Numerical integration/Adaptive Simpson's method is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Lychee (1969)'s Modified Adaptive Simpson's method (doi:10.1145/321526.321537) is a numerical quadrature method that recursively bisects the interval until the precision is high enough.

 ```; Lychee's ASR, Modifications 1, 2, 3 procedure _quad_asr_simpsons(f, a, fa, b, fb) m := (a + b) / 2 fm := f(m) h := b - a ``` ``` return multiple [m, fm, (h / 6) * (f(a) + f(b) + 4*sum1 + 2*sum2)] procedure _quad_asr(f, a, fa, b, fb, tol, whole, m, fm, depth) lm, flm, left  := _quad_asr_simpsons(f, a, fa, m, fm) rm, frm, right := _quad_asr_simpsons(f, m, fm, b, fb) delta := left + right - whole tol' := tol / 2 if depth <= 0 or tol' == tol or abs(delta) <= 15 * tol: return left + right + delta / 15 else: return _quad_asr(f, a, fa, m, fm, tol', left , lm, flm, depth - 1) + _quad_asr(f, m, fm, b, fb, tol', right, rm, frm, depth - 1) procedure quad_asr(f, a, b, tol, depth) fa := f(a) fb := f(b) m, fm, whole := _quad_asr_simpsons(f, a, fa, b, fb) return _quad_asr(f, a, fa, b, fb, tol, whole, m, fm, depth) ```

## C

Translation of: zkl
`#include <stdio.h>#include <math.h> typedef struct { double m; double fm; double simp; } triple; /* "structured" adaptive version, translated from Racket */triple _quad_simpsons_mem(double (*f)(double), double a, double fa, double b, double fb) {    // Evaluates Simpson's Rule, also returning m and f(m) to reuse.    double m = (a + b) / 2;    double fm = f(m);    double simp = fabs(b - a) / 6 * (fa + 4*fm + fb);    triple t = {m, fm, simp};    return t;} double _quad_asr(double (*f)(double), double a, double fa, double b, double fb, double eps, double whole, double m, double fm) {    // Efficient recursive implementation of adaptive Simpson's rule.    // Function values at the start, middle, end of the intervals are retained.    triple lt = _quad_simpsons_mem(f, a, fa, m, fm);    triple rt = _quad_simpsons_mem(f, m, fm, b, fb);    double delta = lt.simp + rt.simp - whole;    if (fabs(delta) <= eps * 15) return lt.simp + rt.simp + delta/15;    return _quad_asr(f, a, fa, m, fm, eps/2, lt.simp, lt.m, lt.fm) +           _quad_asr(f, m, fm, b, fb, eps/2, rt.simp, rt.m, rt.fm);} double quad_asr(double (*f)(double), double a, double b, double eps) {    // Integrate f from a to b using ASR with max error of eps.    double fa = f(a);    double fb = f(b);    triple t = _quad_simpsons_mem(f, a, fa, b, fb);    return _quad_asr(f, a, fa, b, fb, eps, t.simp, t.m, t.fm);} int main(){    double a = 0.0, b = 1.0;    double sinx = quad_asr(sin, a, b, 1e-09);    printf("Simpson's integration of sine from %g to %g = %f\n", a, b, sinx);    return 0;}`
Output:
```Simpson's integration of sine from 0 to 1 = 0.459698
```

## Go

Like the zkl entry, this is also a translation of the Python code in the Wikipedia article.

`package main import (    "fmt"    "math") type F = func(float64) float64 /* "structured" adaptive version, translated from Racket */func quadSimpsonsMem(f F, a, fa, b, fb float64) (m, fm, simp float64) {    // Evaluates Simpson's Rule, also returning m and f(m) to reuse.    m = (a + b) / 2    fm = f(m)    simp = math.Abs(b-a) / 6 * (fa + 4*fm + fb)    return} func quadAsrRec(f F, a, fa, b, fb, eps, whole, m, fm float64) float64 {    // Efficient recursive implementation of adaptive Simpson's rule.    // Function values at the start, middle, end of the intervals are retained.    lm, flm, left := quadSimpsonsMem(f, a, fa, m, fm)    rm, frm, right := quadSimpsonsMem(f, m, fm, b, fb)    delta := left + right - whole    if math.Abs(delta) <= eps*15 {        return left + right + delta/15    }    return quadAsrRec(f, a, fa, m, fm, eps/2, left, lm, flm) +        quadAsrRec(f, m, fm, b, fb, eps/2, right, rm, frm)} func quadAsr(f F, a, b, eps float64) float64 {    // Integrate f from a to b using ASR with max error of eps.    fa, fb := f(a), f(b)    m, fm, whole := quadSimpsonsMem(f, a, fa, b, fb)    return quadAsrRec(f, a, fa, b, fb, eps, whole, m, fm)} func main() {    a, b := 0.0, 1.0    sinx := quadAsr(math.Sin, a, b, 1e-09)    fmt.Printf("Simpson's integration of sine from %g to %g = %f\n", a, b, sinx)   }`
Output:
```Simpson's integration of sine from 0 to 1 = 0.459698
```

## Kotlin

Translation of: Go
`// Version 1.2.71 import kotlin.math.absimport kotlin.math.sin typealias F = (Double) -> Doubletypealias T = Triple<Double, Double, Double> /* "structured" adaptive version, translated from Racket */fun quadSimpsonsMem(f: F, a: Double, fa: Double, b: Double, fb: Double): T {    // Evaluates Simpson's Rule, also returning m and f(m) to reuse    val m = (a + b) / 2    val fm = f(m)    val simp = abs(b - a) / 6 * (fa + 4 * fm + fb)    return T(m, fm, simp)} fun quadAsrRec(f: F, a: Double, fa: Double, b: Double, fb: Double,    eps: Double, whole: Double, m: Double, fm: Double): Double {    // Efficient recursive implementation of adaptive Simpson's rule.    // Function values at the start, middle, end of the intervals are retained.     val (lm, flm, left) = quadSimpsonsMem(f, a, fa, m, fm)    val (rm, frm, right) = quadSimpsonsMem(f, m, fm, b, fb)    val delta = left + right - whole    if (abs(delta) <= eps * 15)  return left + right + delta / 15    return quadAsrRec(f, a, fa, m, fm, eps / 2, left, lm, flm) +        quadAsrRec(f, m, fm, b, fb, eps / 2, right, rm, frm)} fun quadAsr(f: F, a: Double, b: Double, eps: Double): Double {    // Integrate f from a to b using ASR with max error of eps.    val fa = f(a)    val fb = f(b)    val (m, fm, whole) = quadSimpsonsMem(f, a, fa, b, fb)    return quadAsrRec(f, a, fa, b, fb, eps, whole, m, fm)} fun main(args: Array<String>) {    val a = 0.0    val b = 1.0    val sinx = quadAsr(::sin, a, b, 1.0e-09)    println("Simpson's integration of sine from \$a to \$b = \${"%6f".format(sinx)}")}`
Output:
```Simpson's integration of sine from 0.0 to 1.0 = 0.459698
```

## Perl 6

Works with: Rakudo version 2018.10

Fairly direct translation of the Python code.

`sub adaptive-Simpson-quadrature(&f, \$left, \$right, \ε = 1e-9) {    my \$lf = f(\$left);    my \$rf = f(\$right);    my (\$mid, \$midf, \$whole) = Simpson-quadrature-mid(&f, \$left, \$lf, \$right, \$rf);    return recursive-Simpsons-asr(&f, \$left, \$lf, \$right, \$rf, ε, \$whole, \$mid, \$midf);     sub Simpson-quadrature-mid(&g, \$l, \$lf, \$r, \$rf){        my \$mid = (\$l + \$r) / 2;        my \$midf = g(\$mid);        (\$mid, \$midf, (\$r - \$l).abs / 6 * (\$lf + 4 * \$midf + \$rf))    }     sub recursive-Simpsons-asr(&h, \$a, \$fa, \$b, \$fb, \$eps, \$whole, \$m, \$fm){        my (\$lm, \$flm, \$left)  = Simpson-quadrature-mid(&h, \$a, \$fa, \$m, \$fm);        my (\$rm, \$frm, \$right) = Simpson-quadrature-mid(&h, \$m, \$fm, \$b, \$fb);        my \$delta = \$left + \$right - \$whole;        \$delta.abs <= 15 * \$eps            ?? \$left + \$right + \$delta / 15            !! recursive-Simpsons-asr(&h, \$a, \$fa, \$m, \$fm, \$eps/2, \$left,  \$lm, \$flm) +               recursive-Simpsons-asr(&h, \$m, \$fm, \$b, \$fb, \$eps/2, \$right, \$rm, \$frm)    }} my (\$a, \$b) = 0e0, 1e0;my \$sin = adaptive-Simpson-quadrature(&sin, \$a, \$b, 1e-9).round(10**-9);;say "Simpson's integration of sine from \$a to \$b = \$sin";`
Output:
`Simpson's integration of sine from 0 to 1 = 0.459697694`

## Phix

Translation of: Go
`function quadSimpsonsMem(integer f, atom a, fa, b, fb)    -- Evaluates Simpson's Rule, also returning m and f(m) to reuse.    atom m = (a + b) / 2,         fm = call_func(f,{m}),         simp = abs(b-a) / 6 * (fa + 4*fm + fb)    return {m, fm, simp}end function function quadAsrRec(integer f, atom a, fa, b, fb, eps, whole, m, fm)    -- Efficient recursive implementation of adaptive Simpson's rule.    -- Function values at the start, middle, end of the intervals are retained.    atom {lm, flm, left} := quadSimpsonsMem(f, a, fa, m, fm),         {rm, frm, rght} := quadSimpsonsMem(f, m, fm, b, fb),         delta := left + rght - whole    if abs(delta) <= eps*15 then        return left + rght + delta/15    end if    return quadAsrRec(f, a, fa, m, fm, eps/2, left, lm, flm) +           quadAsrRec(f, m, fm, b, fb, eps/2, rght, rm, frm)end function function quadAsr(integer f, atom a, b, eps)    -- Integrate f from a to b using ASR with max error of eps.    atom fa := call_func(f,{a}),         fb := call_func(f,{b}),         {m, fm, whole} := quadSimpsonsMem(f, a, fa, b, fb)    return quadAsrRec(f, a, fa, b, fb, eps, whole, m, fm)end function -- we need a mini wrapper to get a routine_id for sin()-- (because sin() is implemented in low-level assembly)function _sin(atom a)    return sin(a)end function  atom a := 0.0, b := 1.0,     sinx := quadAsr(routine_id("_sin"), a, b, 1e-09)printf(1,"Simpson's integration of sine from %g to %g = %f\n", {a, b, sinx})`
Output:
```Simpson's integration of sine from 0 to 1 = 0.459698
```

## zkl

Translation of: Python
`# "structured" adaptive version, translated from Racketfcn _quad_simpsons_mem(f, a,fa, b,fb){   #Evaluates the Simpson's Rule, also returning m and f(m) to reuse"""   m,fm := (a + b)/2, f(m);   return(m,fm, (b - a).abs()/6*(fa + fm*4 + fb));}fcn _quad_asr(f, a,fa, b,fb, eps, whole, m,fm){  # Efficient recursive implementation of adaptive Simpson's rule.  # Function values at the start, middle, end of the intervals are retained.    lm,flm,left  := _quad_simpsons_mem(f, a,fa, m,fm);   rm,frm,right := _quad_simpsons_mem(f, m,fm, b,fb);   delta:=left + right - whole;   if(delta.abs() <= eps*15) return(left + right + delta/15);   _quad_asr(f, a,fa, m,fm, eps/2, left , lm,flm) +   _quad_asr(f, m,fm, b,fb, eps/2, right, rm,frm)}fcn quad_asr(f,a,b,eps){   #Integrate f from a to b using Adaptive Simpson's Rule with max error of eps   fa,fb      := f(a),f(b);   m,fm,whole := _quad_simpsons_mem(f, a,fa, b,fb);   _quad_asr(f, a,fa, b,fb, eps,whole,m,fm);}`
`sinx:=quad_asr((1.0).sin.unbind(), 0.0, 1.0, 1e-09);println("Simpson's integration of sine from 1 to 2 = ",sinx);`
Output:
```Simpson's integration of sine from 1 to 2 = 0.459698
```