=maxit then 
                        error "Maximum iterations exceeded"
                --if verbose then FORMAT(true,"~,8f ~,8f ~,8f ~,8f~%", xl, xh, fl, fh)$Lisp
                if abs(xh-xl) <= eps then return ans
   msign(x, y) ==
    (abs x) * (if y>0.0::R then 1.0::R else if y<0.0::R then -1.0::R else 0.0::R)
   choose(n, x) ==
    total:Fraction Integer := 1/1
    for denom in 1..x repeat
        total:=total*((n-denom+1)/(denom))::Fraction Integer
    return total
   --chooseNew(n, x) == product((n-i+1)::Fraction Integer/i::Fraction Integer,i=1..x)
   dhyper(x, m, n, k) ==
    choose(m, x) * choose(n, k - x) / choose(m + n, k)
   phyper(x, m, n, k, lowerTail) ==
    i:PositiveInteger
    --total:Fraction Integer:=0/1
    if lowerTail then 
        reduce("+",[dhyper(i, m, n, k) for i in 1..x])
    else 
        reduce("+",[dhyper(i, m, n, k) for i in (x+1)..k])
   fisherTest(a,b,c,d, alternative, OR, confInt, confLevel) ==
        m:I := a+c -- first column
        n:I := b+d -- second column
        k:I := a+b -- first row
        x00:I := a
        lo:I := max(0, k-n)
        hi:I := min(k, m)
        support:List I := [i for i in lo..hi]
        logdc:List R:= [log(dhyper(i, m, n, k)::R) for i in support]
        doubleEps:R := 1.0e-50::R 
        plusInfinity:R := 1.0e6::R -- arbitrary
        dnhyper:(R->List R) :=
             ncp:R := #1
             d:List R := [logdc(i)+log(ncp)*support(i)::R for i in 1..#logdc]
             maxd:R := reduce(max,d)
             d2:List R :=[exp(di-maxd) for di in d]
             sumd2:R := reduce("+",d2)
             [d2i/sumd2 for d2i in d2]
        mnhyper:(R->R) :=
             ncp:R := #1
             if ncp=0.0::R then lo::R
             --else if ncp=%plusInfinity then hi::R
             else 
                d:List R := dnhyper(ncp)
                reduce("+",[support(i)::R*d(i) for i in 1..#d])
        pnhyper:((Integer,R,Boolean)->R) :=
             q:I := #1
             ncp:R := #2
             upperTail:Boolean := #3
             if ncp=1.0 then 
                if upperTail then phyper(q-1, m, n, k, false)::R
                else phyper(q, m, n, k, true)::R
             else if ncp=0.0 then
                if upperTail then 
                    if q<=lo then 1.0::R else 0.0::R
                else if q>=lo then 1.0::R else 0.0::R
--           else if ncp=%plusInfinity then
--              if upperTail then 
--                  if q<=hi then 1.0::R else 0.0::R
--              else if q>= hi then 1.0::R else 0.0::R
             else 
                d:List R := dnhyper(ncp)
                if upperTail then
                      reduce("+",[d(i) for i in 1..#d | support(i)>=q])
                else reduce("+",[d(i) for i in 1..#d | support(i)<=q])
        mle:(I->R) :=
             x:I := #1
             if x=lo then 0.0::R
             else if x=hi then plusInfinity
             else
                mu:R := mnhyper(1.0::R)
                if mu>x::R then 
                        ridder(mnhyper(#1) - x::R,0,1)
                else if mu<x::R then
                        1/ridder(mnhyper(1/#1) - x::R,doubleEps,1.0::R)
                else 1.0::R
        ncpU:(I,R)->R :=
             x:I := #1
             alpha:R := #2
             if x=hi then plusInfinity
             else
                     p:R := pnhyper(x, 1.0::R, false)
                     if p<alpha then 
                        ridder(pnhyper(x,#1,false) - alpha, 0.0::R, 1.0::R)
                     else if p>alpha then 
                                1/ridder(pnhyper(x,1/#1,false) - alpha, doubleEps, 1.0::R)
                     else 1.0::R
        ncpL:(Integer, R)->R :=
             x:I := #1
             alpha:R := #2
             if x=lo then 0.0::R
             else 
                p:R := pnhyper(x, 1, true)
                if p>alpha then
                        ridder(pnhyper(x,#1,true) - alpha, 0,1)
                else if p<alpha then
                        1/ridder(pnhyper(x,1/#1,true) - alpha, doubleEps,1.0::R)
                else 1.0::R
        pValue:R :=
             if alternative="less" then pnhyper(x00, OR,false) 
             else if alternative="greater" then pnhyper(x00, OR,true)
             else if alternative="two-sided" then
                relErr:= 1+1.0e-7::R
                dn:= dnhyper(OR)
                dstar:= dn(x00-lo+1)*relErr
                reduce("+",[di for di in dn | di<dstar])
             else -1.0::R
        cInterval:List R :=
               if confInt then 
                   if alternative="less" then  [0.0::R, ncpU(x00, 1.0::R-confLevel)]
                   else if alternative="greater" then [ncpL(x00, 1.0::R-confLevel), plusInfinity]
                   else if alternative="two-sided" then 
                      alpha:=(1-confLevel)/2
                      [ncpL(x00, alpha), ncpU(x00, alpha)]
                else [-1.0::R,-1.0::R]
                --else []
        estimate:= mle(x00)
        [pValue, cInterval, estimate]
   testTolerance(x, y, atol) ==
    if abs(x-y) <= atol then true else false
   test1() ==
    testTolerance(2*ridder(cos,0.0::R,2.0::R),pi()$Pi::R, 1.0e-18)
   test2() == testTolerance(choose(100, 5)::R, 75287520::R, 0)
   test3() == testTolerance(dhyper(5, 10, 7, 8)::R, 0.3628137::R, 1.0e-7)
   test4() == testTolerance(log(dhyper(5, 10, 7, 8)::R),-1.013866::R,  1.0e-7)
   test5() == testTolerance(phyper(5, 10, 7, 8, true)::R,0.7821884::R, 1.0e-7)
   test6() == testTolerance(phyper(5, 10, 7, 8, false)::R,0.2178116::R, 1.0e-7) 
   test7() == testTolerance(fisherTest(10,10,10,20,"two-sided",1.0,true,0.95).PValue,
       0.2575, 1.0e-3)
   test8() == testTolerance(fisherTest(10,10,10,20,"two-sided",1.0,true,0.95).CI.1,
       0.5383996, 1.0e-6)
   test9() == testTolerance(fisherTest(10,10,10,20,"two-sided",1.0,true,0.95).CI.2,
       7.4363242, 1.0e-4)
   test10() == testTolerance(fisherTest(10,10,10,20,"two-sided",1.0,true,0.95).Estimate,
       1.971640, 1.0e-4)
   alltests() == [test1(), test2(), test3(), test4(), test5(), test6(),
       test7(), test8(), test9(), test10()]

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/2045700826673834242-25px001.spad
      using old system compiler.
   TESTP abbreviates package TestPackage 
------------------------------------------------------------------------
   initializing NRLIB TESTP for TestPackage 
   compiling into NRLIB TESTP 
   importing TrigonometricFunctionCategory
   compiling exported ridder : (Float -> Float,Float,Float) -> Float
Time: 0.02 SEC.

   compiling exported msign : (Float,Float) -> Float
Time: 0.03 SEC.

   compiling exported choose : (Integer,Integer) -> Fraction Integer
Time: 0 SEC.

   compiling exported dhyper : (Integer,Integer,Integer,Integer) -> Fraction Integer
Time: 0 SEC.

   compiling exported phyper : (Integer,Integer,Integer,Integer,Boolean) -> Fraction Integer
Time: 0.02 SEC.

   compiling exported fisherTest : (Integer,Integer,Integer,Integer,String,Float,Boolean,Float) -> Record(PValue: Float,CI: List Float,Estimate: Float)
****** comp fails at level 8 with expression: ******
error in function fisherTest 

(SEQ (|:=| (|:| |m| (|Integer|)) (+ |a| |c|))
     (|:=| (|:| |n| (|Integer|)) (+ |b| |d|))
     (|:=| (|:| |k| (|Integer|)) (+ |a| |b|))
     (|:=| (|:| |x00| (|Integer|)) |a|)
     (|:=| (|:| |lo| (|Integer|)) (|max| 0 (- |k| |n|)))
     (|:=| (|:| |hi| (|Integer|)) (|min| |k| |m|))
     (|:=| (|:| |support| (|List| (|Integer|)))
      (COLLECT (IN |i| (SEGMENT |lo| |hi|)) |i|))
     (|:=| (|:| |logdc| (|List| (|Float|)))
      (COLLECT (IN |i| |support|)
               (|log| (|::| (|dhyper| |i| |m| |n| |k|) (|Float|)))))
     (|:=| (|:| |doubleEps| (|Float|))
      (|::| ((|Sel| (|Float|) |float|) 1 -50 10) (|Float|)))
     (|:=| (|:| |plusInfinity| (|Float|))
      (|::| ((|Sel| (|Float|) |float|) 1 6 10) (|Float|)))
     (|:=| (|:| |dnhyper| (|Mapping| (|List| (|Float|)) (|Float|)))
      (SEQ (|:=| (|:| |ncp| (|Float|)) |#1|)
           (|:=| (|:| |d| (|List| (|Float|)))
            (COLLECT (IN |i| (SEGMENT 1 (|#| |logdc|)))
                     (+ (|logdc| |i|)
                        (* (|log| |ncp|) (|::| (|support| |i|) (|Float|))))))
           (|:=| (|:| |maxd| (|Float|)) (|reduce| |max| |d|))
           (|:=| (|:| |d2| (|List| (|Float|)))
            (COLLECT (IN |di| |d|) (|exp| (- |di| |maxd|))))
           (|:=| (|:| |sumd2| (|Float|)) (|reduce| "+" |d2|))
           (|exit| 1 (COLLECT (IN |d2i| |d2|) (/ |d2i| |sumd2|)))))
     (|:=| (|:| |mnhyper| (|Mapping| (|Float|) (|Float|)))
      (SEQ (|:=| (|:| |ncp| (|Float|)) |#1|)
           (|:=| (|:| #1=#:G682 (|Boolean|))
            (= |ncp| (|::| ((|Sel| (|Float|) |float|) 0 0 10) (|Float|))))
           (|exit| 1
            (IF #1#
                (|::| |lo| (|Float|))
                (SEQ (|:=| (|:| |d| (|List| (|Float|))) (|dnhyper| |ncp|))
                     (|exit| 1
                      (|reduce| "+"
                       (COLLECT (IN |i| (SEGMENT 1 (|#| |d|)))
                                (* (|::| (|support| |i|) (|Float|))
                                   (|d| |i|))))))))))
     (|:=|
      (|:| |pnhyper| (|Mapping| (|Float|) (|Integer|) (|Float|) (|Boolean|)))
      (SEQ (|:=| (|:| |q| (|Integer|)) |#1|)
           (|:=| (|:| |ncp| (|Float|)) (|#| 2))
           (|:=| (|:| |upperTail| (|Boolean|)) (|#| 3))
           (|:=| (|:| #2=#:G684 (|Boolean|))
            (= |ncp| ((|Sel| (|Float|) |float|) 1 0 10)))
           (|exit| 1
            (IF #2#
                (IF |upperTail|
                    (|::| (|phyper| (- |q| 1) |m| |n| |k| |false|) (|Float|))
                    (|::| (|phyper| |q| |m| |n| |k| |true|) (|Float|)))
                (SEQ
                 (|:=| (|:| #3=#:G683 (|Boolean|))
                  (= |ncp| ((|Sel| (|Float|) |float|) 0 0 10)))
                 (|exit| 1
                  (IF #3#
                      (IF |upperTail|
                          (IF (<= |q| |lo|)
                              (|::| ((|Sel| (|Float|) |float|) 1 0 10)
                               (|Float|))
                              (|::| ((|Sel| (|Float|) |float|) 0 0 10)
                               (|Float|)))
                          (IF (>= |q| |lo|)
                              (|::| ((|Sel| (|Float|) |float|) 1 0 10)
                               (|Float|))
                              (|::| ((|Sel| (|Float|) |float|) 0 0 10)
                               (|Float|))))
                      (SEQ
                       (|:=| (|:| |d| (|List| (|Float|))) (|dnhyper| |ncp|))
                       (|exit| 1
                        (IF |upperTail|
                            (|reduce| "+"
                             (COLLECT (IN |i| (SEGMENT 1 (|#| |d|)))
                                      (|\|| (>= (|support| |i|) |q|))
                                      (|d| |i|)))
                            (|reduce| "+"
                             (COLLECT (IN |i| (SEGMENT 1 (|#| |d|)))
                                      (|\|| (<= (|support| |i|) |q|))
                                      (|d| |i|)))))))))))))
     (|:=| (|:| |mle| (|Mapping| (|Float|) (|Integer|)))
      (SEQ (|:=| (|:| |x| (|Integer|)) |#1|)
           (|exit| 1
            (IF (= |x| |lo|)
                (|::| ((|Sel| (|Float|) |float|) 0 0 10) (|Float|))
                (IF (= |x| |hi|)
                    |plusInfinity|
                    (SEQ
                     (|:=| (|:| |mu| (|Float|))
                      (|mnhyper|
                       (|::| ((|Sel| (|Float|) |float|) 1 0 10) (|Float|))))
                     (|exit| 1
                      (IF (> |mu| (|::| |x| (|Float|)))
                          (|ridder| (- (|mnhyper| |#1|) (|::| |x| (|Float|))) 0
                           1)
                          (IF (< |mu| (|::| |x| (|Float|)))
                              (/ 1
                                 (|ridder|
                                  (- (|mnhyper| (/ 1 |#1|))
                                     (|::| |x| (|Float|)))
                                  |doubleEps|
                                  (|::| ((|Sel| (|Float|) |float|) 1 0 10)
                                   (|Float|))))
                              (|::| ((|Sel| (|Float|) |float|) 1 0 10)
                               (|Float|)))))))))))
     (|:=| (|:| |ncpU| (|Mapping| (|Float|) (|Integer|) (|Float|)))
      (SEQ (|:=| (|:| |x| (|Integer|)) |#1|)
           (|:=| (|:| |alpha| (|Float|)) (|#| 2))
           (|exit| 1
            (IF (= |x| |hi|)
                |plusInfinity|
                (SEQ
                 (|:=| (|:| |p| (|Float|))
                  (|pnhyper| |x|
                   (|::| ((|Sel| (|Float|) |float|) 1 0 10) (|Float|))
                   |false|))
                 (|exit| 1
                  (IF (< |p| |alpha|)
                      (|ridder| (- (|pnhyper| |x| |#1| |false|) |alpha|)
                       (|::| ((|Sel| (|Float|) |float|) 0 0 10) (|Float|))
                       (|::| ((|Sel| (|Float|) |float|) 1 0 10) (|Float|)))
                      (IF (> |p| |alpha|)
                          (/ 1
                             (|ridder|
                              (- (|pnhyper| |x| (/ 1 |#1|) |false|) |alpha|)
                              |doubleEps|
                              (|::| ((|Sel| (|Float|) |float|) 1 0 10)
                               (|Float|))))
                          (|::| ((|Sel| (|Float|) |float|) 1 0 10)
                           (|Float|))))))))))
     (|:=| (|:| |ncpL| (|Mapping| (|Float|) (|Integer|) (|Float|)))
      (SEQ (|:=| (|:| |x| (|Integer|)) |#1|)
           (|:=| (|:| |alpha| (|Float|)) (|#| 2))
           (|exit| 1
            (IF (= |x| |lo|)
                (|::| ((|Sel| (|Float|) |float|) 0 0 10) (|Float|))
                (SEQ (|:=| (|:| |p| (|Float|)) (|pnhyper| |x| 1 |true|))
                     (|exit| 1
                      (IF (> |p| |alpha|)
                          (|ridder| (- (|pnhyper| |x| |#1| |true|) |alpha|) 0
                           1)
                          (IF (< |p| |alpha|)
                              (/ 1
                                 (|ridder|
                                  (- (|pnhyper| |x| (/ 1 |#1|) |true|) |alpha|)
                                  |doubleEps|
                                  (|::| ((|Sel| (|Float|) |float|) 1 0 10)
                                   (|Float|))))
                              (|::| ((|Sel| (|Float|) |float|) 1 0 10)
                               (|Float|))))))))))
     (|:=| (|:| |pValue| (|Float|))
      (IF (= |alternative| "less")
          (|pnhyper| |x00| OR |false|)
          (IF (= |alternative| "greater")
              (|pnhyper| |x00| OR |true|)
              (IF (= |alternative| "two-sided")
                  (SEQ
                   (|:=| |relErr|
                    (+ 1 (|::| ((|Sel| (|Float|) |float|) 1 -7 10) (|Float|))))
                   (|:=| |dn| (|dnhyper| OR))
                   (|:=| |dstar| (* (|dn| (+ (- |x00| |lo|) 1)) |relErr|))
                   (|exit| 1
                    (|reduce| "+"
                     (COLLECT (IN |di| |dn|) (|\|| (< |di| |dstar|)) |di|))))
                  (- (|::| ((|Sel| (|Float|) |float|) 1 0 10) (|Float|)))))))
     (|:=| (|:| |cInterval| (|List| (|Float|)))
      (IF |confInt|
          (IF (= |alternative| "less")
              (|construct| (|::| ((|Sel| (|Float|) |float|) 0 0 10) (|Float|))
                           (|ncpU| |x00|
                            (-
                             (|::| ((|Sel| (|Float|) |float|) 1 0 10)
                              (|Float|))
                             |confLevel|)))
              (IF (= |alternative| "greater")
                  (|construct|
                   (|ncpL| |x00|
                    (- (|::| ((|Sel| (|Float|) |float|) 1 0 10) (|Float|))
                       |confLevel|))
                   |plusInfinity|)
                  (IF (= |alternative| "two-sided")
                      (SEQ (|:=| |alpha| (/ (- 1 |confLevel|) 2))
                           (|exit| 1
                            (|construct| (|ncpL| |x00| |alpha|)
                                         (|ncpU| |x00| |alpha|))))
                      |noBranch|)))
          (|construct| (- (|::| ((|Sel| (|Float|) |float|) 1 0 10) (|Float|)))
                       (-
                        (|::| ((|Sel| (|Float|) |float|) 1 0 10) (|Float|))))))
     (|:=| |estimate| (|mle| |x00|))
     (|exit| 1 (|construct| |pValue| |cInterval| |estimate|)))
****** level 8  ******
$x:= 2
$m:= (String)
$f:=
((((|ncp| #) (|q| # #) (|#3| #) (|#2| #) ...)))

   >> Apparent user error:
   no mode found for
   #1

Using this code in Axiom:

-- test code is correct
alltests()

   There are no library operations named alltests 
      Use HyperDoc Browse or issue
                              )what op alltests
      to learn if there is any operation containing " alltests " in its
      name.

   Cannot find a no-argument definition or library operation named 
      alltests .

Second, the Boot translation was more fiddly - but, then again, I had never used Boot before.

doubleFloat(x) == COERCE(x,'DOUBLE_-FLOAT)
DF(x) == COERCE(x,'DOUBLE_-FLOAT)
ridder(func, x1, x2) ==
    --x2:=DF(x2)
    eps:= DF(1.0e-16)
    maxit:= 30
    fl := DF(FUNCALL(func,x1))
    fh := DF(FUNCALL(func,x2))
    xl := x1
    xh := x2
    ans := DF(-1.11e20)
    xnew := 0.0e0
    iterNum:= 0
    if fl=0.0 then return x1
    else if fh=0.0 then return x2
    else if (fl*fh) > 0.0 then error "Initial points are not either side of zero."
    --if (fl*fh) < 0.0 then
    else repeat
                xm:= 0.5 *(xl+xh)
                fm:= FUNCALL(func,xm)
                ss:= SQRT((fm*fm) - (fl*fh))
                if ss =0.0 then return ans
                xnew:= xm + (((xm - xl) * (if (fl>fh) then 1.0 else -1.0) * fm) / ss)
                if ABS(xnew-ans) <= eps then return ans
                ans:= xnew 
                fnew:= DF(FUNCALL(func,ans))
                if fnew=0.0 then return ans
                if msign(fm,fnew) ^= fm then
                    xl:= xm 
                    fl:= fm 
                    xh:= ans 
                    fh:= fnew
                else if msign(fl, fnew) ^= fl then
                    xh:= ans 
                    fh:= fnew
                else if msign(fh, fnew) ^= fh then
                    xl:= ans 
                    fl:= fnew
                iterNum:=iterNum+1
                if iterNum >=maxit then 
                        error "Maximum iterations exceeded"
                --if verbose then FORMAT(true,"~,8f ~,8f ~,8f ~,8f~%", xl, xh, fl, fh)$Lisp
                if ABS(xh-xl) <= eps then return ans
msign(x, y) ==
    (ABS x) * (if y>0.0 then 1.0 else if y<0.0 then -1.0 else 0.0)
choose(n, x) ==
    total := 1
    for denom in 1..x repeat
        total:=total*(n-denom+1)/denom
    return total
   --chooseNew(n, x) == product((n-i+1)::Fraction Integer/i::Fraction Integer,i=1..x)
dhyper(x, m, n, k) ==
    DF(choose(m, x) * choose(n, k - x)) / choose(m + n, k)
-- reduce(func,list) == 
--      value := list.0
--      for i in 1..(#list-1) repeat
--              value:=FUNCALL(func,value,list.i))
--      value
phyper(x, m, n, k, lowerTail) ==
    --total:Fraction Integer:=0/1
    if lowerTail then 
        +/[dhyper(i, m, n, k) for i in 1..x]
    else 
        +/[dhyper(i, m, n, k) for i in (x+1)..k]
dnhyper(ncp,logdc,support) ==
     d := [DF(logdc.i+LOG(ncp)*support.i) for i in 0..(#logdc-1)]
     maxd := APPLY(FUNCTION(MAX),d)
     d2 :=[EXP(di-maxd) for di in d]
     sumd2 := +/d2
     [d2i/sumd2 for d2i in d2]
testTolerance(x, y, atol) ==
    if ABS(x-y) <= atol then true else false
test1() == testTolerance(2*ridder('COS,0.0,2.0),3.1415926535897932385, 1.0e-7)
test2() == testTolerance(choose(100, 5), 75287520, 0)
test3() == testTolerance(dhyper(5, 10, 7, 8), 0.3628137, 1.0e-7)
test4() == testTolerance(LOG(dhyper(5, 10, 7, 8)),-1.013866,  1.0e-7)
test5() == testTolerance(phyper(5, 10, 7, 8, true),0.7821884, 1.0e-7)
test6() == testTolerance(phyper(5, 10, 7, 8, false),0.2178116, 1.0e-7) 
fisherTest(a,b,c,d, alternative, OR, confInt, confLevel) == main where
  main() ==
        $m := a+c -- first column
        $n := b+d -- second column
        $k := a+b -- first row
        $x00 := a
        $lo := MAX(0, $k-$n)
        $hi := MIN($k, $m)
        $support := [i for i in $lo..$hi]
        $logdc := [LOG(dhyper(i, $m, $n, $k)) for i in $support]
        $doubleEps := 1.0e-10
        $plusInfinity := 1.0e10
        pvalue :=
                if alternative='"less" then pnhyper($x00, OR,false) 
                else if alternative='"greater" then pnhyper($x00, OR,true)
                else if alternative='"two-sided" then
                        relErr:= 1+1.0e-7
                        d:= dnhyper(OR,$logdc,$support)
                        dstar:= ELT(d,$x00-$lo)*relErr
                        +/[di for di in d | di<dstar]
                else -1.0 -- no match
        estimate :=
                if $x00=$lo then 0
                -- else if $x00=hi then return($plusInfinity)
                else
                        mu:= mnhyper(1)
                        if mu>$x00 then ridder(FUNCTION(f1),0,1) 
                        else if mu<$x00 then 1/ridder(FUNCTION(f2),$doubleEps,1) 
                        else 1
        interval :=
               if confInt then 
                        $alpha := 1 - confLevel
                        if alternative='"less" then [0, ncpU($x00)]
                        else if alternative='"greater" then [ncpL($x00), $plusInfinity]
                        else if alternative='"two-sided" then 
                                $alpha :=(1-confLevel)/2.0
                                [ncpL($x00), ncpU($x00)]
                        else [-1,-1] 
                else [-2,-2]
        [pvalue,interval,estimate]
  pnhyper (q,ncp,upperTail) ==
             if ncp=1 then 
                if upperTail then phyper(q-1, $m, $n, $k, false)
                else phyper(q, $m, $n, $k, true)
             else if ncp=0 then
                if upperTail then 
                    if q<=$lo then 1 else 0
                else if q>=$lo then 1 else 0
--           else if ncp=$plusInfinity then
--              if upperTail then 
--                  if q<=hi then return(1) else return(0)
--              else if q>= hi then return(1) else return(0)
             else 
                d:= dnhyper(ncp, $logdc, $support)
                if upperTail then
                      +/[d.i for i in 0..(#d-1) | $support.i>=q]
                else +/[d.i for i in 0..(#d-1) | $support.i<=q]
  mnhyper(ncp) ==
             if ncp=0.0 then $lo
             --if ncp=$plusInfinity then return(hi::R)
             else
                d := dnhyper(ncp,$logdc,$support)
                +/[si*di for di in d for si in $support]
  f1(u) == mnhyper(u) - $x00
  f2(u) == mnhyper(1/u) - $x00
  ncpU x ==
             --if x=$hi then $plusInfinity
             p:= pnhyper(x, 1.0, false)
             if p<$alpha then 
                ridder(FUNCTION(fu1),0.0,1.0)
             else if p>$alpha then 
                1/ridder(FUNCTION(fu2), $doubleEps,1)
             else 1
  fu1 u == pnhyper($x00,u,false) - $alpha
  fu2 u == pnhyper($x00,1/u,false) - $alpha
  ncpL x == 
             if x=$lo then 0
             else 
                p:= pnhyper(x, 1, true)
                if p>$alpha then ridder(FUNCTION(fl1), 0,1)
                else if p<$alpha then 1/ridder(FUNCTION(fl2), $doubleEps,1)
                else 1
  fl1 u == pnhyper($x00,u,true) - $alpha
  fl2 u == pnhyper($x00,1/u,true) - $alpha
test7() == fisherTest(10,10,10,20,'"two-sided",1,true,0.95)
alltests() == [test1(), test2(), test3(), test4(), test5(), test6()]

ERROR IN LINE 28
                if msign(fm,fnew) ^= fm then
...................................|
syntax error
ignored from line 28
                if msign(fm,fnew) ^= fm then
................|
ignored through line 38
                    fl:= fnew
.........................|
1874789261577970310-25px003.clisp PRODUCED
; compiling file "/var/aw/var/LatexWiki/1874789261577970310-25px003.clisp" (written 04 APR 2022 07:13:40 PM):

; /var/aw/var/LatexWiki/1874789261577970310-25px003.fasl written
; compilation finished in 0:00:00.065

Using the Boot code from Axiom:

alltests()$Lisp

   >> System error:
   Execution of a form compiled with errors.
Form:
  (("pile syntax error"))
Compile-time error:
  illegal function call

Third, for Reduce (which was also my first Reduce program):

symbolic;

nil

on rounded;

nil

symbolic procedure msign(x, y);
    (abs x) * (if y>0.0 then 1.0 else if y<0.0 then -1.0 else 0.0);

msign

%% Numerical root finding using Ridders method
   %% (Exit criteria hacked: how can one return from a repeat .. until statement?) 
symbolic procedure ridders(func, x1, x2);
   begin scalar eps, maxit, fl, fh, xl, xh, ans, xnew, iterNum, fnew;
      eps:= 1.0e-12;
      maxit:= 100;
      fl := funcall(func, x1);
      fh := funcall(func, x2);
      xl := x1;
      xh := x2;
      ans := -1.0e30;
      xnew := 0.0e0;
      iterNum := 0;
      if (fl*fh) > 0.0 then rederr "Initial points are not either side of zero.";
      if fl=0.0 then x1
      else if fh=0.0 then x2
         %if (fl*fh) < 0.0 then
      else repeat begin scalar xm, fm, ss;
         xm:= 0.5*(xl+xh);
         fm:= funcall(func, xm);
         ss:= sqrt((fm*fm) - (fl*fh));
         %if ss =0.0 then return ans;
         xnew:= xm + (((xm - xl) * (if (fl>fh) then 1.0 else -1.0) * fm) / ss);
         %if abs(xnew-ans) <= eps then return ans;
         ans:= xnew;
         fnew:= funcall(func, ans);
         %write(fnew);
         %if fnew=0.0 then return ans;
         if msign(fm,fnew) neq fm then begin;
            xl:= xm; 
            fl:= fm; 
            xh:= ans; 
            fh:= fnew; 
         end
         else if msign(fl, fnew) neq fl then begin;
            xh:= ans; 
            fh:= fnew;
         end    
         else if msign(fh, fnew) neq fh then begin;
            xl:= ans; 
            fl:= fnew; 
         end;
         iterNum:=iterNum+1;
         if iterNum >=maxit then rederr "Maximum iterations exceeded";
         %if verbose then write xl, xh, fl, fh;
      end until abs(fnew)<eps or abs(xh-xl) <= eps;
      return ans
   end;

ridders

symbolic procedure choose(n, x);
   begin scalar total, denom;
    total := 1.0;
    for denom:=1:x do <<
       total:=total/denom*(n-denom+1) >>;
    return total
   end;
*** SMACRO choose redefined 

choose

symbolic procedure dhyper(x, m, n, k);
   choose(m, x) * choose(n, k - x) / choose(m + n, k);

dhyper

procedure phyper(x, m, n, k, lowerTail);
   if lowerTail then 
      for i:=1:x sum dhyper(i, m, n, k)
   else 
      for i:=(x+1):k sum dhyper(i, m, n, k);

phyper

symbolic procedure testTolerance(x, y, atol);
   if abs(x-y) <= atol then t else nil;

testtolerance

symbolic procedure test1(); testTolerance(2*ridders(function(cos),0,2),
      cdr reval(algebraic pi),1e-8);

test1

symbolic procedure test2(); testTolerance(choose(100, 5), 75287520, 0);

test2

symbolic procedure test3(); testTolerance(dhyper(5, 10, 7, 8), 0.3628137, 1.0e-7);

test3

symbolic procedure test4(); testTolerance(log(dhyper(5, 10, 7, 8)),-1.013866,  1.0e-7);

test4

symbolic procedure test5(); testTolerance(phyper(5, 10, 7, 8, t),0.7821884, 1.0e-7);

test5

symbolic procedure test6(); testTolerance(phyper(5, 10, 7, 8, nil),0.2178116, 1.0e-7);

test6

symbolic procedure testSet1(); {test1(), test2(), test3(), test4(), test5(), test6()};

testset1

testSet1();

+++ error: 0 ((100 5) " invalid for " perm)

***** Continuing with parsing only ...

in "/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/6644752855287583515-25px.009.rin";

in "/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/2179316222567577690-25px.010.rin";

in "/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/5632302405673461694-25px.011.rin";

in "/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/5155894888540326291-25px.012.rin";

Fourth, the Maxima implementation was fairly brief:

Fifth, the implementation in Common Lisp was a more direct translation of the R code:

;; from cl-statistics.lisp
(defun safe-exp (x)
  "Eliminates floating point underflow for the exponential function.
Instead, it just returns 0.0d0"
  (setf x (coerce x 'double-float))
  (if (< x (log least-positive-double-float))
      0.0d0
      (exp x)))
(defun ridder (func x1 x2 &key (eps 1.0d-16) (maxit 30) (verbose nil))
  (let (
        (fl (funcall func x1))
        (fh (funcall func x2))
        (xl x1)
        (xh x2)
        (ans -1.11d30)
        (xnew 0.0d0)
        (iter-num 0)
        )
    (cond
     ((= fl 0) x1)
     ((= fh 0) x2)
     ((> (* fl fh) 0.0d0) 
      (error "Functions of the start points are not either side of zero."))
     ((< (* fl fh) 0.0d0) 
      (loop
       (let* (
              (xm (* 0.5d0 (+ xl xh)))
              (fm (funcall func xm))
              (ss (sqrt (- (* fm fm) (* fl fh))))
              )
         (if (= ss 0.0d0) (return ans))
         (setf xnew (+ xm (/ (* (- xm xl) (if (> fl fh) 1.0d0 -1.0d0) fm) ss)))
         (if (<= (abs (- xnew ans)) eps) (return ans))
         (setf ans xnew fnew (funcall func ans))
         (if (= fnew 0.0d0) (return ans))
         (cond ((not (= (msign fm fnew) fm))
                (setf xl xm fl fm xh ans fh fnew))
               ((not (= (msign fl fnew) fl))
                (setf xh ans fh fnew))
               ((not (= (msign fh fnew) fh))
                (setf xl ans fl fnew)))
         (incf iter-num)
         (if (>= iter-num maxit) 
             (return (values nil "Maximum iterations exceeded"))) ;; (error)?
         (if verbose (format t "~,8f ~,8f ~,8f ~,8f~%" xl xh fl fh))
         (if (<= (abs (- xh xl)) eps) (return ans))))))))
(defun msign (x y)
  (* (abs x) (cond ((> y 0.0d0) 1.0d0) ((< y 0.0d0) -1.0d0) (t 0.0d0))))
;;(- (* (ridder #'cos 0.0d0 2.0d0) 2.0d0) pi)

(defun choose (n x)
  (loop 
     for denom from 1 to x 
     and numerator from n downto (- n (1- x)) 
     and total = 1 then (* total (/ numerator denom)) 
     finally (return total)))
(defun dhyper (x m n k &key (log nil))
  (let ((val
         (/ (* (choose m x) (choose n (- k x))) (choose (+ m n) k))))
    (if log 
        (log (coerce val 'double-float)) 
        val)))
(defun phyper (x m n k &key (lower-tail t))
  (if lower-tail 
      (loop for i from 1 to x summing (dhyper i m n k))
      (loop for i from (1+ x) to k summing (dhyper i m n k))))

(defun fisher-test (x &key (alternative 'two-sided) (or 1.0d0)
                    (conf-int t)
                    (conf-level 0.95d0) (uniroot #'ridder))
    "Fisher's exact test for a 2x2 integer array.
This is a hand translation of R's fisher.test() 
making use of CL's large integers for the hypergeometric distribution"
    (let* ((m (loop for i upto 1 summing (aref x i 0)))
           (n (loop for i upto 1 summing (aref x i 1)))
           (k (loop for i upto 1 summing (aref x 0 i)))
           (x00 (aref x 0 0)) ; cf replacing x by (aref x 0 0)
           (lo (max 0 (- k n)))
           (hi (min k m))
           (support (loop for i from lo to hi collect i))
           (log-dc (loop for i in support 
                      collect (dhyper i m n k :log t)))
           (double-eps 1.0d-50))
      (labels 
          ((dnhyper (ncp)
             (setf ncp (coerce ncp 'double-float))
             (let* ((d (loop for i in log-dc and j in support
                          collect (+ i (* (log  ncp) j))))
                    (max-d (apply #'max d))
                    (d2 (loop for i in d collect 
                             (safe-exp (- i max-d)))) ;; NB: safe-exp used here
                    (sum-d2 (reduce #'+ d2)))
               (loop for i in d2 collect (/ i sum-d2))))
           (mnhyper (ncp)
             (cond
               ((= ncp 0) lo)
               ((equal ncp 'infinity) hi)
               (t (loop for i in support and j in (dnhyper ncp)
                       summing (* i j)))))
           (pnhyper (q ncp &key (upper-tail nil))
             (cond 
               ((= ncp 1)
                (if upper-tail 
                    (coerce (phyper (1- x00) m n k :lower-tail nil) 'double-float)
                    (coerce (phyper x00 m n k) 'double-float)))
               ((= ncp 0)
                (if upper-tail 
                    (if (<= q lo) 1 0)
                    (if (>= q lo) 1 0)))
               ((equal ncp 'infinity)
                (if upper-tail 
                    (if (<= q hi) 1 0)
                    (if (>= q hi) 1 0)))
               (t 
                (let ((d (dnhyper ncp)))
                  (if upper-tail
                      (loop for d-i in d and support-i in support
                           when (>= support-i q)
                           summing d-i)
                      (loop for d-i in d and support-i in support
                           when (<= support-i q)
                           summing d-i))))))
           (mle (x)
             (cond
               ((= x lo) 0)
               ((= x hi) 'infinity)
               (t
                (let ((mu (mnhyper 1)))
                  (cond 
                    ((> mu x)
                     (funcall uniroot (lambda (u) (- (mnhyper u) x))
                              0 1))
                    ((< mu x)
                     (/ (funcall uniroot (lambda (u) (- (mnhyper (/ u)) x))
                              double-eps 1)))
                    (t 1))))))
           (ncp-u (x alpha)
             (and (= x hi) 'infinity)
             (let ((p (pnhyper x 1)))
               (cond 
                 ((< p alpha)
                  (funcall uniroot (lambda (u) (- (pnhyper x u) alpha))
                         0 1))
                 ((> p alpha)
                  (/ (funcall uniroot (lambda (u) (- (pnhyper x (/ u)) alpha))
                            double-eps 1)))
                 (t 1))))
           (ncp-l (x alpha)
             (and (= x lo) 0)
             (let ((p (pnhyper x 1 :upper-tail t)))
               (cond 
                 ((> p alpha)
                  (funcall uniroot (lambda (u) (- (pnhyper x u :upper-tail t) alpha))
                         0 1))
                 ((< p alpha)
                  (/ (funcall uniroot (lambda (u) (- (pnhyper x (/ u) :upper-tail t) alpha))
                            double-eps 1)))
                 (t 1)))))
        (let ((p-value 
               (ecase alternative
                 (less (pnhyper x00 or))
                 (greater (pnhyper x00 or :upper-tail t)) 
                 (two-sided 
                  (let* ((relErr  (1+ 1.0d-7))
                         (d (dnhyper or))
                         (dstar (* (elt d (- x00 lo)) relErr)))
                    (loop for di in d when (< di dstar) summing di)))))
              (c-interval
               (if conf-int
                   (ecase alternative
                     (less (list 0 (ncp-u x00 (- 1 conf-level))))
                     (greater (list (ncp-l x00 (- 1 conf-level)) 'infinity))
                     (two-sided 
                      (let ((alpha (/ (- 1 conf-level) 2)))
                        (list (ncp-l x00 alpha) (ncp-u x00 alpha)))))
                   nil))
              (estimate (mle x00)))
          (values p-value c-interval estimate)))))
;;(fisher-test #2a((10 10) (10 20)))

; compiling file "/var/aw/var/LatexWiki/3567673564658662221-25px005.lisp" (written 04 APR 2022 07:13:40 PM):

; /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3567673564658662221-25px005.fasl written
; compilation finished in 0:00:00.082
Value = T

With output:

Finally, I have also included an implementation in Sage and commented out early code that would allow this to be used in Python.

?) else: return sum([dhyper(i, m, n, k) for i in range(x+1,k+1)]?) def fishertest(a,b,c,d,alternative,oddsratio,confLevel): m = a+c n = b+d k = a+b x00 = a lo = max(0, k-n) hi = min(k, m) support = range(lo,hi+1) logdc = [log(dhyper(i, m, n, k)) for i in support]? doubleEps = 1.0e-10 def dnhyper(ncp): d = [logdc[i]?+log(ncp)*support[i]? for i in range(len(logdc))] d2 = [exp(di-max(d)) for di in d]? return [d2i/sum(d2) for d2i in d2]? def mnhyper(ncp): if ncp==0: return lo if ncp==infinity: return hi else: d = dnhyper(ncp) return sum([support[i]?*d[i]? for i in range(len(d))]) def pnhyper(q,ncp,upperTail): if ncp==1: if upperTail: return phyper(q-1, m, n, k, False) else: return phyper(q, m, n, k, True) elif ncp==0: if upperTail: return 1 if q<=lo else 0 else: return 1 if q>=lo else 0 elif ncp==infinity: if upperTail: return 1 if q<=hi else 0 else: return 1 if q>= hi else 0 else: d = dnhyper(ncp) if upperTail: return sum([d[i]? for i in range(len(d)) if support[i]?>=q]) else: return sum([d[i]? for i in range(1,len(d)) if support[i]?<=q]) def ncpL(alpha): if x00==lo: return 0 else: p = pnhyper(x00, 1, True) if p>alpha: return brentq(lambda y: pnhyper(x00,y,True) - alpha, 0, 1) elif p<alpha: return 1/brentq(lambda y: pnhyper(x00,1/y,True) - alpha, doubleEps, 1) else: return 1 def ncpU(alpha): if x00==hi: return infinity else: p = pnhyper(x00, 1, False) if p<alpha: return brentq(lambda y: pnhyper(x00,y,False) - alpha, 0, 1) elif p>alpha: return 1/brentq(lambda y: pnhyper(x00,1/y,False) - alpha, doubleEps, 1) else: return 1 def pvalue(): if alternative == "less": return pnhyper(x00,oddsratio,False) elif alternative == "greater": return pnhyper(x00,oddsratio,True) elif alternative == "two-sided": relErr=1+1.0e-7 dn=dnhyper(oddsratio) dstar=dn[x00-lo]?*relErr return sum([di for di in dn if di<dstar]?) else: return -1 def cInterval(): if alternative == "less": return [0, ncpU(1-confLevel)]? elif alternative == "greater": return [ncpL(1-confLevel), infinity]? elif alternative == "two-sided": return [ncpL((1-confLevel)/2.0), ncpU((1-confLevel)/2.0)]? else: return [-1,-1]? def mle(): if x00==lo: return 0 elif x00==hi: return infinity else: mu = mnhyper(1) if mu>x00: return brentq(lambda y: mnhyper(y) - x00,0,1) elif mu<x00: return 1/brentq(lambda y: mnhyper(1/y) - x00,doubleEps,1) else: return 1 return dict(pvalue=pvalue(),cInterval=cInterval(),mle=mle())

## do some checks def testTolerance(x,y,tol): return abs(x-y)<=tol fit=fishertest(10,10,10,20,"two-sided",1,0.95) print [testTolerance(brentq(cos,0,2)*2,pi,0), testTolerance(binomial(100, 5), 75287520, 0), testTolerance(binomial(100, 95), 75287520, 0), testTolerance(dhyper(5, 10, 7, 8), 0.3628137, 1.0e-7), testTolerance(log(dhyper(5, 10, 7, 8)),-1.013866, 1.0e-7), testTolerance(phyper(5, 10, 7, 8, True),0.7821884, 1.0e-7), testTolerance(phyper(5, 10, 7, 8, False),0.2178116, 1.0e-7), testTolerance(fit['pvalue']?, 0.2575, 1.0e-3), testTolerance(fit['cInterval']?[0]?, 0.5383996, 1.0e-6), testTolerance(fit['cInterval']?[1]?, 7.4363242, 1.0e-4), testTolerance(fit['mle']?, 1.971640, 1.0e-4)] ## print some tests print fishertest(10,10,10,20,"two-sided",1,0.95) print fishertest(10,10,10,20,"less",1,0.95) print fishertest(10,10,10,20,"greater",1,0.95) " title=" \begin{sageblock} import scipy from scipy.optimize import brentq

## the following is required for use in Python/Scipy outside of Sage # from math import log, exp, cos, pi # infinity="infinity" # def binomial(n,x): # total=1 # for i in range(min(x,n-x)): # total=total*(n-i)/(i+1) # return total

def dhyper(x, m, n, k): return 1.0 binomial(m, x) binomial(n, k - x) / binomial(m + n, k) def phyper(x, m, n, k, lowerTail): if lowerTail: return sum([dhyper(i, m, n, k) for i in range(1,x+1)]?) else: return sum([dhyper(i, m, n, k) for i in range(x+1,k+1)]?) def fishertest(a,b,c,d,alternative,oddsratio,confLevel): m = a+c n = b+d k = a+b x00 = a lo = max(0, k-n) hi = min(k, m) support = range(lo,hi+1) logdc = [log(dhyper(i, m, n, k)) for i in support]? doubleEps = 1.0e-10 def dnhyper(ncp): d = [logdc[i]?+log(ncp)*support[i]? for i in range(len(logdc))] d2 = [exp(di-max(d)) for di in d]? return [d2i/sum(d2) for d2i in d2]? def mnhyper(ncp): if ncp==0: return lo if ncp==infinity: return hi else: d = dnhyper(ncp) return sum([support[i]?*d[i]? for i in range(len(d))]) def pnhyper(q,ncp,upperTail): if ncp==1: if upperTail: return phyper(q-1, m, n, k, False) else: return phyper(q, m, n, k, True) elif ncp==0: if upperTail: return 1 if q<=lo else 0 else: return 1 if q>=lo else 0 elif ncp==infinity: if upperTail: return 1 if q<=hi else 0 else: return 1 if q>= hi else 0 else: d = dnhyper(ncp) if upperTail: return sum([d[i]? for i in range(len(d)) if support[i]?>=q]) else: return sum([d[i]? for i in range(1,len(d)) if support[i]?<=q]) def ncpL(alpha): if x00==lo: return 0 else: p = pnhyper(x00, 1, True) if p>alpha: return brentq(lambda y: pnhyper(x00,y,True) - alpha, 0, 1) elif p<alpha: return 1/brentq(lambda y: pnhyper(x00,1/y,True) - alpha, doubleEps, 1) else: return 1 def ncpU(alpha): if x00==hi: return infinity else: p = pnhyper(x00, 1, False) if p<alpha: return brentq(lambda y: pnhyper(x00,y,False) - alpha, 0, 1) elif p>alpha: return 1/brentq(lambda y: pnhyper(x00,1/y,False) - alpha, doubleEps, 1) else: return 1 def pvalue(): if alternative == "less": return pnhyper(x00,oddsratio,False) elif alternative == "greater": return pnhyper(x00,oddsratio,True) elif alternative == "two-sided": relErr=1+1.0e-7 dn=dnhyper(oddsratio) dstar=dn[x00-lo]?*relErr return sum([di for di in dn if di<dstar]?) else: return -1 def cInterval(): if alternative == "less": return [0, ncpU(1-confLevel)]? elif alternative == "greater": return [ncpL(1-confLevel), infinity]? elif alternative == "two-sided": return [ncpL((1-confLevel)/2.0), ncpU((1-confLevel)/2.0)]? else: return [-1,-1]? def mle(): if x00==lo: return 0 elif x00==hi: return infinity else: mu = mnhyper(1) if mu>x00: return brentq(lambda y: mnhyper(y) - x00,0,1) elif mu<x00: return 1/brentq(lambda y: mnhyper(1/y) - x00,doubleEps,1) else: return 1 return dict(pvalue=pvalue(),cInterval=cInterval(),mle=mle())

## do some checks def testTolerance(x,y,tol): return abs(x-y)<=tol fit=fishertest(10,10,10,20,"two-sided",1,0.95) print [testTolerance(brentq(cos,0,2)*2,pi,0), testTolerance(binomial(100, 5), 75287520, 0), testTolerance(binomial(100, 95), 75287520, 0), testTolerance(dhyper(5, 10, 7, 8), 0.3628137, 1.0e-7), testTolerance(log(dhyper(5, 10, 7, 8)),-1.013866, 1.0e-7), testTolerance(phyper(5, 10, 7, 8, True),0.7821884, 1.0e-7), testTolerance(phyper(5, 10, 7, 8, False),0.2178116, 1.0e-7), testTolerance(fit['pvalue']?, 0.2575, 1.0e-3), testTolerance(fit['cInterval']?[0]?, 0.5383996, 1.0e-6), testTolerance(fit['cInterval']?[1]?, 7.4363242, 1.0e-4), testTolerance(fit['mle']?, 1.971640, 1.0e-4)] ## print some tests print fishertest(10,10,10,20,"two-sided",1,0.95) print fishertest(10,10,10,20,"less",1,0.95) print fishertest(10,10,10,20,"greater",1,0.95) " class="equation" src="images/7581108121597389083-16.0px.png" align="bottom" Style="vertical-align:text-bottom" width="522" height="728"/>

This would give the following output:

  [0 <= 0, True, True, True, True, True, True, 4.92428109829e-05 <= 0.00100000000000000, 
     True, True, True]
  {'mle': 1.9716269432603342, 'pvalue': 0.257549242811, 
   'cInterval': [0.53839938167817269, 7.4363408387619616]}
  {'mle': 1.9716269432603342, 'pvalue': 0.929480494158038, 
   'cInterval': [0, 6.1438085831669706]}
  {'mle': 1.9716269432603342, 'pvalue': 0.188301375769915, 
   'cInterval': [0.64599378194777934, +Infinity]}