# Thiele's interpolation formula

Thiele's interpolation formula
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Thiele's interpolation formula is an interpolation formula for a function f(•) of a single variable.   It is expressed as a continued fraction:

${\displaystyle f(x)=f(x_{1})+{\cfrac {x-x_{1}}{\rho _{1}(x_{1},x_{2})+{\cfrac {x-x_{2}}{\rho _{2}(x_{1},x_{2},x_{3})-f(x_{1})+{\cfrac {x-x_{3}}{\rho _{3}(x_{1},x_{2},x_{3},x_{4})-\rho _{1}(x_{1},x_{2})+\cdots }}}}}}}$

${\displaystyle \rho }$   represents the   reciprocal difference,   demonstrated here for reference:

${\displaystyle \rho _{1}(x_{0},x_{1})={\frac {x_{0}-x_{1}}{f(x_{0})-f(x_{1})}}}$
${\displaystyle \rho _{2}(x_{0},x_{1},x_{2})={\frac {x_{0}-x_{2}}{\rho _{1}(x_{0},x_{1})-\rho _{1}(x_{1},x_{2})}}+f(x_{1})}$
${\displaystyle \rho _{n}(x_{0},x_{1},\ldots ,x_{n})={\frac {x_{0}-x_{n}}{\rho _{n-1}(x_{0},x_{1},\ldots ,x_{n-1})-\rho _{n-1}(x_{1},x_{2},\ldots ,x_{n})}}+\rho _{n-2}(x_{1},\ldots ,x_{n-1})}$

Demonstrate Thiele's interpolation function by:

1. Building a   32   row trig table of values   for   ${\displaystyle x}$   from   0   by   0.05   to   1.55   of the trig functions:
•   sin
•   cos
•   tan
2. Using columns from this table define an inverse - using Thiele's interpolation - for each trig function;
3. Finally: demonstrate the following well known trigonometric identities:
•   6 × sin-1 ½ = ${\displaystyle \pi }$
•   3 × cos-1 ½ = ${\displaystyle \pi }$
•   4 × tan-1 1 = ${\displaystyle \pi }$

## Contents

with Ada.Numerics.Generic_Real_Arrays; generic   type Real is digits <>;package Thiele is   package Real_Arrays is new Ada.Numerics.Generic_Real_Arrays (Real);   subtype Real_Array is Real_Arrays.Real_Vector;    type Thiele_Interpolation (Length : Natural) is private;    function Create (X, Y : Real_Array) return Thiele_Interpolation;   function Inverse (T : Thiele_Interpolation; X : Real) return Real;private   type Thiele_Interpolation (Length : Natural) is record      X, Y, RhoX : Real_Array (1 .. Length);   end record;end Thiele;

package body Thiele is   use type Real_Array;    function "/" (Left, Right : Real_Array) return Real_Array is      Result : Real_Array (Left'Range);   begin      if Left'Length /= Right'Length then         raise Constraint_Error with "arrays not same size";      end if;      for I in Result'Range loop         Result (I) := Left (I) / Right (I);      end loop;      return Result;   end "/";    function Rho (X, Y : Real_Array) return Real_Array is      N      : constant Natural                      := X'Length;      P      : array (1 .. N) of Real_Array (1 .. N) :=        (others => (others => 9.9));      Result : Real_Array (1 .. N);   begin      P (1) (1 .. N)      := Y (1 .. N);      P (2) (1 .. N - 1)  := (X (1 .. N - 1) - X (2 .. N)) /        (P (1) (1 .. N - 1) - P (1) (2 .. N));      for I in 3 .. N loop         P (I) (1 .. N - I + 1)  := P (I - 2) (2 .. N - I + 2) +           (X (1 .. N - I + 1) - X (I .. N)) /           (P (I - 1) (1 .. N - I + 1) - P (I - 1) (2 .. N - I + 2));      end loop;      for I in X'Range loop         Result (I) := P (I) (1);      end loop;      return Result;   end Rho;    function Create (X, Y : Real_Array) return Thiele_Interpolation is   begin      if X'Length < 3 then         raise Constraint_Error with "at least 3 values";      end if;      if X'Length /= Y'Length then         raise Constraint_Error with "input arrays not same size";      end if;      return (Length => X'Length, X => X, Y => Y, RhoX => Rho (X, Y));   end Create;    function Inverse (T : Thiele_Interpolation; X : Real) return Real is      A : Real := 0.0;   begin      for I in reverse 3 .. T.Length loop         A := (X - T.X (I - 1)) / (T.RhoX (I) - T.RhoX (I - 2) + A);      end loop;      return T.Y (1) + (X - T.X (1)) / (T.RhoX (2) + A);   end Inverse; end Thiele;

example:

with Ada.Text_IO;with Ada.Numerics.Generic_Elementary_Functions;with Thiele; procedure Main is   package Math is new Ada.Numerics.Generic_Elementary_Functions     (Long_Float);   package Float_Thiele is new Thiele (Long_Float);   use Float_Thiele;    Row_Count : Natural := 32;    X_Values   : Real_Array (1 .. Row_Count);   Sin_Values : Real_Array (1 .. Row_Count);   Cos_Values : Real_Array (1 .. Row_Count);   Tan_Values : Real_Array (1 .. Row_Count);begin   -- build table   for I in 1 .. Row_Count loop      X_Values (I)   := Long_Float (I) * 0.05 - 0.05;      Sin_Values (I) := Math.Sin (X_Values (I));      Cos_Values (I) := Math.Cos (X_Values (I));      Tan_Values (I) := Math.Tan (X_Values (I));   end loop;   declare      Sin : Thiele_Interpolation := Create (Sin_Values, X_Values);      Cos : Thiele_Interpolation := Create (Cos_Values, X_Values);      Tan : Thiele_Interpolation := Create (Tan_Values, X_Values);   begin      Ada.Text_IO.Put_Line        ("Internal Math.Pi:    " &         Long_Float'Image (Ada.Numerics.Pi));      Ada.Text_IO.Put_Line        ("Thiele 6*InvSin(0.5):" &         Long_Float'Image (6.0 * Inverse (Sin, 0.5)));      Ada.Text_IO.Put_Line        ("Thiele 3*InvCos(0.5):" &         Long_Float'Image (3.0 * Inverse (Cos, 0.5)));      Ada.Text_IO.Put_Line        ("Thiele 4*InvTan(1):  " &         Long_Float'Image (4.0 * Inverse (Tan, 1.0)));   end;end Main;

output:

Internal Math.Pi:     3.14159265358979E+00
Thiele 6*InvSin(0.5): 3.14159265358979E+00
Thiele 3*InvCos(0.5): 3.14159265358979E+00
Thiele 4*InvTan(1):   3.14159265358979E+00

## ALGOL 68

Works with: ALGOL 68 version Revision 1 - except the Currying (aka partial parametrisation) in test block is a proposal for ALGOL 68 Rev2
Works with: ALGOL 68G version Any - tested with release 1.18.0-9h.tiny - Currying is supported.
PROC raise exception = ([]STRING msg)VOID: ( putf(stand error,("Exception:", $" "g$, msg, $l$)); stop ); # The MODE of lx and ly here should really be a UNION of "something REAL","something COMPLex", and "something SYMBOLIC" ... # PROC thiele=([]REAL lx,ly, REAL x) REAL:BEGIN  []REAL xx=lx[@1],yy=ly[@1];  INT n=UPB xx;  IF UPB yy=n THEN# Assuming that the values of xx are distinct ... #    [0:n-1,1:n]REAL p;    p[0,]:=yy[];    FOR i TO n-1 DO p[1,i]:=(xx[i]-xx[1+i])/(p[0,i]-p[0,1+i]) OD;    FOR i FROM 2 TO n-1 DO      FOR j TO n-i DO        p[i,j]:=(xx[j]-xx[j+i])/(p[i-1,j]-p[i-1,j+1])+p[i-2,j+1]      OD    OD;    REAL a:=0;    FOR i FROM n-1 BY -1 TO 2 DO a:=(x-xx[i])/(p[i,1]-p[i-2,1]+a) OD;    yy[1]+(x-xx[1])/(p[1,1]+a)  ELSE    raise exception(("Unequal length arrays supplied: ",whole(UPB xx,0)," NE ",whole(UPB yy,0))); SKIP  FIEND; test:(  FORMAT real fmt = $g(0,real width-2)$;   REAL lwb x=0, upb x=1.55, delta x = 0.05;    [0:ENTIER ((upb x-lwb x)/delta x)]STRUCT(REAL x, sin x, cos x, tan x) trig table;   PROC init trig table = VOID:     FOR i FROM LWB trig table TO UPB trig table DO       REAL x = lwb x+i*delta x;       trig table[i]:=(x, sin(x), cos(x), tan(x))    OD;   init trig table; # Curry the thiele function to create matching inverse trigonometric functions #  PROC (REAL)REAL inv sin = thiele(sin x OF trig table, x OF trig table,),                  inv cos = thiele(cos x OF trig table, x OF trig table,),                  inv tan = thiele(tan x OF trig table, x OF trig table,);   printf(($"pi estimate using "g" interpolation: "f(real fmt)l$,     "sin", 6*inv sin(1/2),    "cos", 3*inv cos(1/2),    "tan", 4*inv tan(1)  )))

Output:

pi estimate using sin interpolation: 3.1415926535898
pi estimate using cos interpolation: 3.1415926535898
pi estimate using tan interpolation: 3.1415926535898


## C

The recursive relations of ${\displaystyle \rho }$s can be made clearer: Given ${\displaystyle N+1}$ sampled points ${\displaystyle x_{0},x_{1},\cdots x_{N}}$, rewrite the symbol ${\displaystyle \rho }$ as

${\displaystyle \rho _{n,i}=\rho _{n}(x_{i},x_{i+1},\cdots x_{i+n})}$ where ${\displaystyle 1\leq n\leq N}$,

with suplements

${\displaystyle \rho _{0,i}=f(x_{i})\qquad {\text{and}}\qquad \rho _{n,i}=0\quad {\text{for}}\quad n<0}$.

Now the recursive relation is simply

${\displaystyle \rho _{n,i}=\displaystyle {x_{i}-x_{i+n} \over \rho _{n-1,i}-\rho _{n-1,i+1}}+\rho _{n-2,i+1},\ 0\leq i+n\leq N}$

Also note how ${\displaystyle f(x_{1})}$ in the interpolation formula can be replaced by ${\displaystyle \rho _{0,1}}$; define Thiele interpolation at step ${\displaystyle n}$ as

${\displaystyle \displaystyle {F_{n}(x)=\rho _{n,1}-\rho _{n-2,1}+{x-x_{n+1} \over F_{n+1}(x)}}}$

with the termination ${\displaystyle F_{N}(x)=1}$, and the interpolation formula is now ${\displaystyle f(x)=F_{0}(x)}$, easily implemented as a recursive function.

Note that each ${\displaystyle \rho _{n}}$ needs to look up ${\displaystyle \rho _{n-1}}$ twice, so the total look ups go up as ${\displaystyle O(2^{N})}$ while there are only ${\displaystyle O(N^{2})}$ values. This is a text book situation for memoization.

#include <stdio.h>#include <string.h>#include <math.h> #define N 32#define N2 (N * (N - 1) / 2)#define STEP .05 double xval[N], t_sin[N], t_cos[N], t_tan[N]; /* rho tables, layout:	rho_{n-1}(x0)	rho_{n-2}(x0), rho_{n-1}(x1),	....	rho_0(x0), rho_0(x1), ... rho_0(x_{n-1})   rho_i row starts at index (n - 1 - i) * (n - i) / 2  	*/double r_sin[N2], r_cos[N2], r_tan[N2]; /* both rho and thiele functions recursively resolve values as decribed by   formulas.  rho is cached, thiele is not. */ /* rho_n(x_i, x_{i+1}, ..., x_{i + n}) */double rho(double *x, double *y, double *r, int i, int n){	if (n < 0) return 0;	if (!n) return y[i]; 	int idx = (N - 1 - n) * (N - n) / 2 + i;	if (r[idx] != r[idx]) /* only happens if value not computed yet */		r[idx] = (x[i] - x[i + n])			/ (rho(x, y, r, i, n - 1) - rho(x, y, r, i + 1, n - 1))			+ rho(x, y, r, i + 1, n - 2);	return r[idx];} double thiele(double *x, double *y, double *r, double xin, int n){	if (n > N - 1) return 1;	return rho(x, y, r, 0, n) - rho(x, y, r, 0, n - 2)		+ (xin - x[n]) / thiele(x, y, r, xin, n + 1);} #define i_sin(x) thiele(t_sin, xval, r_sin, x, 0)#define i_cos(x) thiele(t_cos, xval, r_cos, x, 0)#define i_tan(x) thiele(t_tan, xval, r_tan, x, 0) int main(){	int i;	for (i = 0; i < N; i++) {		xval[i] = i * STEP;		t_sin[i] = sin(xval[i]);		t_cos[i] = cos(xval[i]);		t_tan[i] = t_sin[i] / t_cos[i];	}	for (i = 0; i < N2; i++)		/* init rho tables to NaN */		r_sin[i] = r_cos[i] = r_tan[i] = 0/0.; 	printf("%16.14f\n", 6 * i_sin(.5));	printf("%16.14f\n", 3 * i_cos(.5));	printf("%16.14f\n", 4 * i_tan(1.));	return 0;}
output
3.141592653589793.141592653589793.14159265358979

## Common Lisp

Using the notations from above the C code instead of task desc.

;; 256 is heavy overkill, but hey, we memoized(defparameter *thiele-length* 256)(defparameter *rho-cache* (make-hash-table :test #'equal)) (defmacro make-thele-func (f name xx0 xx1)  (let ((xv (gensym)) (yv (gensym))	(x0 (gensym)) (x1 (gensym)))    (let* ((,xv (make-array (1+ *thiele-length*)))	    (,yv (make-array (1+ *thiele-length*)))	    (,x0 ,xx0)	    (,x1 ,xx1))       (loop for i to *thiele-length* with x do	     (setf x (+ ,x0 (* (/ (- ,x1 ,x0) *thiele-length*) i))		   (aref ,yv i) x		   (aref ,xv i) (funcall ,f x)))       (defun ,name (x) (thiele x ,yv ,xv, 0))))) (defun rho (yv xv n i)  (let (hit (key (list yv xv n i)))    (if (setf hit (gethash key *rho-cache*))      hit      (setf (gethash key *rho-cache*)	    (cond ((zerop n) (aref yv i))		  ((minusp n) 0)		  (t (+ (rho yv xv (- n 2) (1+ i))			(/  (- (aref xv i) 			       (aref xv (+ i n)))			    (- (rho yv xv (1- n) i)			       (rho yv xv (1- n) (1+ i))))))))))) (defun thiele (x yv xv n)  (if (= n *thiele-length*)    1    (+ (- (rho yv xv n 1) (rho yv xv (- n 2) 1))       (/ (- x (aref xv (1+ n)))	  (thiele x yv xv (1+ n)))))) (make-thele-func #'sin inv-sin 0 (/ pi 2))(make-thele-func #'cos inv-cos 0 (/ pi 2))(make-thele-func #'tan inv-tan 0 (/ pi 2.1)) ; tan(pi/2) is INF (format t "~f~%" (* 6 (inv-sin .5)))(format t "~f~%" (* 3 (inv-cos .5)))(format t "~f~%" (* 4 (inv-tan 1)))
output (SBCL):
3.1415926535897933.14159265358851723.141592653589819

import std.stdio, std.range, std.array, std.algorithm, std.math; struct Domain {    const real b, e, s;     auto range() const pure /*nothrow*/ @safe /*@nogc*/ {        return iota(b, e + s, s);    }} real eval0(alias RY, alias X, alias Y)(in real x) pure nothrow @safe @nogc {    real a = 0.0L;    foreach_reverse (immutable i; 2 .. X.length - 3)        a = (x - X[i]) / (RY[i] - RY[i-2] + a);    return Y[1] + (x - X[1]) / (RY[1] + a);} immutable struct Thiele {    immutable real[] Y, X, rhoY, rhoX;     this(real[] y, real[] x) immutable pure nothrow /*@safe*/    in {        assert(x.length > 2, "at leat 3 values");        assert(x.length == y.length, "input arrays not of same size");    } body {        this.Y = y.idup;        this.X = x.idup;        rhoY = rhoN(Y, X);        rhoX = rhoN(X, Y);    }     this(in real function(real) pure nothrow @safe @nogc f,         Domain d = Domain(0.0L, 1.55L, 0.05L))    immutable pure /*nothrow @safe*/ {        auto xrng = d.range.array;        this(xrng.map!f.array, xrng);    }     auto rhoN(immutable real[] y, immutable real[] x)    pure nothrow @safe {        immutable int N = x.length;        auto p = new real[][](N, N);        p[0][] = y[];        p[1][0 .. $- 1] = (x[0 ..$-1] - x[1 .. $]) / (p[0][0 ..$-1] - p[0][1 .. $]); foreach (immutable int j; 2 .. N - 1) { immutable M = N - j - 1; p[j][0..M] = p[j-2][1..M+1] + (x[0..M] - x[j..M+j]) / (p[j-1][0 .. M] - p[j-1][1 .. M+1]); } return p.map!q{ a[1] }.array; } alias eval = eval0!(rhoY, X, Y); alias inverse = eval0!(rhoX, Y, X);} void main() { // Can't pass sin, cos and tan directly. immutable tsin = Thiele(x => x.sin); immutable tcos = Thiele(x => x.cos); immutable ttan = Thiele(x => x.tan); writefln(" %d interpolating points\n", tsin.X.length); writefln("std.math.sin(0.5): %20.18f", 0.5L.sin); writefln(" Thiele sin(0.5): %20.18f\n", tsin.eval(0.5L)); writefln("*%20.19f library constant", PI); writefln(" %20.19f 6 * inv_sin(0.5)", tsin.inverse(0.5L) * 6.0L); writefln(" %20.19f 3 * inv_cos(0.5)", tcos.inverse(0.5L) * 3.0L); writefln(" %20.19f 4 * inv_tan(1.0)", ttan.inverse(1.0L) * 4.0L);} Output:  32 interpolating points std.math.sin(0.5): 0.479425538604203000 Thiele sin(0.5): 0.479425538604203000 *3.1415926535897932385 library constant 3.1415926535897932380 6 * inv_sin(0.5) 3.1415926535897932382 3 * inv_cos(0.5) 3.1415926535897932382 4 * inv_tan(1.0) ## Go Translation of: ALGOL 68 package main import ( "fmt" "math") func main() { // task 1: build 32 row trig table const nn = 32 const step = .05 xVal := make([]float64, nn) tSin := make([]float64, nn) tCos := make([]float64, nn) tTan := make([]float64, nn) for i := range xVal { xVal[i] = float64(i) * step tSin[i], tCos[i] = math.Sincos(xVal[i]) tTan[i] = tSin[i] / tCos[i] } // task 2: define inverses iSin := thieleInterpolator(tSin, xVal) iCos := thieleInterpolator(tCos, xVal) iTan := thieleInterpolator(tTan, xVal) // task 3: demonstrate identities fmt.Printf("%16.14f\n", 6*iSin(.5)) fmt.Printf("%16.14f\n", 3*iCos(.5)) fmt.Printf("%16.14f\n", 4*iTan(1))} func thieleInterpolator(x, y []float64) func(float64) float64 { n := len(x) ρ := make([][]float64, n) for i := range ρ { ρ[i] = make([]float64, n-i) ρ[i][0] = y[i] } for i := 0; i < n-1; i++ { ρ[i][1] = (x[i] - x[i+1]) / (ρ[i][0] - ρ[i+1][0]) } for i := 2; i < n; i++ { for j := 0; j < n-i; j++ { ρ[j][i] = (x[j]-x[j+i])/(ρ[j][i-1]-ρ[j+1][i-1]) + ρ[j+1][i-2] } } // ρ0 used in closure. the rest of ρ becomes garbage. ρ0 := ρ[0] return func(xin float64) float64 { var a float64 for i := n - 1; i > 1; i-- { a = (xin - x[i-1]) / (ρ0[i] - ρ0[i-2] + a) } return y[0] + (xin-x[0])/(ρ0[1]+a) }} Output: 3.14159265358979 3.14159265358979 3.14159265358980  ## Haskell Caching of rho is automatic due to lazy lists. thiele :: [Double] -> [Double] -> Double -> Doublethiele xs ys = f rho1 (tail xs) where f _ [] _ = 1 f r@(r0:r1:r2:rs) (x:xs) v = r2 - r0 + (v - x) / f (tail r) xs v rho1 = ((!! 1) . (++ [0])) <$> rho    rho = [0,0 ..] : [0,0 ..] : ys : rnext (tail rho) xs (tail xs)      where        rnext _ _ [] = []        rnext r@(r0:r1:rs) x xn =          let z_ = zipWith          in z_ (+) (tail r0) (z_ (/) (z_ (-) x xn) (z_ (-) r1 (tail r1))) :             rnext (tail r) x (tail xn) -- Inverted interpolation function of finvInterp :: (Double -> Double) -> [Double] -> Double -> DoubleinvInterp f xs = thiele (map f xs) xs main :: IO ()main =  mapM_    print    [ 3.21 * inv_sin (sin (pi / 3.21))    , pi / 1.2345 * inv_cos (cos 1.2345)    , 7 * inv_tan (tan (pi / 7))    ]  where    [inv_sin, inv_cos, inv_tan] =      uncurry ((. div_pi) . invInterp) <$> [(sin, (2, 31)), (cos, (2, 100)), (tan, (4, 1000))] -- N points taken uniformly from 0 to Pi/d div_pi (d, n) = (* (pi / (d * n))) <$> [0 .. n]
Output:
3.141592653589795
3.141592653589791
3.141592653587783

## J

 span =: {. - {:  NB. head - tailspans =: span\   NB. apply span to successive infixes 
   span 12 888 6 4 8 3
9
4 spans 12 888 6 4 8 3
8 880 3

 NB. abscissae_of_knots coef ordinates_of_knotsNB. returns the interpolation coefficients for evalcoef =: 4 : 0 p =. _2 _{.,:y for_i. i. # x do.   p =. (p , ([: }. - }. p {~ _2:) + (x spans~ 2+]) % 2 spans - }. [: {: p"_) i end. x; , _ 1 {. p) NB. unknown_abscissae eval coefficientseval =: 4 : 0 'xx p' =. y a =. 0 i =. <: # xx while. 0 < i=.<:i do.   a =. (x-i{xx)%-/(p{~i+2),(i{p),a end. (p{~>:i)+(x-i{xx)%(p{~i+2)+a) 
   trig_table =: 1 2 3 o./ angles =: 5r100*i.32

0 _1 }. ": (;:'angle sin cos tan'),.<"1] 8j4": _ 5{.angles,trig_table
┌─────┬────────────────────────────────────────
│angle│  0.0000  0.0500  0.1000  0.1500  0.2000
├─────┼────────────────────────────────────────
│sin  │  0.0000  0.0500  0.0998  0.1494  0.1987
├─────┼────────────────────────────────────────
│cos  │  1.0000  0.9988  0.9950  0.9888  0.9801
├─────┼────────────────────────────────────────
│tan  │  0.0000  0.0500  0.1003  0.1511  0.2027
└─────┴────────────────────────────────────────

('Thiele pi';'error'),;/"1(,. 1p1&-)6 3 4 * 1r2 1r2 1 eval"0 1 trig_table coef"1 angles
┌─────────┬────────────┐
│Thiele pi│error       │
├─────────┼────────────┤
│3.14159  │_4.44089e_15│
├─────────┼────────────┤
│3.14159  │_4.44089e_16│
├─────────┼────────────┤
│3.14159  │_7.10543e_15│
└─────────┴────────────┘

 thiele =: 2 : 0 p =. _2 _{.,:n for_i. i.#m do.   p =. (p , ([: }. - }. p {~ _2:) + (m spans~ 2+]) % 2 spans - }. [: {: p"_) i end. p =. , _ 1 {. p a =. 0 i =. <:#m while. 0 < i=.<:i do.   a =. (y-i{m)%-/(p{~i+2),(i{p),a end. (p{~>:i)+(y-i{m)%a+p{~i+2) 
   's c t' =: trig_table
asin =: s thiele angles

6*asin 0.5
3.14159

1r5 * i.6
0 1r5 2r5 3r5 4r5 1
100*(_1&o. %~ _1&o. - asin) 1r5*i.6   NB. % error arcsin
0 1.4052 4.50319 9.32495 16.9438 39.321


## Java

Translation of: C
import static java.lang.Math.*; public class Test {    final static int N = 32;    final static int N2 = (N * (N - 1) / 2);    final static double STEP = 0.05;     static double[] xval = new double[N];    static double[] t_sin = new double[N];    static double[] t_cos = new double[N];    static double[] t_tan = new double[N];     static double[] r_sin = new double[N2];    static double[] r_cos = new double[N2];    static double[] r_tan = new double[N2];     static double rho(double[] x, double[] y, double[] r, int i, int n) {        if (n < 0)            return 0;         if (n == 0)            return y[i];         int idx = (N - 1 - n) * (N - n) / 2 + i;        if (r[idx] != r[idx])            r[idx] = (x[i] - x[i + n])                    / (rho(x, y, r, i, n - 1) - rho(x, y, r, i + 1, n - 1))                    + rho(x, y, r, i + 1, n - 2);         return r[idx];    }     static double thiele(double[] x, double[] y, double[] r, double xin, int n) {        if (n > N - 1)            return 1;        return rho(x, y, r, 0, n) - rho(x, y, r, 0, n - 2)                + (xin - x[n]) / thiele(x, y, r, xin, n + 1);    }     public static void main(String[] args) {        for (int i = 0; i < N; i++) {            xval[i] = i * STEP;            t_sin[i] = sin(xval[i]);            t_cos[i] = cos(xval[i]);            t_tan[i] = t_sin[i] / t_cos[i];        }         for (int i = 0; i < N2; i++)            r_sin[i] = r_cos[i] = r_tan[i] = Double.NaN;         System.out.printf("%16.14f%n", 6 * thiele(t_sin, xval, r_sin, 0.5, 0));        System.out.printf("%16.14f%n", 3 * thiele(t_cos, xval, r_cos, 0.5, 0));        System.out.printf("%16.14f%n", 4 * thiele(t_tan, xval, r_tan, 1.0, 0));    }}
3.14159265358979
3.14159265358979
3.14159265358980

## Julia

Accuracy improves with a larger table and smaller step size.

Translation of: C
const N = 256const N2 = N * div(N - 1, 2)const step = 0.01const xval_table = zeros(Float64, N)const tsin_table = zeros(Float64, N)const tcos_table = zeros(Float64, N)const ttan_table = zeros(Float64, N)const rsin_cache = Dict{Float64, Float64}()const rcos_cache = Dict{Float64, Float64}()const rtan_cache = Dict{Float64, Float64}() function rho(x, y, rhocache, i, n)    if n < 0         return 0.0    elseif n == 0         return y[i+1]    end    idx = (N - 1 - n) * div(N - n, 2) + i    if !haskey(rhocache, idx)        rhocache[idx] = (x[i+1] - x[i + n+1]) / (rho(x, y, rhocache, i, n - 1) -             rho(x, y, rhocache, i + 1, n - 1)) + rho(x, y, rhocache, i + 1, n - 2)    end    rhocache[idx] end function thiele(x, y, r, xin, n)    if n > N - 1        return 1.0    end    rho(x, y, r, 0, n) - rho(x, y, r, 0, n - 2) + (xin - x[n + 1]) / thiele(x, y, r, xin, n + 1)end function thiele_tables()    for i in 1:N        xval_table[i] = (i-1) * step        tsin_table[i] = sin(xval_table[i])        tcos_table[i] = cos(xval_table[i])        ttan_table[i] = tsin_table[i] / tcos_table[i]    end    println(6 * thiele(tsin_table, xval_table, rsin_cache, 0.5, 0))    println(3 * thiele(tcos_table, xval_table, rcos_cache, 0.5, 0))    println(4 * thiele(ttan_table, xval_table, rtan_cache, 1.0, 0))end thiele_tables() 
Output:

3.1415926535898335
3.141592653589818
3.141592653589824



## Kotlin

Translation of: C
// version 1.1.2 const val N = 32const val N2 = N * (N - 1) / 2const val STEP = 0.05 val xval = DoubleArray(N)val tsin = DoubleArray(N)val tcos = DoubleArray(N)val ttan = DoubleArray(N)val rsin = DoubleArray(N2) { Double.NaN }val rcos = DoubleArray(N2) { Double.NaN }val rtan = DoubleArray(N2) { Double.NaN } fun rho(x: DoubleArray, y: DoubleArray, r: DoubleArray, i: Int, n: Int): Double {    if (n < 0) return 0.0    if (n == 0) return y[i]    val idx = (N - 1 - n) * (N - n) / 2 + i    if (r[idx].isNaN()) {        r[idx] = (x[i] - x[i + n]) /                  (rho(x, y, r, i, n - 1) - rho(x, y, r, i + 1, n - 1)) +                  rho(x, y, r, i + 1, n - 2)    }    return r[idx] } fun thiele(x: DoubleArray, y: DoubleArray, r: DoubleArray, xin: Double, n: Int): Double {     if (n > N - 1) return 1.0    return rho(x, y, r, 0, n) - rho(x, y, r, 0, n - 2) +            (xin - x[n]) / thiele(x, y, r, xin, n + 1)} fun main(args: Array<String>) {    for (i in 0 until N) {        xval[i] = i * STEP        tsin[i] = Math.sin(xval[i])        tcos[i] = Math.cos(xval[i])        ttan[i] = tsin[i] / tcos[i]    }    println("%16.14f".format(6 * thiele(tsin, xval, rsin, 0.5, 0)))    println("%16.14f".format(3 * thiele(tcos, xval, rcos, 0.5, 0)))    println("%16.14f".format(4 * thiele(ttan, xval, rtan, 1.0, 0)))}
Output:
3.14159265358979
3.14159265358979
3.14159265358980


## OCaml

This example shows how the accuracy changes with the degree of interpolation. The table 'columns' are only constructed implicitly during the recursive calculation of rdiff and thiele, but (as mentioned in the C code example) using memoization or explicit tabulation would speed up the calculation. The interpolation uses the nearest points around x for accuracy.

let xv, fv = fst, snd let rec rdiff a l r =   if l > r then 0.0 else   if l = r then fv a.(l) else   if l+1 = r then (xv a.(l) -. xv a.(r)) /. (fv a.(l) -. fv a.(r)) else   (xv a.(l) -. xv a.(r)) /. (rdiff a l (r-1) -. rdiff a (l+1) r) +. rdiff a (l+1) (r-1) let rec thiele x a a0 k n =   if k = n then 1.0 else   rdiff a a0 (a0+k) -. rdiff a a0 (a0+k-2) +. (x -. xv a.(a0+k)) /. thiele x a a0 (k+1) n let interpolate x a n =   let m = Array.length a in   let dist i = abs_float (x -. xv a.(i)) in   let nearer i j = if dist j < dist i then j else i in   let rec closest i j = if j = m then i else closest (nearer i j) (j+1) in   let c = closest 0 1 in   let c' = if c < n/2 then 0 else if c > m-n then m-n else c-(n/2) in   thiele x a c' 0 n let table a b n f =   let g i =      let x = a +. (b-.a)*.(float i)/.(float (n-1)) in      (f x, x) in   Array.init n g let [sin_tab; cos_tab; tan_tab] = List.map (table 0.0 1.55 32) [sin; cos; tan] let test n =   Printf.printf "\nDegree %d interpolation:\n" n;   Printf.printf "6*arcsin(0.5) = %.15f\n" (6.0*.(interpolate 0.5 sin_tab n));   Printf.printf "3*arccos(0.5) = %.15f\n" (3.0*.(interpolate 0.5 cos_tab n));   Printf.printf "4*arctan(1.0) = %.15f\n" (4.0*.(interpolate 1.0 tan_tab n));; List.iter test [8; 12; 16]

Output:

Degree 8 interpolation:
6*arcsin(0.5) = 3.141592654456238
3*arccos(0.5) = 3.141592653520809
4*arctan(1.0) = 3.141592653437432

Degree 12 interpolation:
6*arcsin(0.5) = 3.141592653587590
3*arccos(0.5) = 3.141592653562618
4*arctan(1.0) = 3.141592653589756

Degree 16 interpolation:
6*arcsin(0.5) = 3.141592653589793
3*arccos(0.5) = 3.141592653589793
4*arctan(1.0) = 3.141592653589793

## Perl 6

Works with: rakudo version 2018.09

Implemented to parallel the generalized formula, making for clearer, but slower, code. Offsetting that, the use of Promise allows concurrent calculations, so running all three types of interpolation should not take any longer than running just one (presuming available cores).

# reciprocal difference:multi sub ρ(&f, @x where * < 1) { 0 } # Identitymulti sub ρ(&f, @x where * == 1) { &f(@x[0]) }multi sub ρ(&f, @x where * > 1) {    ( @x[0] - @x[* - 1] )       # ( x - x[n] )    / (ρ(&f, @x[^(@x - 1)])     # / ( ρ[n-1](x[0], ..., x[n-1])    - ρ(&f, @x[1..^@x]) )       # - ρ[n-1](x[1], ..., x[n]) )    + ρ(&f, @x[1..^(@x - 1)]);  # + ρ[n-2](x[1], ..., x[n-1])} # Thiele:multi sub thiele($x, %f,$ord where { $ord == +%f }) { 1 } # Identitymulti sub thiele($x, %f, $ord) { my &f = {%f{$^a}};                # f(x) as a table lookup   # must sort hash keys to maintain order between invocations  my $a = ρ(&f, %f.keys.sort[^($ord +1)]);  my $b = ρ(&f, %f.keys.sort[^($ord -1)]);   my $num =$x - %f.keys.sort[$ord]; my$cont = thiele($x, %f,$ord +1);   # Thiele always takes this form:  return $a -$b + ( $num /$cont );} ## Demosub mk-inv(&fn, $d,$lim) {  my %h;  for 0..$lim { %h{ &fn($_ * $d) } =$_ * $d } return %h;} sub MAIN($tblsz = 12) {   my ($sin_pi,$cos_pi, $tan_pi); my$p1 = Promise.start( { my %invsin = mk-inv(&sin, 0.05, $tblsz);$sin_pi = 6 * thiele(0.5, %invsin, 0) } );  my $p2 = Promise.start( { my %invcos = mk-inv(&cos, 0.05,$tblsz); $cos_pi = 3 * thiele(0.5, %invcos, 0) } ); my$p3 = Promise.start( { my %invtan = mk-inv(&tan, 0.05, $tblsz);$tan_pi = 4 * thiele(1.0, %invtan, 0) } );  await $p1,$p2, $p3; say "pi = {pi}"; say "estimations using a table of$tblsz elements:";  say "sin interpolation: $sin_pi"; say "cos interpolation:$cos_pi";  say "tan interpolation: $tan_pi";} Output: pi = 3.14159265358979 estimations using a table of 12 elements: sin interpolation: 3.14159265358961 cos interpolation: 3.1387286696692 tan interpolation: 3.14159090545243 ## Phix Translation of: C To be honest I was slightly wary of this, what with tables being passed by reference and fairly heavy use of closures in other languages, but in the end all it took was a simple enum (R_SIN..R_TRIG). constant N = 32, N2 = (N * (N - 1) / 2), STEP = 0.05 constant inf = 1e300*1e300, nan = -(inf/inf) sequence {xval, t_sin, t_cos, t_tan} @= repeat(0,N) for i=1 to N do xval[i] = (i-1) * STEP t_sin[i] = sin(xval[i]) t_cos[i] = cos(xval[i]) t_tan[i] = t_sin[i] / t_cos[i]end for enum R_SIN, R_COS, R_TAN, R_TRIG=$ sequence rhot = repeat(repeat(nan,N2),R_TRIG) function rho(sequence x, y, integer rdx, int i, int n)    if n<0 then return 0 end if    if n=0 then return y[i+1] end if     integer idx = (N - 1 - n) * (N - n) / 2 + i + 1;    if rhot[rdx][idx]=nan then -- value not computed yet        rhot[rdx][idx] = (x[i+1] - x[i+1 + n])                        / (rho(x, y, rdx, i, n-1) - rho(x, y, rdx, i+1, n-1))                        + rho(x, y, rdx, i+1, n-2)    end if    return rhot[rdx][idx]end function function thiele(sequence x, y, integer rdx, atom xin, integer n)    if n>N-1 then return 1 end if    return rho(x, y, rdx, 0, n) - rho(x, y, rdx, 0, n-2)            + (xin-x[n+1]) / thiele(x, y, rdx, xin, n+1)end function constant fmt = iff(machine_bits()=32?"%32s : %.14f\n"                                    :"%32s : %.17f\n")printf(1,fmt,{"PI",PI})printf(1,fmt,{"6*arcsin(0.5)",6*arcsin(0.5)})printf(1,fmt,{"3*arccos(0.5)",3*arccos(0.5)})printf(1,fmt,{"4*arctan(1)",4*arctan(1)}) printf(1,fmt,{"6*thiele(t_sin,xval,R_SIN,0.5,0)",6*thiele(t_sin,xval,R_SIN,0.5,0)})printf(1,fmt,{"3*thiele(t_cos,xval,R_COS,0.5,0)",3*thiele(t_cos,xval,R_COS,0.5,0)})printf(1,fmt,{"4*thiele(t_tan,xval,R_TAN,1,0)",4*thiele(t_tan,xval,R_TAN,1,0)})
Output:

(64 bit, obviously 3 fewer digits on 32 bit)

                              PI : 3.14159265358979324
6*arcsin(0.5) : 3.14159265358979324
3*arccos(0.5) : 3.14159265358979324
4*arctan(1) : 3.14159265358979324
6*thiele(t_sin,xval,R_SIN,0.5,0) : 3.14159265358979324
3*thiele(t_cos,xval,R_COS,0.5,0) : 3.14159265358979324
4*thiele(t_tan,xval,R_TAN,1,0) : 3.14159265358979324


## PicoLisp

Translation of: C
(scl 17)(load "@lib/math.l") (setq   *X-Table (range 0.0 1.55 0.05)   *SinTable (mapcar sin *X-Table)   *CosTable (mapcar cos *X-Table)   *TanTable (mapcar tan *X-Table)   *TrigRows (length *X-Table) ) (let N2 (>> 1 (* *TrigRows (dec *TrigRows)))   (setq      *InvSinTable (need N2)      *InvCosTable (need N2)      *InvTanTable (need N2) ) ) (de rho (Tbl Inv I N)   (cond      ((lt0 N) 0)      ((=0 N) (get *X-Table I))      (T         (let Idx (+ I (>> 1 (* (- *TrigRows 1 N) (- *TrigRows N))))            (or               (get Inv Idx)               (set (nth Inv Idx)  # only happens if value not computed yet                  (+                     (rho Tbl Inv (inc I) (- N 2))                     (*/                        (- (get Tbl I) (get Tbl (+ I N)))                        1.0                        (-                           (rho Tbl Inv I (dec N))                           (rho Tbl Inv (inc I) (dec N)) ) ) ) ) ) ) ) ) ) (de thiele (Tbl Inv X N)   (if (> N *TrigRows)      1.0      (+         (-            (rho Tbl Inv 1 (dec N))            (rho Tbl Inv 1 (- N 3)) )         (*/            (- X (get Tbl N))            1.0            (thiele Tbl Inv X (inc N)) ) ) ) ) (de iSin (X)   (thiele *SinTable *InvSinTable X 1) ) (de iCos (X)   (thiele *CosTable *InvCosTable X 1) ) (de iTan (X)   (thiele *TanTable *InvTanTable 1.0 1) )

Test:

(prinl (round (* 6 (iSin 0.5)) 15))(prinl (round (* 3 (iCos 0.5)) 15))(prinl (round (* 4 (iTan 1.0)) 15))

Output:

3.141592653589793
3.141592653589793
3.141592653589793

## PowerShell

Function Reciprocal-Difference( [Double[][]] $function ){$rho[email protected]()    $rho+=0$funcl = $function.length if($funcl -gt 0 )    {        -2..($funcl-1) | ForEach-Object {$i=$_ #Write-Host "$($i+1) -$($rho[$i+1]) - $($rho[$i+1].GetType())"$rho[$i+2] =$( 0..($funcl-$i-1) | Where-Object {$_ -lt$funcl} | ForEach-Object {                $j=$_                switch ($i) { {$_ -lt 0 } { 0 }                    {$_ -eq 0 } {$function[$j][1] } {$_ -gt 0 } { ( $function[$j][0] - $function[$j+$i][0] ) / ($rho[$i+1][$j] - $rho[$i+1][$j+1] ) +$rho[$i][$j+1] }                }            if( $_ -lt$funcl )            {                $rho += 0 } }) } }$rho} Function Thiele-Interpolation ( [Double[][]] $function ){$funcl = $function.length$invoke = "{ntparam([Double] $x)n" if($funcl -gt 1)    {        $rho = Reciprocal-Difference$function        ($funcl-1)..0 | ForEach-Object {$invoke += "t"            $invoke += '$x{0} = {1} - {2}' -f $_, @($rho[$_+2])[0], @($rho[$_])[0] if($_ -lt ($funcl-1)) {$invoke += ' + ( $x - {0} ) /$x{1} ' -f $function[$_][0], ($_+1) }$invoke += "n"        }        $invoke+="t$x0n}"    } else {        $invoke += "t$xn}"    }    invoke-expression $invoke}$sint[email protected]{}; 0..31 | ForEach-Object { $_ * 0.05 } | ForEach-Object {$sint[$_] = [Math]::sin($_) }$cost[email protected]{}; 0..31 | ForEach-Object {$_ * 0.05 } | ForEach-Object { $cost[$_] = [Math]::cos($_) }$tant[email protected]{}; 0..31 | ForEach-Object { $_ * 0.05 } | ForEach-Object {$tant[$_] = [Math]::tan($_) }$asint=New-Object 'Double[][]' 32,2;$sint.GetEnumerator() | Sort-Object Value | ForEach-Object {$i=0}{$asint[$i][0] =$_.Value; $asint[$i][1] = $_.Name;$i++ }$acost=New-Object 'Double[][]' 32,2;$cost.GetEnumerator() | Sort-Object Value | ForEach-Object { $i=0 }{$acost[$i][0] =$_.Value; $acost[$i][1] = $_.Name;$i++ }$atant=New-Object 'Double[][]' 32,2;$tant.GetEnumerator() | Sort-Object Value | ForEach-Object {$i=0}{$atant[$i][0] =$_.Value; $atant[$i][1] = $_.Name;$i++ } $asin = (Thiele-Interpolation$asint)#uncomment to see the function#"{$asin}"6*$asin.InvokeReturnAsIs(.5)$acos = (Thiele-Interpolation$acost)#uncomment to see the function#"{$acos}"3*$acos.InvokeReturnAsIs(.5)$atan = (Thiele-Interpolation$atant)#uncomment to see the function#"{$atan}"4*$atan.InvokeReturnAsIs(1)

## Python

Translation of: Go
#!/usr/bin/env python3 import math def thieleInterpolator(x, y):    ρ = [[yi]*(len(y)-i) for i, yi in enumerate(y)]    for i in range(len(ρ)-1):        ρ[i][1] = (x[i] - x[i+1]) / (ρ[i][0] - ρ[i+1][0])    for i in range(2, len(ρ)):        for j in range(len(ρ)-i):            ρ[j][i] = (x[j]-x[j+i]) / (ρ[j][i-1]-ρ[j+1][i-1]) + ρ[j+1][i-2]    ρ0 = ρ[0]    def t(xin):        a = 0        for i in range(len(ρ0)-1, 1, -1):            a = (xin - x[i-1]) / (ρ0[i] - ρ0[i-2] + a)        return y[0] + (xin-x[0]) / (ρ0[1]+a)    return t # task 1: build 32 row trig tablexVal = [i*.05 for i in range(32)]tSin = [math.sin(x) for x in xVal]tCos = [math.cos(x) for x in xVal]tTan = [math.tan(x) for x in xVal]# task 2: define inversesiSin = thieleInterpolator(tSin, xVal)iCos = thieleInterpolator(tCos, xVal)iTan = thieleInterpolator(tTan, xVal)# task 3: demonstrate identitiesprint('{:16.14f}'.format(6*iSin(.5)))print('{:16.14f}'.format(3*iCos(.5)))print('{:16.14f}'.format(4*iTan(1)))
Output:
3.14159265358979
3.14159265358979
3.14159265358980


## Racket

 #lang racket(define xs (for/vector ([x (in-range 0.0 1.6 0.05)]) x))(define (x i) (vector-ref xs i)) (define-syntax define-table   (syntax-rules ()    [(_ f tf rf if)      (begin (define tab (for/vector ([x xs]) (f x)))            (define (tf n) (vector-ref tab n))            (define cache (make-vector (/ (* 32 31) 2) #f))            (define (rf n thunk)              (or (vector-ref cache n)                  (let ([v (thunk)])                    (vector-set! cache n v)                    v)))            (define (if t) (thiele tf x rf t 0)))])) (define-table sin tsin rsin isin)(define-table cos tcos rcos icos)(define-table tan ttan rtan itan) (define (rho x y r i n)  (cond    [(< n 0) 0]    [(= n 0) (y i)]    [else (r (+ (/ (* (- 32 1 n) (- 32 n)) 2) i)             (λ() (+ (/ (- (x i) (x (+ i n)))                        (- (rho x y r i (- n 1)) (rho x y r (+ i 1) (- n 1))))                     (rho x y r (+ i 1) (- n 2)))))])) (define (thiele x y r xin n)  (cond    [(> n 31) 1]    [(+ (rho x y r 0 n) (- (rho x y r 0 (- n 2)))         (/ (- xin (x n)) (thiele x y r xin (+ n 1))))])) (* 6 (isin 0.5)) (* 3 (icos 0.5)) (* 4 (itan 1.))   

Output:

 3.1415926535897933.14159265358979363.1415926535897953 

## Sidef

Translation of: Python
func thiele(x, y) {    var ρ = {|i| [y[i]]*(y.len-i) }.map(^y)     for i in ^(ρ.end) {        ρ[i][1] = ((x[i] - x[i+1]) / (ρ[i][0] - ρ[i+1][0]))    }    for i (2 .. ρ.end) {        for j (0 .. ρ.end-i) {            ρ[j][i] = (((x[j]-x[j+i]) / (ρ[j][i-1]-ρ[j+1][i-1])) + ρ[j+1][i-2])        }    }     var ρ0 = ρ[0]     func t(xin) {        var a = 0        for i (ρ0.len ^.. 2) {            a = ((xin - x[i-1]) / (ρ0[i] - ρ0[i-2] + a))        }        y[0] + ((xin-x[0]) / (ρ0[1]+a))    }    return t} # task 1: build 32 row trig tablevar xVal = {|k| k * 0.05 }.map(^32)var tSin = xVal.map { .sin }var tCos = xVal.map { .cos }var tTan = xVal.map { .tan } # task 2: define inversesvar iSin = thiele(tSin, xVal)var iCos = thiele(tCos, xVal)var iTan = thiele(tTan, xVal) # task 3: demonstrate identitiessay 6*iSin(0.5)say 3*iCos(0.5)say 4*iTan(1)
Output:
3.14159265358979323846438729976818601771260734312
3.14159265358979323846157620314930763214337987744
3.14159265358979323846264318595256260456200366896


## Tcl

Works with: Tcl version 8.5
Translation of: D
#### Create a thiele-interpretation function with the given name that interpolates### off the given table.#proc thiele {name : X -> F} {    # Sanity check    if {[llength $X] != [llength$F]} {	error "unequal length lists supplied: [llength $X] != [llength$F]"    }     #    ### Compute the table of reciprocal differences    #    set p [lrepeat [llength $X] [lrepeat [llength$X] 0.0]]    set i 0    foreach x0 [lrange $X 0 end-1] x1 [lrange$X 1 end] \	    f0 [lrange $F 0 end-1] f1 [lrange$F 1 end] {	lset p $i 0$f0	lset p $i 1 [expr {($x0 - $x1) / ($f0 - $f1)}] lset p [incr i] 0$f1    }    for {set j 2} {$j<[llength$X]-1} {incr j} {	for {set i 0} {$i<[llength$X]-$j} {incr i} { lset p$i $j [expr { [lindex$p $i+1$j-2] +		([lindex $X$i] - [lindex $X$i+$j]) / ([lindex$p $i$j-1] - [lindex $p$i+1 $j-1]) }] } } # ### Make pseudo-curried function that actually evaluates Thiele's formula # interp alias {}$name {} apply {{X rho f1 x} {	set a 0.0	foreach Xi  [lreverse [lrange $X 2 end]] \ Ri [lreverse [lrange$rho 2 end]] \		Ri2 [lreverse [lrange $rho 0 end-2]] { set a [expr {($x - $Xi) / ($Ri - $Ri2 +$a)}]	}	expr {$f1 + ($x - [lindex $X 1]) / ([lindex$rho 1] + $a)} }}$X [lindex $p 1] [lindex$F 1]}

Demonstration code:

proc initThieleTest {} {    for {set i 0} {$i < 32} {incr i} { lappend trigTable(x) [set x [expr {0.05 *$i}]]	lappend trigTable(sin) [expr {sin($x)}] lappend trigTable(cos) [expr {cos($x)}]	lappend trigTable(tan) [expr {tan($x)}] } thiele invSin :$trigTable(sin) -> $trigTable(x) thiele invCos :$trigTable(cos) -> $trigTable(x) thiele invTan :$trigTable(tan) -> \$trigTable(x)}initThieleTestputs "pi estimate using sin interpolation: [expr {6 * [invSin 0.5]}]"puts "pi estimate using cos interpolation: [expr {3 * [invCos 0.5]}]"puts "pi estimate using tan interpolation: [expr {4 * [invTan 1.0]}]"

Output:

pi estimate using sin interpolation: 3.1415926535897936
pi estimate using cos interpolation: 3.141592653589793
pi estimate using tan interpolation: 3.141592653589794


## zkl

Translation of: C

Please see the C example for the comments I've removed (this is an as pure-as-I-make-it translation).

const N=32, N2=(N * (N - 1) / 2), STEP=0.05; fcn rho(xs,ys,rs, i,n){   if (n < 0) return(0.0);   if (not n) return(ys[i]);    idx := (N - 1 - n) * (N - n) / 2 + i;   if (Void==rs[idx])      rs[idx] = (xs[i] - xs[i + n])		/ (rho(xs, ys, rs, i, n - 1) - rho(xs, ys, rs, i + 1, n - 1))		+ rho(xs, ys, rs, i + 1, n - 2);   return(rs[idx]);} fcn thiele(xs,ys,rs, xin, n){   if (n > N - 1) return(1.0);   rho(xs, ys, rs, 0, n) - rho(xs, ys, rs, 0, n - 2)      + (xin - xs[n]) / thiele(xs, ys, rs, xin, n + 1);} /////////// reg t_sin=L(), t_cos=L(), t_tan=L(),    r_sin=L(), r_cos=L(), r_tan=L(),  xval=L(); i_sin := thiele.fpM("11101",t_sin, xval, r_sin, 0);i_cos := thiele.fpM("11101",t_cos, xval, r_cos, 0);i_tan := thiele.fpM("11101",t_tan, xval, r_tan, 0); foreach i in (N){   xval.append(x:=STEP*i);   t_sin.append(x.sin());   t_cos.append(x.cos());   t_tan.append(t_sin[i] / t_cos[i]);}foreach i in (N2){ r_sin+Void; r_cos+Void; r_tan+Void; } print("%16.14f\n".fmt( 6.0 * i_sin(0.5)));print("%16.14f\n".fmt( 3.0 * i_cos(0.5)));print("%16.14f\n".fmt( 4.0 * i_tan(1.0)));
Output:
3.14159265358979
3.14159265358979
3.14159265358979
`