Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.5% → 98.8%

Time: 56.7s

Alternatives: 11

Speedup: 1.4×

• could not determine a ground truth

  1. z = -1.567283312585191e+289

Specification

\[z \leq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := t\_0 + 7\\ t_2 := t\_1 + 0.5\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {t\_2}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{-t\_2}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{t\_0 + 1}\right) + \frac{-1259.1392167224028}{t\_0 + 2}\right) + \frac{771.3234287776531}{t\_0 + 3}\right) + \frac{-176.6150291621406}{t\_0 + 4}\right) + \frac{12.507343278686905}{t\_0 + 5}\right) + \frac{-0.13857109526572012}{t\_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_1}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)) (t_1 (+ t_0 7.0)) (t_2 (+ t_1 0.5)))
   (*
    (/ PI (sin (* PI z)))
    (*
     (* (* (sqrt (* PI 2.0)) (pow t_2 (+ t_0 0.5))) (exp (- t_2)))
     (+
      (+
       (+
        (+
         (+
          (+
           (+
            (+ 0.9999999999998099 (/ 676.5203681218851 (+ t_0 1.0)))
            (/ -1259.1392167224028 (+ t_0 2.0)))
           (/ 771.3234287776531 (+ t_0 3.0)))
          (/ -176.6150291621406 (+ t_0 4.0)))
         (/ 12.507343278686905 (+ t_0 5.0)))
        (/ -0.13857109526572012 (+ t_0 6.0)))
       (/ 9.984369578019572e-6 t_1))
      (/ 1.5056327351493116e-7 (+ t_0 8.0)))))))

C

double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = t_0 + 7.0;
	double t_2 = t_1 + 0.5;
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt((((double) M_PI) * 2.0)) * pow(t_2, (t_0 + 0.5))) * exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = t_0 + 7.0;
	double t_2 = t_1 + 0.5;
	return (Math.PI / Math.sin((Math.PI * z))) * (((Math.sqrt((Math.PI * 2.0)) * Math.pow(t_2, (t_0 + 0.5))) * Math.exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
}

Python

def code(z):
	t_0 = (1.0 - z) - 1.0
	t_1 = t_0 + 7.0
	t_2 = t_1 + 0.5
	return (math.pi / math.sin((math.pi * z))) * (((math.sqrt((math.pi * 2.0)) * math.pow(t_2, (t_0 + 0.5))) * math.exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - 1.0)
	t_1 = Float64(t_0 + 7.0)
	t_2 = Float64(t_1 + 0.5)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (t_2 ^ Float64(t_0 + 0.5))) * exp(Float64(-t_2))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(t_0 + 1.0))) + Float64(-1259.1392167224028 / Float64(t_0 + 2.0))) + Float64(771.3234287776531 / Float64(t_0 + 3.0))) + Float64(-176.6150291621406 / Float64(t_0 + 4.0))) + Float64(12.507343278686905 / Float64(t_0 + 5.0))) + Float64(-0.13857109526572012 / Float64(t_0 + 6.0))) + Float64(9.984369578019572e-6 / t_1)) + Float64(1.5056327351493116e-7 / Float64(t_0 + 8.0)))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - 1.0;
	t_1 = t_0 + 7.0;
	t_2 = t_1 + 0.5;
	tmp = (pi / sin((pi * z))) * (((sqrt((pi * 2.0)) * (t_2 ^ (t_0 + 0.5))) * exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 7.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + 0.5), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$2, N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$2)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(t$95$0 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(t$95$0 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(t$95$0 + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(t$95$0 + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(t$95$0 + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Initial Program: 96.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := t\_0 + 7\\ t_2 := t\_1 + 0.5\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {t\_2}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{-t\_2}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{t\_0 + 1}\right) + \frac{-1259.1392167224028}{t\_0 + 2}\right) + \frac{771.3234287776531}{t\_0 + 3}\right) + \frac{-176.6150291621406}{t\_0 + 4}\right) + \frac{12.507343278686905}{t\_0 + 5}\right) + \frac{-0.13857109526572012}{t\_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_1}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)) (t_1 (+ t_0 7.0)) (t_2 (+ t_1 0.5)))
   (*
    (/ PI (sin (* PI z)))
    (*
     (* (* (sqrt (* PI 2.0)) (pow t_2 (+ t_0 0.5))) (exp (- t_2)))
     (+
      (+
       (+
        (+
         (+
          (+
           (+
            (+ 0.9999999999998099 (/ 676.5203681218851 (+ t_0 1.0)))
            (/ -1259.1392167224028 (+ t_0 2.0)))
           (/ 771.3234287776531 (+ t_0 3.0)))
          (/ -176.6150291621406 (+ t_0 4.0)))
         (/ 12.507343278686905 (+ t_0 5.0)))
        (/ -0.13857109526572012 (+ t_0 6.0)))
       (/ 9.984369578019572e-6 t_1))
      (/ 1.5056327351493116e-7 (+ t_0 8.0)))))))

C

double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = t_0 + 7.0;
	double t_2 = t_1 + 0.5;
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt((((double) M_PI) * 2.0)) * pow(t_2, (t_0 + 0.5))) * exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = t_0 + 7.0;
	double t_2 = t_1 + 0.5;
	return (Math.PI / Math.sin((Math.PI * z))) * (((Math.sqrt((Math.PI * 2.0)) * Math.pow(t_2, (t_0 + 0.5))) * Math.exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
}

Python

def code(z):
	t_0 = (1.0 - z) - 1.0
	t_1 = t_0 + 7.0
	t_2 = t_1 + 0.5
	return (math.pi / math.sin((math.pi * z))) * (((math.sqrt((math.pi * 2.0)) * math.pow(t_2, (t_0 + 0.5))) * math.exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - 1.0)
	t_1 = Float64(t_0 + 7.0)
	t_2 = Float64(t_1 + 0.5)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (t_2 ^ Float64(t_0 + 0.5))) * exp(Float64(-t_2))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(t_0 + 1.0))) + Float64(-1259.1392167224028 / Float64(t_0 + 2.0))) + Float64(771.3234287776531 / Float64(t_0 + 3.0))) + Float64(-176.6150291621406 / Float64(t_0 + 4.0))) + Float64(12.507343278686905 / Float64(t_0 + 5.0))) + Float64(-0.13857109526572012 / Float64(t_0 + 6.0))) + Float64(9.984369578019572e-6 / t_1)) + Float64(1.5056327351493116e-7 / Float64(t_0 + 8.0)))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - 1.0;
	t_1 = t_0 + 7.0;
	t_2 = t_1 + 0.5;
	tmp = (pi / sin((pi * z))) * (((sqrt((pi * 2.0)) * (t_2 ^ (t_0 + 0.5))) * exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 7.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + 0.5), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$2, N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$2)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(t$95$0 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(t$95$0 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(t$95$0 + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(t$95$0 + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(t$95$0 + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Alternative 1: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) + -1\\ t_1 := \sin \left(\pi \cdot z\right)\\ t_2 := \frac{\pi}{t\_1}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \left(z + -1\right) - -1\\ t_5 := t\_4 - 7\\ \mathbf{if}\;t\_2 \cdot \left(\left(\left(t\_3 \cdot {\left(\left(t\_0 + 7\right) + 0.5\right)}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{t\_5 - 0.5}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8} - \left(\frac{9.984369578019572 \cdot 10^{-6}}{t\_5} - \left(\left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{-1 - t\_0}\right) - \frac{-1259.1392167224028}{t\_4 - 2}\right) + \frac{771.3234287776531}{t\_0 + 3}\right) + \frac{-176.6150291621406}{t\_0 + 4}\right) + \frac{12.507343278686905}{t\_0 + 5}\right) + \frac{-0.13857109526572012}{t\_0 + 6}\right)\right)\right)\right) \leq 4 \cdot 10^{+283}:\\ \;\;\;\;\pi \cdot \frac{\left(t\_3 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} - \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - 7} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - 8}\right)\right)\right) - \left(\left(\left(\frac{-1259.1392167224028}{z - 2} + \frac{771.3234287776531}{z - 3}\right) - 0.9999999999998099\right) + \frac{-176.6150291621406}{z - 4}\right)\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_3 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \left(t\_2 \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ (- 1.0 z) -1.0))
        (t_1 (sin (* PI z)))
        (t_2 (/ PI t_1))
        (t_3 (sqrt (* PI 2.0)))
        (t_4 (- (+ z -1.0) -1.0))
        (t_5 (- t_4 7.0)))
   (if (<=
        (*
         t_2
         (*
          (* (* t_3 (pow (+ (+ t_0 7.0) 0.5) (+ t_0 0.5))) (exp (- t_5 0.5)))
          (-
           (/ 1.5056327351493116e-7 (+ t_0 8.0))
           (-
            (/ 9.984369578019572e-6 t_5)
            (+
             (+
              (+
               (+
                (-
                 (- 0.9999999999998099 (/ 676.5203681218851 (- -1.0 t_0)))
                 (/ -1259.1392167224028 (- t_4 2.0)))
                (/ 771.3234287776531 (+ t_0 3.0)))
               (/ -176.6150291621406 (+ t_0 4.0)))
              (/ 12.507343278686905 (+ t_0 5.0)))
             (/ -0.13857109526572012 (+ t_0 6.0)))))))
        4e+283)
     (*
      PI
      (/
       (*
        (* t_3 (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5))))
        (+
         (/ 676.5203681218851 (- 1.0 z))
         (-
          (+
           (/ -0.13857109526572012 (- 6.0 z))
           (-
            (/ 12.507343278686905 (- 5.0 z))
            (+
             (/ 9.984369578019572e-6 (- z 7.0))
             (/ 1.5056327351493116e-7 (- z 8.0)))))
          (+
           (-
            (+
             (/ -1259.1392167224028 (- z 2.0))
             (/ 771.3234287776531 (- z 3.0)))
            0.9999999999998099)
           (/ -176.6150291621406 (- z 4.0))))))
       t_1))
     (*
      (* t_3 (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp -7.5)))
      (*
       t_2
       (+
        263.3831869810514
        (* z (+ 436.8961725563396 (* z 545.0353078428827)))))))))

C

double code(double z) {
	double t_0 = (1.0 - z) + -1.0;
	double t_1 = sin((((double) M_PI) * z));
	double t_2 = ((double) M_PI) / t_1;
	double t_3 = sqrt((((double) M_PI) * 2.0));
	double t_4 = (z + -1.0) - -1.0;
	double t_5 = t_4 - 7.0;
	double tmp;
	if ((t_2 * (((t_3 * pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5))) * exp((t_5 - 0.5))) * ((1.5056327351493116e-7 / (t_0 + 8.0)) - ((9.984369578019572e-6 / t_5) - ((((((0.9999999999998099 - (676.5203681218851 / (-1.0 - t_0))) - (-1259.1392167224028 / (t_4 - 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))))))) <= 4e+283) {
		tmp = ((double) M_PI) * (((t_3 * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5)))) * ((676.5203681218851 / (1.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) - ((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0))))) - ((((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0))) - 0.9999999999998099) + (-176.6150291621406 / (z - 4.0)))))) / t_1);
	} else {
		tmp = (t_3 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(-7.5))) * (t_2 * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) + -1.0;
	double t_1 = Math.sin((Math.PI * z));
	double t_2 = Math.PI / t_1;
	double t_3 = Math.sqrt((Math.PI * 2.0));
	double t_4 = (z + -1.0) - -1.0;
	double t_5 = t_4 - 7.0;
	double tmp;
	if ((t_2 * (((t_3 * Math.pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5))) * Math.exp((t_5 - 0.5))) * ((1.5056327351493116e-7 / (t_0 + 8.0)) - ((9.984369578019572e-6 / t_5) - ((((((0.9999999999998099 - (676.5203681218851 / (-1.0 - t_0))) - (-1259.1392167224028 / (t_4 - 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))))))) <= 4e+283) {
		tmp = Math.PI * (((t_3 * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5)))) * ((676.5203681218851 / (1.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) - ((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0))))) - ((((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0))) - 0.9999999999998099) + (-176.6150291621406 / (z - 4.0)))))) / t_1);
	} else {
		tmp = (t_3 * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(-7.5))) * (t_2 * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = (1.0 - z) + -1.0
	t_1 = math.sin((math.pi * z))
	t_2 = math.pi / t_1
	t_3 = math.sqrt((math.pi * 2.0))
	t_4 = (z + -1.0) - -1.0
	t_5 = t_4 - 7.0
	tmp = 0
	if (t_2 * (((t_3 * math.pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5))) * math.exp((t_5 - 0.5))) * ((1.5056327351493116e-7 / (t_0 + 8.0)) - ((9.984369578019572e-6 / t_5) - ((((((0.9999999999998099 - (676.5203681218851 / (-1.0 - t_0))) - (-1259.1392167224028 / (t_4 - 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))))))) <= 4e+283:
		tmp = math.pi * (((t_3 * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5)))) * ((676.5203681218851 / (1.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) - ((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0))))) - ((((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0))) - 0.9999999999998099) + (-176.6150291621406 / (z - 4.0)))))) / t_1)
	else:
		tmp = (t_3 * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(-7.5))) * (t_2 * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))))
	return tmp

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) + -1.0)
	t_1 = sin(Float64(pi * z))
	t_2 = Float64(pi / t_1)
	t_3 = sqrt(Float64(pi * 2.0))
	t_4 = Float64(Float64(z + -1.0) - -1.0)
	t_5 = Float64(t_4 - 7.0)
	tmp = 0.0
	if (Float64(t_2 * Float64(Float64(Float64(t_3 * (Float64(Float64(t_0 + 7.0) + 0.5) ^ Float64(t_0 + 0.5))) * exp(Float64(t_5 - 0.5))) * Float64(Float64(1.5056327351493116e-7 / Float64(t_0 + 8.0)) - Float64(Float64(9.984369578019572e-6 / t_5) - Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 - Float64(676.5203681218851 / Float64(-1.0 - t_0))) - Float64(-1259.1392167224028 / Float64(t_4 - 2.0))) + Float64(771.3234287776531 / Float64(t_0 + 3.0))) + Float64(-176.6150291621406 / Float64(t_0 + 4.0))) + Float64(12.507343278686905 / Float64(t_0 + 5.0))) + Float64(-0.13857109526572012 / Float64(t_0 + 6.0))))))) <= 4e+283)
		tmp = Float64(pi * Float64(Float64(Float64(t_3 * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5)))) * Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(Float64(12.507343278686905 / Float64(5.0 - z)) - Float64(Float64(9.984369578019572e-6 / Float64(z - 7.0)) + Float64(1.5056327351493116e-7 / Float64(z - 8.0))))) - Float64(Float64(Float64(Float64(-1259.1392167224028 / Float64(z - 2.0)) + Float64(771.3234287776531 / Float64(z - 3.0))) - 0.9999999999998099) + Float64(-176.6150291621406 / Float64(z - 4.0)))))) / t_1));
	else
		tmp = Float64(Float64(t_3 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(-7.5))) * Float64(t_2 * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * 545.0353078428827))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = (1.0 - z) + -1.0;
	t_1 = sin((pi * z));
	t_2 = pi / t_1;
	t_3 = sqrt((pi * 2.0));
	t_4 = (z + -1.0) - -1.0;
	t_5 = t_4 - 7.0;
	tmp = 0.0;
	if ((t_2 * (((t_3 * (((t_0 + 7.0) + 0.5) ^ (t_0 + 0.5))) * exp((t_5 - 0.5))) * ((1.5056327351493116e-7 / (t_0 + 8.0)) - ((9.984369578019572e-6 / t_5) - ((((((0.9999999999998099 - (676.5203681218851 / (-1.0 - t_0))) - (-1259.1392167224028 / (t_4 - 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))))))) <= 4e+283)
		tmp = pi * (((t_3 * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5)))) * ((676.5203681218851 / (1.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) - ((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0))))) - ((((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0))) - 0.9999999999998099) + (-176.6150291621406 / (z - 4.0)))))) / t_1);
	else
		tmp = (t_3 * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(-7.5))) * (t_2 * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(Pi / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(z + -1.0), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 - 7.0), $MachinePrecision]}, If[LessEqual[N[(t$95$2 * N[(N[(N[(t$95$3 * N[Power[N[(N[(t$95$0 + 7.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(t$95$5 - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(1.5056327351493116e-7 / N[(t$95$0 + 8.0), $MachinePrecision]), $MachinePrecision] - N[(N[(9.984369578019572e-6 / t$95$5), $MachinePrecision] - N[(N[(N[(N[(N[(N[(0.9999999999998099 - N[(676.5203681218851 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1259.1392167224028 / N[(t$95$4 - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(t$95$0 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(t$95$0 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(t$95$0 + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(t$95$0 + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+283], N[(Pi * N[(N[(N[(t$95$3 * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] - N[(N[(9.984369578019572e-6 / N[(z - 7.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(z - 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(-1259.1392167224028 / N[(z - 2.0), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.9999999999998099), $MachinePrecision] + N[(-176.6150291621406 / N[(z - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$3 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * 545.0353078428827), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (PI.f64) (sin.f64 (*.f64 (PI.f64) z))) (*.f64 (*.f64 (*.f64 (sqrt.f64 (*.f64 (PI.f64) #s(literal 2 binary64))) (pow.f64 (+.f64 (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)) #s(literal 1/2 binary64)) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 1/2 binary64)))) (exp.f64 (neg.f64 (+.f64 (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)) #s(literal 1/2 binary64))))) (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 #s(literal 9999999999998099/10000000000000000 binary64) (/.f64 #s(literal 6765203681218851/10000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 1 binary64)))) (/.f64 #s(literal -3147848041806007/2500000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 2 binary64)))) (/.f64 #s(literal 7713234287776531/10000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 3 binary64)))) (/.f64 #s(literal -883075145810703/5000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 4 binary64)))) (/.f64 #s(literal 2501468655737381/200000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 5 binary64)))) (/.f64 #s(literal -3464277381643003/25000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 6 binary64)))) (/.f64 #s(literal 2496092394504893/250000000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)))) (/.f64 #s(literal 3764081837873279/25000000000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 8 binary64)))))) < 3.99999999999999982e283

   1. Initial program 97.3%

      \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. Simplified 96.2%

      \[\leadsto \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\frac{\pi \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{\left(-\left(1 - z\right)\right) + -6.5} \cdot \left(\left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{7 + \left(1 - z\right)}\right)\right) + \left(\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) + \left(\frac{-1259.1392167224028}{1 + \left(1 - z\right)} + \left(\frac{771.3234287776531}{2 + \left(1 - z\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right)\right)\right)\right)}{\sin \left(\pi \cdot z\right)}} \]

   8. Simplified 99.3%

      \[\leadsto \color{blue}{\pi \cdot \frac{\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)}{\sin \left(z \cdot \pi\right)}} \]

   if 3.99999999999999982e283 < (*.f64 (/.f64 (PI.f64) (sin.f64 (*.f64 (PI.f64) z))) (*.f64 (*.f64 (*.f64 (sqrt.f64 (*.f64 (PI.f64) #s(literal 2 binary64))) (pow.f64 (+.f64 (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)) #s(literal 1/2 binary64)) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 1/2 binary64)))) (exp.f64 (neg.f64 (+.f64 (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)) #s(literal 1/2 binary64))))) (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 #s(literal 9999999999998099/10000000000000000 binary64) (/.f64 #s(literal 6765203681218851/10000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 1 binary64)))) (/.f64 #s(literal -3147848041806007/2500000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 2 binary64)))) (/.f64 #s(literal 7713234287776531/10000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 3 binary64)))) (/.f64 #s(literal -883075145810703/5000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 4 binary64)))) (/.f64 #s(literal 2501468655737381/200000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 5 binary64)))) (/.f64 #s(literal -3464277381643003/25000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 6 binary64)))) (/.f64 #s(literal 2496092394504893/250000000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)))) (/.f64 #s(literal 3764081837873279/25000000000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 8 binary64))))))

   1. Initial program 64.9%

      \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. Simplified 65.1%

      \[\leadsto \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 66.1%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative 66.1%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot \left(436.8961725563396 + \color{blue}{z \cdot 545.0353078428827}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   7. Simplified 66.1%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. Taylor expanded in z around 0 99.5%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{e^{-7.5}}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) + -1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) + -1\right) + 0.5\right)}\right) \cdot e^{\left(\left(\left(z + -1\right) - -1\right) - 7\right) - 0.5}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) + -1\right) + 8} - \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(z + -1\right) - -1\right) - 7} - \left(\left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{-1 - \left(\left(1 - z\right) + -1\right)}\right) - \frac{-1259.1392167224028}{\left(\left(z + -1\right) - -1\right) - 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) + -1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) + -1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) + -1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) + -1\right) + 6}\right)\right)\right)\right) \leq 4 \cdot 10^{+283}:\\ \;\;\;\;\pi \cdot \frac{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} - \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - 7} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - 8}\right)\right)\right) - \left(\left(\left(\frac{-1259.1392167224028}{z - 2} + \frac{771.3234287776531}{z - 3}\right) - 0.9999999999998099\right) + \frac{-176.6150291621406}{z - 4}\right)\right)\right)}{\sin \left(\pi \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ t_1 := \sin \left(\pi \cdot z\right)\\ \mathbf{if}\;z \leq -1000:\\ \;\;\;\;\left(t\_0 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \left(\frac{\pi}{t\_1} \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\left(t\_0 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} - \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - 7} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - 8}\right)\right)\right) + \left(z \cdot \left(z \cdot \left(z \cdot -69.86359203642401 - 131.58447752178645\right) - 240.12064030571747\right) - 415.6155560591857\right)\right)\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 2.0))) (t_1 (sin (* PI z))))
   (if (<= z -1000.0)
     (*
      (* t_0 (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp -7.5)))
      (*
       (/ PI t_1)
       (+
        263.3831869810514
        (* z (+ 436.8961725563396 (* z 545.0353078428827))))))
     (*
      PI
      (/
       (*
        (* t_0 (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5))))
        (+
         (/ 676.5203681218851 (- 1.0 z))
         (+
          (+
           (/ -0.13857109526572012 (- 6.0 z))
           (-
            (/ 12.507343278686905 (- 5.0 z))
            (+
             (/ 9.984369578019572e-6 (- z 7.0))
             (/ 1.5056327351493116e-7 (- z 8.0)))))
          (-
           (*
            z
            (-
             (* z (- (* z -69.86359203642401) 131.58447752178645))
             240.12064030571747))
           415.6155560591857))))
       t_1)))))

C

double code(double z) {
	double t_0 = sqrt((((double) M_PI) * 2.0));
	double t_1 = sin((((double) M_PI) * z));
	double tmp;
	if (z <= -1000.0) {
		tmp = (t_0 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(-7.5))) * ((((double) M_PI) / t_1) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))));
	} else {
		tmp = ((double) M_PI) * (((t_0 * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5)))) * ((676.5203681218851 / (1.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) - ((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0))))) + ((z * ((z * ((z * -69.86359203642401) - 131.58447752178645)) - 240.12064030571747)) - 415.6155560591857)))) / t_1);
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.sqrt((Math.PI * 2.0));
	double t_1 = Math.sin((Math.PI * z));
	double tmp;
	if (z <= -1000.0) {
		tmp = (t_0 * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(-7.5))) * ((Math.PI / t_1) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))));
	} else {
		tmp = Math.PI * (((t_0 * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5)))) * ((676.5203681218851 / (1.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) - ((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0))))) + ((z * ((z * ((z * -69.86359203642401) - 131.58447752178645)) - 240.12064030571747)) - 415.6155560591857)))) / t_1);
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.sqrt((math.pi * 2.0))
	t_1 = math.sin((math.pi * z))
	tmp = 0
	if z <= -1000.0:
		tmp = (t_0 * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(-7.5))) * ((math.pi / t_1) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))))
	else:
		tmp = math.pi * (((t_0 * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5)))) * ((676.5203681218851 / (1.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) - ((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0))))) + ((z * ((z * ((z * -69.86359203642401) - 131.58447752178645)) - 240.12064030571747)) - 415.6155560591857)))) / t_1)
	return tmp

Julia

function code(z)
	t_0 = sqrt(Float64(pi * 2.0))
	t_1 = sin(Float64(pi * z))
	tmp = 0.0
	if (z <= -1000.0)
		tmp = Float64(Float64(t_0 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(-7.5))) * Float64(Float64(pi / t_1) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * 545.0353078428827))))));
	else
		tmp = Float64(pi * Float64(Float64(Float64(t_0 * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5)))) * Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(Float64(12.507343278686905 / Float64(5.0 - z)) - Float64(Float64(9.984369578019572e-6 / Float64(z - 7.0)) + Float64(1.5056327351493116e-7 / Float64(z - 8.0))))) + Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * -69.86359203642401) - 131.58447752178645)) - 240.12064030571747)) - 415.6155560591857)))) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = sqrt((pi * 2.0));
	t_1 = sin((pi * z));
	tmp = 0.0;
	if (z <= -1000.0)
		tmp = (t_0 * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(-7.5))) * ((pi / t_1) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))));
	else
		tmp = pi * (((t_0 * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5)))) * ((676.5203681218851 / (1.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) - ((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0))))) + ((z * ((z * ((z * -69.86359203642401) - 131.58447752178645)) - 240.12064030571747)) - 415.6155560591857)))) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1000.0], N[(N[(t$95$0 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / t$95$1), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * 545.0353078428827), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(N[(N[(t$95$0 * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] - N[(N[(9.984369578019572e-6 / N[(z - 7.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(z - 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z * N[(N[(z * N[(N[(z * -69.86359203642401), $MachinePrecision] - 131.58447752178645), $MachinePrecision]), $MachinePrecision] - 240.12064030571747), $MachinePrecision]), $MachinePrecision] - 415.6155560591857), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < -1e3

   1. Initial program 0.0%

      \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot \left(436.8961725563396 + \color{blue}{z \cdot 545.0353078428827}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   7. Simplified 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. Taylor expanded in z around 0 100.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{e^{-7.5}}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   if -1e3 < z

   1. Initial program 97.3%

      \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. Simplified 96.2%

      \[\leadsto \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\frac{\pi \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{\left(-\left(1 - z\right)\right) + -6.5} \cdot \left(\left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{7 + \left(1 - z\right)}\right)\right) + \left(\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) + \left(\frac{-1259.1392167224028}{1 + \left(1 - z\right)} + \left(\frac{771.3234287776531}{2 + \left(1 - z\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right)\right)\right)\right)}{\sin \left(\pi \cdot z\right)}} \]

   8. Simplified 99.3%

      \[\leadsto \color{blue}{\pi \cdot \frac{\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)}{\sin \left(z \cdot \pi\right)}} \]

   9. Taylor expanded in z around 0 99.0%

      \[\leadsto \pi \cdot \frac{\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\color{blue}{\left(z \cdot \left(z \cdot \left(-69.86359203642401 \cdot z - 131.58447752178645\right) - 240.12064030571747\right) - 415.6155560591857\right)} + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)}{\sin \left(z \cdot \pi\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1000:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} - \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - 7} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - 8}\right)\right)\right) + \left(z \cdot \left(z \cdot \left(z \cdot -69.86359203642401 - 131.58447752178645\right) - 240.12064030571747\right) - 415.6155560591857\right)\right)\right)}{\sin \left(\pi \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ t_1 := \sin \left(\pi \cdot z\right)\\ \mathbf{if}\;z \leq -1.15:\\ \;\;\;\;\left(t\_0 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \left(\frac{\pi}{t\_1} \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\left(t\_0 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} - \left(\left(415.6155560591857 - z \cdot \left(z \cdot -131.58447752178645 - 240.12064030571747\right)\right) + \left(\frac{-0.13857109526572012}{z - 6} + \left(\frac{12.507343278686905}{z - 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - 7} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - 8}\right)\right)\right)\right)\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 2.0))) (t_1 (sin (* PI z))))
   (if (<= z -1.15)
     (*
      (* t_0 (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp -7.5)))
      (*
       (/ PI t_1)
       (+
        263.3831869810514
        (* z (+ 436.8961725563396 (* z 545.0353078428827))))))
     (*
      PI
      (/
       (*
        (* t_0 (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5))))
        (-
         (/ 676.5203681218851 (- 1.0 z))
         (+
          (-
           415.6155560591857
           (* z (- (* z -131.58447752178645) 240.12064030571747)))
          (+
           (/ -0.13857109526572012 (- z 6.0))
           (+
            (/ 12.507343278686905 (- z 5.0))
            (+
             (/ 9.984369578019572e-6 (- z 7.0))
             (/ 1.5056327351493116e-7 (- z 8.0))))))))
       t_1)))))

C

double code(double z) {
	double t_0 = sqrt((((double) M_PI) * 2.0));
	double t_1 = sin((((double) M_PI) * z));
	double tmp;
	if (z <= -1.15) {
		tmp = (t_0 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(-7.5))) * ((((double) M_PI) / t_1) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))));
	} else {
		tmp = ((double) M_PI) * (((t_0 * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5)))) * ((676.5203681218851 / (1.0 - z)) - ((415.6155560591857 - (z * ((z * -131.58447752178645) - 240.12064030571747))) + ((-0.13857109526572012 / (z - 6.0)) + ((12.507343278686905 / (z - 5.0)) + ((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0)))))))) / t_1);
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.sqrt((Math.PI * 2.0));
	double t_1 = Math.sin((Math.PI * z));
	double tmp;
	if (z <= -1.15) {
		tmp = (t_0 * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(-7.5))) * ((Math.PI / t_1) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))));
	} else {
		tmp = Math.PI * (((t_0 * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5)))) * ((676.5203681218851 / (1.0 - z)) - ((415.6155560591857 - (z * ((z * -131.58447752178645) - 240.12064030571747))) + ((-0.13857109526572012 / (z - 6.0)) + ((12.507343278686905 / (z - 5.0)) + ((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0)))))))) / t_1);
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.sqrt((math.pi * 2.0))
	t_1 = math.sin((math.pi * z))
	tmp = 0
	if z <= -1.15:
		tmp = (t_0 * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(-7.5))) * ((math.pi / t_1) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))))
	else:
		tmp = math.pi * (((t_0 * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5)))) * ((676.5203681218851 / (1.0 - z)) - ((415.6155560591857 - (z * ((z * -131.58447752178645) - 240.12064030571747))) + ((-0.13857109526572012 / (z - 6.0)) + ((12.507343278686905 / (z - 5.0)) + ((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0)))))))) / t_1)
	return tmp

Julia

function code(z)
	t_0 = sqrt(Float64(pi * 2.0))
	t_1 = sin(Float64(pi * z))
	tmp = 0.0
	if (z <= -1.15)
		tmp = Float64(Float64(t_0 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(-7.5))) * Float64(Float64(pi / t_1) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * 545.0353078428827))))));
	else
		tmp = Float64(pi * Float64(Float64(Float64(t_0 * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5)))) * Float64(Float64(676.5203681218851 / Float64(1.0 - z)) - Float64(Float64(415.6155560591857 - Float64(z * Float64(Float64(z * -131.58447752178645) - 240.12064030571747))) + Float64(Float64(-0.13857109526572012 / Float64(z - 6.0)) + Float64(Float64(12.507343278686905 / Float64(z - 5.0)) + Float64(Float64(9.984369578019572e-6 / Float64(z - 7.0)) + Float64(1.5056327351493116e-7 / Float64(z - 8.0)))))))) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = sqrt((pi * 2.0));
	t_1 = sin((pi * z));
	tmp = 0.0;
	if (z <= -1.15)
		tmp = (t_0 * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(-7.5))) * ((pi / t_1) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))));
	else
		tmp = pi * (((t_0 * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5)))) * ((676.5203681218851 / (1.0 - z)) - ((415.6155560591857 - (z * ((z * -131.58447752178645) - 240.12064030571747))) + ((-0.13857109526572012 / (z - 6.0)) + ((12.507343278686905 / (z - 5.0)) + ((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0)))))))) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1.15], N[(N[(t$95$0 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / t$95$1), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * 545.0353078428827), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(N[(N[(t$95$0 * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[(N[(415.6155560591857 - N[(z * N[(N[(z * -131.58447752178645), $MachinePrecision] - 240.12064030571747), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(z - 6.0), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(z - 5.0), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(z - 7.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(z - 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.1499999999999999

   1. Initial program 0.0%

      \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot \left(436.8961725563396 + \color{blue}{z \cdot 545.0353078428827}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   7. Simplified 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. Taylor expanded in z around 0 100.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{e^{-7.5}}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   if -1.1499999999999999 < z

   1. Initial program 97.3%

      \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. Simplified 96.2%

      \[\leadsto \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\frac{\pi \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{\left(-\left(1 - z\right)\right) + -6.5} \cdot \left(\left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{7 + \left(1 - z\right)}\right)\right) + \left(\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) + \left(\frac{-1259.1392167224028}{1 + \left(1 - z\right)} + \left(\frac{771.3234287776531}{2 + \left(1 - z\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right)\right)\right)\right)}{\sin \left(\pi \cdot z\right)}} \]

   8. Simplified 99.3%

      \[\leadsto \color{blue}{\pi \cdot \frac{\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)}{\sin \left(z \cdot \pi\right)}} \]

   9. Taylor expanded in z around 0 98.8%

      \[\leadsto \pi \cdot \frac{\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\color{blue}{\left(z \cdot \left(-131.58447752178645 \cdot z - 240.12064030571747\right) - 415.6155560591857\right)} + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)}{\sin \left(z \cdot \pi\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} - \left(\left(415.6155560591857 - z \cdot \left(z \cdot -131.58447752178645 - 240.12064030571747\right)\right) + \left(\frac{-0.13857109526572012}{z - 6} + \left(\frac{12.507343278686905}{z - 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - 7} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - 8}\right)\right)\right)\right)\right)}{\sin \left(\pi \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ t_1 := \sin \left(\pi \cdot z\right)\\ \mathbf{if}\;z \leq -0.68:\\ \;\;\;\;\left(t\_0 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \left(\frac{\pi}{t\_1} \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \frac{t\_0 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)}{t\_1}\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 2.0))) (t_1 (sin (* PI z))))
   (if (<= z -0.68)
     (*
      (* t_0 (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp -7.5)))
      (*
       (/ PI t_1)
       (+
        263.3831869810514
        (* z (+ 436.8961725563396 (* z 545.0353078428827))))))
     (*
      (* PI (/ (* t_0 (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5)))) t_1))
      (+
       (+
        263.3831855358925
        (*
         z
         (+
          436.8961723502244
          (* z (+ 545.0353078134797 (* z 606.6766809125655))))))
       (+
        (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))
        (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0))))))))

C

double code(double z) {
	double t_0 = sqrt((((double) M_PI) * 2.0));
	double t_1 = sin((((double) M_PI) * z));
	double tmp;
	if (z <= -0.68) {
		tmp = (t_0 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(-7.5))) * ((((double) M_PI) / t_1) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))));
	} else {
		tmp = (((double) M_PI) * ((t_0 * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5)))) / t_1)) * ((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.sqrt((Math.PI * 2.0));
	double t_1 = Math.sin((Math.PI * z));
	double tmp;
	if (z <= -0.68) {
		tmp = (t_0 * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(-7.5))) * ((Math.PI / t_1) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))));
	} else {
		tmp = (Math.PI * ((t_0 * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5)))) / t_1)) * ((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.sqrt((math.pi * 2.0))
	t_1 = math.sin((math.pi * z))
	tmp = 0
	if z <= -0.68:
		tmp = (t_0 * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(-7.5))) * ((math.pi / t_1) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))))
	else:
		tmp = (math.pi * ((t_0 * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5)))) / t_1)) * ((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))))
	return tmp

Julia

function code(z)
	t_0 = sqrt(Float64(pi * 2.0))
	t_1 = sin(Float64(pi * z))
	tmp = 0.0
	if (z <= -0.68)
		tmp = Float64(Float64(t_0 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(-7.5))) * Float64(Float64(pi / t_1) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * 545.0353078428827))))));
	else
		tmp = Float64(Float64(pi * Float64(Float64(t_0 * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5)))) / t_1)) * Float64(Float64(263.3831855358925 + Float64(z * Float64(436.8961723502244 + Float64(z * Float64(545.0353078134797 + Float64(z * 606.6766809125655)))))) + Float64(Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0)) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = sqrt((pi * 2.0));
	t_1 = sin((pi * z));
	tmp = 0.0;
	if (z <= -0.68)
		tmp = (t_0 * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(-7.5))) * ((pi / t_1) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))));
	else
		tmp = (pi * ((t_0 * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5)))) / t_1)) * ((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -0.68], N[(N[(t$95$0 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / t$95$1), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * 545.0353078428827), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * N[(N[(t$95$0 * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(263.3831855358925 + N[(z * N[(436.8961723502244 + N[(z * N[(545.0353078134797 + N[(z * 606.6766809125655), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.680000000000000049

   1. Initial program 0.0%

      \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot \left(436.8961725563396 + \color{blue}{z \cdot 545.0353078428827}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   7. Simplified 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. Taylor expanded in z around 0 100.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{e^{-7.5}}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   if -0.680000000000000049 < z

   1. Initial program 97.3%

      \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. Simplified 99.0%

      \[\leadsto \color{blue}{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 98.6%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 98.6%

         \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + \color{blue}{z \cdot 606.6766809125655}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   7. Simplified 98.6%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   9. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\frac{\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right)}{\sin \left(\pi \cdot z\right)}} \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   11. Simplified 98.6%

       \[\leadsto \color{blue}{\left(\pi \cdot \frac{\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right)}{\sin \left(z \cdot \pi\right)}\right)} \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.68:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \frac{\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq -0.9:\\ \;\;\;\;\left(t\_0 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_0 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} - \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - 7} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - 8}\right)\right)\right) - \left(\left(\left(\frac{-1259.1392167224028}{z - 2} + \frac{771.3234287776531}{z - 3}\right) - 0.9999999999998099\right) + \frac{-176.6150291621406}{z - 4}\right)\right)\right)\right) \cdot \frac{1}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 2.0))))
   (if (<= z -0.9)
     (*
      (* t_0 (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp -7.5)))
      (*
       (/ PI (sin (* PI z)))
       (+
        263.3831869810514
        (* z (+ 436.8961725563396 (* z 545.0353078428827))))))
     (*
      (*
       (* t_0 (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5))))
       (+
        (/ 676.5203681218851 (- 1.0 z))
        (-
         (+
          (/ -0.13857109526572012 (- 6.0 z))
          (-
           (/ 12.507343278686905 (- 5.0 z))
           (+
            (/ 9.984369578019572e-6 (- z 7.0))
            (/ 1.5056327351493116e-7 (- z 8.0)))))
         (+
          (-
           (+
            (/ -1259.1392167224028 (- z 2.0))
            (/ 771.3234287776531 (- z 3.0)))
           0.9999999999998099)
          (/ -176.6150291621406 (- z 4.0))))))
      (/ 1.0 z)))))

C

double code(double z) {
	double t_0 = sqrt((((double) M_PI) * 2.0));
	double tmp;
	if (z <= -0.9) {
		tmp = (t_0 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(-7.5))) * ((((double) M_PI) / sin((((double) M_PI) * z))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))));
	} else {
		tmp = ((t_0 * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5)))) * ((676.5203681218851 / (1.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) - ((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0))))) - ((((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0))) - 0.9999999999998099) + (-176.6150291621406 / (z - 4.0)))))) * (1.0 / z);
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.sqrt((Math.PI * 2.0));
	double tmp;
	if (z <= -0.9) {
		tmp = (t_0 * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(-7.5))) * ((Math.PI / Math.sin((Math.PI * z))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))));
	} else {
		tmp = ((t_0 * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5)))) * ((676.5203681218851 / (1.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) - ((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0))))) - ((((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0))) - 0.9999999999998099) + (-176.6150291621406 / (z - 4.0)))))) * (1.0 / z);
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.sqrt((math.pi * 2.0))
	tmp = 0
	if z <= -0.9:
		tmp = (t_0 * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(-7.5))) * ((math.pi / math.sin((math.pi * z))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))))
	else:
		tmp = ((t_0 * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5)))) * ((676.5203681218851 / (1.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) - ((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0))))) - ((((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0))) - 0.9999999999998099) + (-176.6150291621406 / (z - 4.0)))))) * (1.0 / z)
	return tmp

Julia

function code(z)
	t_0 = sqrt(Float64(pi * 2.0))
	tmp = 0.0
	if (z <= -0.9)
		tmp = Float64(Float64(t_0 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(-7.5))) * Float64(Float64(pi / sin(Float64(pi * z))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * 545.0353078428827))))));
	else
		tmp = Float64(Float64(Float64(t_0 * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5)))) * Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(Float64(12.507343278686905 / Float64(5.0 - z)) - Float64(Float64(9.984369578019572e-6 / Float64(z - 7.0)) + Float64(1.5056327351493116e-7 / Float64(z - 8.0))))) - Float64(Float64(Float64(Float64(-1259.1392167224028 / Float64(z - 2.0)) + Float64(771.3234287776531 / Float64(z - 3.0))) - 0.9999999999998099) + Float64(-176.6150291621406 / Float64(z - 4.0)))))) * Float64(1.0 / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = sqrt((pi * 2.0));
	tmp = 0.0;
	if (z <= -0.9)
		tmp = (t_0 * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(-7.5))) * ((pi / sin((pi * z))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))));
	else
		tmp = ((t_0 * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5)))) * ((676.5203681218851 / (1.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) - ((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0))))) - ((((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0))) - 0.9999999999998099) + (-176.6150291621406 / (z - 4.0)))))) * (1.0 / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -0.9], N[(N[(t$95$0 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * 545.0353078428827), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] - N[(N[(9.984369578019572e-6 / N[(z - 7.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(z - 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(-1259.1392167224028 / N[(z - 2.0), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.9999999999998099), $MachinePrecision] + N[(-176.6150291621406 / N[(z - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.900000000000000022

   1. Initial program 0.0%

      \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot \left(436.8961725563396 + \color{blue}{z \cdot 545.0353078428827}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   7. Simplified 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. Taylor expanded in z around 0 100.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{e^{-7.5}}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   if -0.900000000000000022 < z

   1. Initial program 97.3%

      \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 98.5%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right) \cdot \left(\left(\left(\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) + \left(\frac{-1259.1392167224028}{1 + \left(1 - z\right)} + \left(\frac{771.3234287776531}{2 + \left(1 - z\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6}\right) + \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right) \cdot \frac{1.5056327351493116 \cdot 10^{-7}}{7 + \left(1 - z\right)}\right)} \]

   6. Simplified 99.2%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)\right)} \]

   7. Taylor expanded in z around 0 98.5%

      \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.9:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} - \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - 7} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - 8}\right)\right)\right) - \left(\left(\left(\frac{-1259.1392167224028}{z - 2} + \frac{771.3234287776531}{z - 3}\right) - 0.9999999999998099\right) + \frac{-176.6150291621406}{z - 4}\right)\right)\right)\right) \cdot \frac{1}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(\left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} - \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - 7} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - 8}\right)\right)\right) - \left(\left(\left(\frac{-1259.1392167224028}{z - 2} + \frac{771.3234287776531}{z - 3}\right) - 0.9999999999998099\right) + \frac{-176.6150291621406}{z - 4}\right)\right)\right)\right) \cdot \frac{1}{z} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (* (sqrt (* PI 2.0)) (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5))))
   (+
    (/ 676.5203681218851 (- 1.0 z))
    (-
     (+
      (/ -0.13857109526572012 (- 6.0 z))
      (-
       (/ 12.507343278686905 (- 5.0 z))
       (+
        (/ 9.984369578019572e-6 (- z 7.0))
        (/ 1.5056327351493116e-7 (- z 8.0)))))
     (+
      (-
       (+ (/ -1259.1392167224028 (- z 2.0)) (/ 771.3234287776531 (- z 3.0)))
       0.9999999999998099)
      (/ -176.6150291621406 (- z 4.0))))))
  (/ 1.0 z)))

C

double code(double z) {
	return ((sqrt((((double) M_PI) * 2.0)) * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5)))) * ((676.5203681218851 / (1.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) - ((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0))))) - ((((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0))) - 0.9999999999998099) + (-176.6150291621406 / (z - 4.0)))))) * (1.0 / z);
}

Java

public static double code(double z) {
	return ((Math.sqrt((Math.PI * 2.0)) * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5)))) * ((676.5203681218851 / (1.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) - ((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0))))) - ((((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0))) - 0.9999999999998099) + (-176.6150291621406 / (z - 4.0)))))) * (1.0 / z);
}

Python

def code(z):
	return ((math.sqrt((math.pi * 2.0)) * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5)))) * ((676.5203681218851 / (1.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) - ((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0))))) - ((((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0))) - 0.9999999999998099) + (-176.6150291621406 / (z - 4.0)))))) * (1.0 / z)

Julia

function code(z)
	return Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5)))) * Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(Float64(12.507343278686905 / Float64(5.0 - z)) - Float64(Float64(9.984369578019572e-6 / Float64(z - 7.0)) + Float64(1.5056327351493116e-7 / Float64(z - 8.0))))) - Float64(Float64(Float64(Float64(-1259.1392167224028 / Float64(z - 2.0)) + Float64(771.3234287776531 / Float64(z - 3.0))) - 0.9999999999998099) + Float64(-176.6150291621406 / Float64(z - 4.0)))))) * Float64(1.0 / z))
end

MATLAB

function tmp = code(z)
	tmp = ((sqrt((pi * 2.0)) * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5)))) * ((676.5203681218851 / (1.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) - ((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0))))) - ((((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0))) - 0.9999999999998099) + (-176.6150291621406 / (z - 4.0)))))) * (1.0 / z);
end

Wolfram

code[z_] := N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] - N[(N[(9.984369578019572e-6 / N[(z - 7.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(z - 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(-1259.1392167224028 / N[(z - 2.0), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.9999999999998099), $MachinePrecision] + N[(-176.6150291621406 / N[(z - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.8%

   \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
2. Add Preprocessing

4. Applied egg-rr 97.0%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right) \cdot \left(\left(\left(\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) + \left(\frac{-1259.1392167224028}{1 + \left(1 - z\right)} + \left(\frac{771.3234287776531}{2 + \left(1 - z\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6}\right) + \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right) \cdot \frac{1.5056327351493116 \cdot 10^{-7}}{7 + \left(1 - z\right)}\right)} \]

6. Simplified 97.7%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)\right)} \]

7. Taylor expanded in z around 0 96.9%

   \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)\right) \]

8. Final simplification 96.9%

   \[\leadsto \left(\left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} - \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - 7} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - 8}\right)\right)\right) - \left(\left(\left(\frac{-1259.1392167224028}{z - 2} + \frac{771.3234287776531}{z - 3}\right) - 0.9999999999998099\right) + \frac{-176.6150291621406}{z - 4}\right)\right)\right)\right) \cdot \frac{1}{z} \]
9. Add Preprocessing

Alternative 7: 96.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(\frac{1}{z} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{-0.5 + \left(-6 + \left(z + -1\right)\right)}\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (+
   (+
    263.3831855358925
    (*
     z
     (+
      436.8961723502244
      (* z (+ 545.0353078134797 (* z 606.6766809125655))))))
   (+
    (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))
    (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0))))
  (*
   (/ 1.0 z)
   (*
    (sqrt (* PI 2.0))
    (*
     (pow (+ (+ (- 1.0 z) -1.0) 7.5) (- (- 1.0 z) 0.5))
     (exp (+ -0.5 (+ -6.0 (+ z -1.0)))))))))

C

double code(double z) {
	return ((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)))) * ((1.0 / z) * (sqrt((((double) M_PI) * 2.0)) * (pow((((1.0 - z) + -1.0) + 7.5), ((1.0 - z) - 0.5)) * exp((-0.5 + (-6.0 + (z + -1.0)))))));
}

Java

public static double code(double z) {
	return ((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)))) * ((1.0 / z) * (Math.sqrt((Math.PI * 2.0)) * (Math.pow((((1.0 - z) + -1.0) + 7.5), ((1.0 - z) - 0.5)) * Math.exp((-0.5 + (-6.0 + (z + -1.0)))))));
}

Python

def code(z):
	return ((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)))) * ((1.0 / z) * (math.sqrt((math.pi * 2.0)) * (math.pow((((1.0 - z) + -1.0) + 7.5), ((1.0 - z) - 0.5)) * math.exp((-0.5 + (-6.0 + (z + -1.0)))))))

Julia

function code(z)
	return Float64(Float64(Float64(263.3831855358925 + Float64(z * Float64(436.8961723502244 + Float64(z * Float64(545.0353078134797 + Float64(z * 606.6766809125655)))))) + Float64(Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0)) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0)))) * Float64(Float64(1.0 / z) * Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(Float64(Float64(1.0 - z) + -1.0) + 7.5) ^ Float64(Float64(1.0 - z) - 0.5)) * exp(Float64(-0.5 + Float64(-6.0 + Float64(z + -1.0))))))))
end

MATLAB

function tmp = code(z)
	tmp = ((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)))) * ((1.0 / z) * (sqrt((pi * 2.0)) * (((((1.0 - z) + -1.0) + 7.5) ^ ((1.0 - z) - 0.5)) * exp((-0.5 + (-6.0 + (z + -1.0)))))));
end

Wolfram

code[z_] := N[(N[(N[(263.3831855358925 + N[(z * N[(436.8961723502244 + N[(z * N[(545.0353078134797 + N[(z * 606.6766809125655), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / z), $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(N[(N[(1.0 - z), $MachinePrecision] + -1.0), $MachinePrecision] + 7.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-0.5 + N[(-6.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.8%

   \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

3. Simplified 97.4%

   \[\leadsto \color{blue}{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in z around 0 97.0%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
6. Step-by-step derivation

   1. *-commutative 97.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + \color{blue}{z \cdot 606.6766809125655}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

7. Simplified 97.0%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

8. Taylor expanded in z around 0 96.6%

   \[\leadsto \left(\color{blue}{\frac{1}{z}} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

9. Final simplification 96.6%

   \[\leadsto \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(\frac{1}{z} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{-0.5 + \left(-6 + \left(z + -1\right)\right)}\right)\right)\right) \]
10. Add Preprocessing

Alternative 8: 95.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{\pi} \cdot \left(e^{-7.5} \cdot \frac{\sqrt{15}}{z}\right)\right) \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (* (sqrt PI) (* (exp -7.5) (/ (sqrt 15.0) z)))
  (+ 263.3831869810514 (* z 436.8961725563396))))

C

double code(double z) {
	return (sqrt(((double) M_PI)) * (exp(-7.5) * (sqrt(15.0) / z))) * (263.3831869810514 + (z * 436.8961725563396));
}

Java

public static double code(double z) {
	return (Math.sqrt(Math.PI) * (Math.exp(-7.5) * (Math.sqrt(15.0) / z))) * (263.3831869810514 + (z * 436.8961725563396));
}

Python

def code(z):
	return (math.sqrt(math.pi) * (math.exp(-7.5) * (math.sqrt(15.0) / z))) * (263.3831869810514 + (z * 436.8961725563396))

Julia

function code(z)
	return Float64(Float64(sqrt(pi) * Float64(exp(-7.5) * Float64(sqrt(15.0) / z))) * Float64(263.3831869810514 + Float64(z * 436.8961725563396)))
end

MATLAB

function tmp = code(z)
	tmp = (sqrt(pi) * (exp(-7.5) * (sqrt(15.0) / z))) * (263.3831869810514 + (z * 436.8961725563396));
end

Wolfram

code[z_] := N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] * N[(N[Sqrt[15.0], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * 436.8961725563396), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.8%

   \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

3. Simplified 97.4%

   \[\leadsto \color{blue}{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in z around 0 95.6%

   \[\leadsto \color{blue}{\left(\frac{e^{-7.5} \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)}{z} \cdot \sqrt{\pi}\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
6. Step-by-step derivation

   1. associate-*r* 95.6%

      \[\leadsto \left(\frac{\color{blue}{\left(e^{-7.5} \cdot \sqrt{2}\right) \cdot \sqrt{7.5}}}{z} \cdot \sqrt{\pi}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

7. Simplified 95.6%

   \[\leadsto \color{blue}{\left(\frac{\left(e^{-7.5} \cdot \sqrt{2}\right) \cdot \sqrt{7.5}}{z} \cdot \sqrt{\pi}\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

8. Taylor expanded in z around 0 95.7%

   \[\leadsto \left(\frac{\left(e^{-7.5} \cdot \sqrt{2}\right) \cdot \sqrt{7.5}}{z} \cdot \sqrt{\pi}\right) \cdot \color{blue}{\left(263.3831869810514 + 436.8961725563396 \cdot z\right)} \]
9. Step-by-step derivation

   1. *-commutative 95.7%

      \[\leadsto \left(\frac{\left(e^{-7.5} \cdot \sqrt{2}\right) \cdot \sqrt{7.5}}{z} \cdot \sqrt{\pi}\right) \cdot \left(263.3831869810514 + \color{blue}{z \cdot 436.8961725563396}\right) \]

10. Simplified 95.7%

    \[\leadsto \left(\frac{\left(e^{-7.5} \cdot \sqrt{2}\right) \cdot \sqrt{7.5}}{z} \cdot \sqrt{\pi}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot 436.8961725563396\right)} \]
11. Step-by-step derivation

    1. associate-*l/ 95.5%

       \[\leadsto \color{blue}{\frac{\left(\left(e^{-7.5} \cdot \sqrt{2}\right) \cdot \sqrt{7.5}\right) \cdot \sqrt{\pi}}{z}} \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right) \]

    2. associate-*l* 95.5%

       \[\leadsto \frac{\color{blue}{\left(e^{-7.5} \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)\right)} \cdot \sqrt{\pi}}{z} \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right) \]

    3. pow1/2 95.5%

       \[\leadsto \frac{\left(e^{-7.5} \cdot \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{7.5}\right)\right) \cdot \sqrt{\pi}}{z} \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right) \]

    4. pow1/2 95.5%

       \[\leadsto \frac{\left(e^{-7.5} \cdot \left({2}^{0.5} \cdot \color{blue}{{7.5}^{0.5}}\right)\right) \cdot \sqrt{\pi}}{z} \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right) \]

    5. pow-prod-down 95.5%

       \[\leadsto \frac{\left(e^{-7.5} \cdot \color{blue}{{\left(2 \cdot 7.5\right)}^{0.5}}\right) \cdot \sqrt{\pi}}{z} \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right) \]

    6. metadata-eval 95.5%

       \[\leadsto \frac{\left(e^{-7.5} \cdot {\color{blue}{15}}^{0.5}\right) \cdot \sqrt{\pi}}{z} \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right) \]

12. Applied egg-rr 95.5%

    \[\leadsto \color{blue}{\frac{\left(e^{-7.5} \cdot {15}^{0.5}\right) \cdot \sqrt{\pi}}{z}} \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right) \]
13. Step-by-step derivation

    1. associate-*l/ 95.7%

       \[\leadsto \color{blue}{\left(\frac{e^{-7.5} \cdot {15}^{0.5}}{z} \cdot \sqrt{\pi}\right)} \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right) \]

    2. *-commutative 95.7%

       \[\leadsto \color{blue}{\left(\sqrt{\pi} \cdot \frac{e^{-7.5} \cdot {15}^{0.5}}{z}\right)} \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right) \]

    3. associate-/l* 95.8%

       \[\leadsto \left(\sqrt{\pi} \cdot \color{blue}{\left(e^{-7.5} \cdot \frac{{15}^{0.5}}{z}\right)}\right) \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right) \]

    4. unpow1/2 95.8%

       \[\leadsto \left(\sqrt{\pi} \cdot \left(e^{-7.5} \cdot \frac{\color{blue}{\sqrt{15}}}{z}\right)\right) \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right) \]

14. Simplified 95.8%

    \[\leadsto \color{blue}{\left(\sqrt{\pi} \cdot \left(e^{-7.5} \cdot \frac{\sqrt{15}}{z}\right)\right)} \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right) \]
15. Add Preprocessing

Alternative 9: 95.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{263.3831869810514 \cdot \left(e^{-7.5} \cdot {\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}\right)}{z} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (/ (* 263.3831869810514 (* (exp -7.5) (pow (* (* PI 2.0) 7.5) 0.5))) z))

C

double code(double z) {
	return (263.3831869810514 * (exp(-7.5) * pow(((((double) M_PI) * 2.0) * 7.5), 0.5))) / z;
}

Java

public static double code(double z) {
	return (263.3831869810514 * (Math.exp(-7.5) * Math.pow(((Math.PI * 2.0) * 7.5), 0.5))) / z;
}

Python

def code(z):
	return (263.3831869810514 * (math.exp(-7.5) * math.pow(((math.pi * 2.0) * 7.5), 0.5))) / z

Julia

function code(z)
	return Float64(Float64(263.3831869810514 * Float64(exp(-7.5) * (Float64(Float64(pi * 2.0) * 7.5) ^ 0.5))) / z)
end

MATLAB

function tmp = code(z)
	tmp = (263.3831869810514 * (exp(-7.5) * (((pi * 2.0) * 7.5) ^ 0.5))) / z;
end

Wolfram

code[z_] := N[(N[(263.3831869810514 * N[(N[Exp[-7.5], $MachinePrecision] * N[Power[N[(N[(Pi * 2.0), $MachinePrecision] * 7.5), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Derivation

1. Initial program 95.8%

   \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

3. Simplified 94.7%

   \[\leadsto \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)} \]
4. Add Preprocessing

5. Taylor expanded in z around 0 94.1%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\frac{263.3831869810514}{z}} \]

6. Taylor expanded in z around 0 94.3%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\sqrt{7.5}} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \frac{263.3831869810514}{z} \]

7. Taylor expanded in z around 0 94.2%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left(\sqrt{7.5} \cdot \color{blue}{e^{-7.5}}\right)\right) \cdot \frac{263.3831869810514}{z} \]
8. Step-by-step derivation

   1. associate-*r/ 95.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\pi \cdot 2} \cdot \left(\sqrt{7.5} \cdot e^{-7.5}\right)\right) \cdot 263.3831869810514}{z}} \]

   2. associate-*r* 95.2%

      \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{\pi \cdot 2} \cdot \sqrt{7.5}\right) \cdot e^{-7.5}\right)} \cdot 263.3831869810514}{z} \]

   3. pow1/2 95.2%

      \[\leadsto \frac{\left(\left(\color{blue}{{\left(\pi \cdot 2\right)}^{0.5}} \cdot \sqrt{7.5}\right) \cdot e^{-7.5}\right) \cdot 263.3831869810514}{z} \]

   4. pow1/2 95.2%

      \[\leadsto \frac{\left(\left({\left(\pi \cdot 2\right)}^{0.5} \cdot \color{blue}{{7.5}^{0.5}}\right) \cdot e^{-7.5}\right) \cdot 263.3831869810514}{z} \]

   5. pow-prod-down 95.2%

      \[\leadsto \frac{\left(\color{blue}{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}} \cdot e^{-7.5}\right) \cdot 263.3831869810514}{z} \]

   6. *-commutative 95.2%

      \[\leadsto \frac{\left({\left(\color{blue}{\left(2 \cdot \pi\right)} \cdot 7.5\right)}^{0.5} \cdot e^{-7.5}\right) \cdot 263.3831869810514}{z} \]

9. Applied egg-rr 95.2%

   \[\leadsto \color{blue}{\frac{\left({\left(\left(2 \cdot \pi\right) \cdot 7.5\right)}^{0.5} \cdot e^{-7.5}\right) \cdot 263.3831869810514}{z}} \]

10. Final simplification 95.2%

    \[\leadsto \frac{263.3831869810514 \cdot \left(e^{-7.5} \cdot {\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}\right)}{z} \]
11. Add Preprocessing

Alternative 10: 94.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \left(e^{-7.5} \cdot \sqrt{\pi \cdot 15}\right) \cdot \frac{263.3831869810514}{z} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (* (* (exp -7.5) (sqrt (* PI 15.0))) (/ 263.3831869810514 z)))

C

double code(double z) {
	return (exp(-7.5) * sqrt((((double) M_PI) * 15.0))) * (263.3831869810514 / z);
}

Java

public static double code(double z) {
	return (Math.exp(-7.5) * Math.sqrt((Math.PI * 15.0))) * (263.3831869810514 / z);
}

Python

def code(z):
	return (math.exp(-7.5) * math.sqrt((math.pi * 15.0))) * (263.3831869810514 / z)

Julia

function code(z)
	return Float64(Float64(exp(-7.5) * sqrt(Float64(pi * 15.0))) * Float64(263.3831869810514 / z))
end

MATLAB

function tmp = code(z)
	tmp = (exp(-7.5) * sqrt((pi * 15.0))) * (263.3831869810514 / z);
end

Wolfram

code[z_] := N[(N[(N[Exp[-7.5], $MachinePrecision] * N[Sqrt[N[(Pi * 15.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.8%

   \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

3. Simplified 94.7%

   \[\leadsto \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)} \]
4. Add Preprocessing

5. Taylor expanded in z around 0 94.1%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\frac{263.3831869810514}{z}} \]

6. Taylor expanded in z around 0 94.3%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\sqrt{7.5}} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \frac{263.3831869810514}{z} \]

7. Taylor expanded in z around 0 94.2%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left(\sqrt{7.5} \cdot \color{blue}{e^{-7.5}}\right)\right) \cdot \frac{263.3831869810514}{z} \]
8. Step-by-step derivation

   1. pow1 94.2%

      \[\leadsto \color{blue}{{\left(\sqrt{\pi \cdot 2} \cdot \left(\sqrt{7.5} \cdot e^{-7.5}\right)\right)}^{1}} \cdot \frac{263.3831869810514}{z} \]

   2. associate-*r* 94.2%

      \[\leadsto {\color{blue}{\left(\left(\sqrt{\pi \cdot 2} \cdot \sqrt{7.5}\right) \cdot e^{-7.5}\right)}}^{1} \cdot \frac{263.3831869810514}{z} \]

   3. pow1/2 94.2%

      \[\leadsto {\left(\left(\color{blue}{{\left(\pi \cdot 2\right)}^{0.5}} \cdot \sqrt{7.5}\right) \cdot e^{-7.5}\right)}^{1} \cdot \frac{263.3831869810514}{z} \]

   4. pow1/2 94.2%

      \[\leadsto {\left(\left({\left(\pi \cdot 2\right)}^{0.5} \cdot \color{blue}{{7.5}^{0.5}}\right) \cdot e^{-7.5}\right)}^{1} \cdot \frac{263.3831869810514}{z} \]

   5. pow-prod-down 94.2%

      \[\leadsto {\left(\color{blue}{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}} \cdot e^{-7.5}\right)}^{1} \cdot \frac{263.3831869810514}{z} \]

   6. *-commutative 94.2%

      \[\leadsto {\left({\left(\color{blue}{\left(2 \cdot \pi\right)} \cdot 7.5\right)}^{0.5} \cdot e^{-7.5}\right)}^{1} \cdot \frac{263.3831869810514}{z} \]

9. Applied egg-rr 94.2%

   \[\leadsto \color{blue}{{\left({\left(\left(2 \cdot \pi\right) \cdot 7.5\right)}^{0.5} \cdot e^{-7.5}\right)}^{1}} \cdot \frac{263.3831869810514}{z} \]
10. Step-by-step derivation

    1. unpow1 94.2%

       \[\leadsto \color{blue}{\left({\left(\left(2 \cdot \pi\right) \cdot 7.5\right)}^{0.5} \cdot e^{-7.5}\right)} \cdot \frac{263.3831869810514}{z} \]

    2. *-commutative 94.2%

       \[\leadsto \color{blue}{\left(e^{-7.5} \cdot {\left(\left(2 \cdot \pi\right) \cdot 7.5\right)}^{0.5}\right)} \cdot \frac{263.3831869810514}{z} \]

    3. unpow1/2 94.2%

       \[\leadsto \left(e^{-7.5} \cdot \color{blue}{\sqrt{\left(2 \cdot \pi\right) \cdot 7.5}}\right) \cdot \frac{263.3831869810514}{z} \]

    4. *-commutative 94.2%

       \[\leadsto \left(e^{-7.5} \cdot \sqrt{\color{blue}{\left(\pi \cdot 2\right)} \cdot 7.5}\right) \cdot \frac{263.3831869810514}{z} \]

    5. associate-*l* 94.2%

       \[\leadsto \left(e^{-7.5} \cdot \sqrt{\color{blue}{\pi \cdot \left(2 \cdot 7.5\right)}}\right) \cdot \frac{263.3831869810514}{z} \]

    6. metadata-eval 94.2%

       \[\leadsto \left(e^{-7.5} \cdot \sqrt{\pi \cdot \color{blue}{15}}\right) \cdot \frac{263.3831869810514}{z} \]

11. Simplified 94.2%

    \[\leadsto \color{blue}{\left(e^{-7.5} \cdot \sqrt{\pi \cdot 15}\right)} \cdot \frac{263.3831869810514}{z} \]
12. Add Preprocessing

Alternative 11: 94.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\pi \cdot 15} \cdot \left(e^{-7.5} \cdot \frac{263.3831869810514}{z}\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (* (sqrt (* PI 15.0)) (* (exp -7.5) (/ 263.3831869810514 z))))

C

double code(double z) {
	return sqrt((((double) M_PI) * 15.0)) * (exp(-7.5) * (263.3831869810514 / z));
}

Java

public static double code(double z) {
	return Math.sqrt((Math.PI * 15.0)) * (Math.exp(-7.5) * (263.3831869810514 / z));
}

Python

def code(z):
	return math.sqrt((math.pi * 15.0)) * (math.exp(-7.5) * (263.3831869810514 / z))

Julia

function code(z)
	return Float64(sqrt(Float64(pi * 15.0)) * Float64(exp(-7.5) * Float64(263.3831869810514 / z)))
end

MATLAB

function tmp = code(z)
	tmp = sqrt((pi * 15.0)) * (exp(-7.5) * (263.3831869810514 / z));
end

Wolfram

code[z_] := N[(N[Sqrt[N[(Pi * 15.0), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] * N[(263.3831869810514 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.8%

   \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

3. Simplified 94.7%

   \[\leadsto \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)} \]
4. Add Preprocessing

5. Taylor expanded in z around 0 94.1%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\frac{263.3831869810514}{z}} \]

6. Taylor expanded in z around 0 94.3%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\sqrt{7.5}} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \frac{263.3831869810514}{z} \]

7. Taylor expanded in z around 0 94.2%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left(\sqrt{7.5} \cdot \color{blue}{e^{-7.5}}\right)\right) \cdot \frac{263.3831869810514}{z} \]
8. Step-by-step derivation

   1. associate-*r/ 95.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\pi \cdot 2} \cdot \left(\sqrt{7.5} \cdot e^{-7.5}\right)\right) \cdot 263.3831869810514}{z}} \]

   2. associate-*r* 95.2%

      \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{\pi \cdot 2} \cdot \sqrt{7.5}\right) \cdot e^{-7.5}\right)} \cdot 263.3831869810514}{z} \]

   3. pow1/2 95.2%

      \[\leadsto \frac{\left(\left(\color{blue}{{\left(\pi \cdot 2\right)}^{0.5}} \cdot \sqrt{7.5}\right) \cdot e^{-7.5}\right) \cdot 263.3831869810514}{z} \]

   4. pow1/2 95.2%

      \[\leadsto \frac{\left(\left({\left(\pi \cdot 2\right)}^{0.5} \cdot \color{blue}{{7.5}^{0.5}}\right) \cdot e^{-7.5}\right) \cdot 263.3831869810514}{z} \]

   5. pow-prod-down 95.2%

      \[\leadsto \frac{\left(\color{blue}{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}} \cdot e^{-7.5}\right) \cdot 263.3831869810514}{z} \]

   6. *-commutative 95.2%

      \[\leadsto \frac{\left({\left(\color{blue}{\left(2 \cdot \pi\right)} \cdot 7.5\right)}^{0.5} \cdot e^{-7.5}\right) \cdot 263.3831869810514}{z} \]

9. Applied egg-rr 95.2%

   \[\leadsto \color{blue}{\frac{\left({\left(\left(2 \cdot \pi\right) \cdot 7.5\right)}^{0.5} \cdot e^{-7.5}\right) \cdot 263.3831869810514}{z}} \]
10. Step-by-step derivation

    1. associate-*r/ 94.2%

       \[\leadsto \color{blue}{\left({\left(\left(2 \cdot \pi\right) \cdot 7.5\right)}^{0.5} \cdot e^{-7.5}\right) \cdot \frac{263.3831869810514}{z}} \]

    2. associate-*l* 94.1%

       \[\leadsto \color{blue}{{\left(\left(2 \cdot \pi\right) \cdot 7.5\right)}^{0.5} \cdot \left(e^{-7.5} \cdot \frac{263.3831869810514}{z}\right)} \]

    3. unpow1/2 94.1%

       \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \pi\right) \cdot 7.5}} \cdot \left(e^{-7.5} \cdot \frac{263.3831869810514}{z}\right) \]

    4. *-commutative 94.1%

       \[\leadsto \sqrt{\color{blue}{\left(\pi \cdot 2\right)} \cdot 7.5} \cdot \left(e^{-7.5} \cdot \frac{263.3831869810514}{z}\right) \]

    5. associate-*l* 94.1%

       \[\leadsto \sqrt{\color{blue}{\pi \cdot \left(2 \cdot 7.5\right)}} \cdot \left(e^{-7.5} \cdot \frac{263.3831869810514}{z}\right) \]

    6. metadata-eval 94.1%

       \[\leadsto \sqrt{\pi \cdot \color{blue}{15}} \cdot \left(e^{-7.5} \cdot \frac{263.3831869810514}{z}\right) \]

11. Simplified 94.1%

    \[\leadsto \color{blue}{\sqrt{\pi \cdot 15} \cdot \left(e^{-7.5} \cdot \frac{263.3831869810514}{z}\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024146 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z less than 0.5"
  :precision binary64
  :pre (<= z 0.5)
  (* (/ PI (sin (* PI z))) (* (* (* (sqrt (* PI 2.0)) (pow (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5) (+ (- (- 1.0 z) 1.0) 0.5))) (exp (- (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- (- 1.0 z) 1.0) 1.0))) (/ -1259.1392167224028 (+ (- (- 1.0 z) 1.0) 2.0))) (/ 771.3234287776531 (+ (- (- 1.0 z) 1.0) 3.0))) (/ -176.6150291621406 (+ (- (- 1.0 z) 1.0) 4.0))) (/ 12.507343278686905 (+ (- (- 1.0 z) 1.0) 5.0))) (/ -0.13857109526572012 (+ (- (- 1.0 z) 1.0) 6.0))) (/ 9.984369578019572e-6 (+ (- (- 1.0 z) 1.0) 7.0))) (/ 1.5056327351493116e-7 (+ (- (- 1.0 z) 1.0) 8.0))))))