Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.4% → 99.0%

Time: 1.5min

Alternatives: 14

Speedup: 1.6×

• could not determine a ground truth

  1. z = -1.2228942639164068e+274

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: 96.4% 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: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \left(1 - z\right)\\ t_1 := t_0 + 7\\ t_2 := \frac{\pi}{\sin \left(\pi \cdot z\right)}\\ t_3 := \left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\\ t_4 := \sqrt{\pi \cdot 2}\\ t_5 := \left(z + -1\right) - -1\\ \mathbf{if}\;t_2 \cdot \left(\left(\left(t_4 \cdot {\left(0.5 + t_1\right)}^{\left(0.5 + t_0\right)}\right) \cdot e^{\left(t_5 - 7\right) - 0.5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 + t_0}\right) + \frac{-1259.1392167224028}{2 + t_0}\right) + \frac{771.3234287776531}{t_0 + 3}\right) + \frac{-176.6150291621406}{t_0 + 4}\right) + \frac{12.507343278686905}{t_0 + 5}\right) + \frac{-0.13857109526572012}{t_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t_1}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t_0 + 8}\right)\right) \leq 2 \cdot 10^{+307}:\\ \;\;\;\;t_3 \cdot \left(t_2 \cdot \left(\left(t_4 \cdot {\left(7.5 + \mathsf{expm1}\left(\mathsf{log1p}\left(-z\right)\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{t_5 - 7.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(t_2 \cdot e^{\log \left(\sqrt{2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi}\right)\right) + \left(z - 7.5\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ -1.0 (- 1.0 z)))
        (t_1 (+ t_0 7.0))
        (t_2 (/ PI (sin (* PI z))))
        (t_3
         (+
          (+
           (+
            (+
             (/ -1259.1392167224028 (- (- 1.0 z) -1.0))
             (+ 0.9999999999998099 (/ 676.5203681218851 (- 1.0 z))))
            (+
             (/ 771.3234287776531 (- (- 1.0 z) -2.0))
             (/ -176.6150291621406 (- (- 1.0 z) -3.0))))
           (+
            (/ 12.507343278686905 (- (- 1.0 z) -4.0))
            (/ -0.13857109526572012 (- (- 1.0 z) -5.0))))
          (+
           (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))
           (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0)))))
        (t_4 (sqrt (* PI 2.0)))
        (t_5 (- (+ z -1.0) -1.0)))
   (if (<=
        (*
         t_2
         (*
          (* (* t_4 (pow (+ 0.5 t_1) (+ 0.5 t_0))) (exp (- (- t_5 7.0) 0.5)))
          (+
           (+
            (+
             (+
              (+
               (+
                (+
                 (+ 0.9999999999998099 (/ 676.5203681218851 (+ 1.0 t_0)))
                 (/ -1259.1392167224028 (+ 2.0 t_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)))))
        2e+307)
     (*
      t_3
      (*
       t_2
       (*
        (* t_4 (pow (+ 7.5 (expm1 (log1p (- z)))) (- (- 1.0 z) 0.5)))
        (exp (- t_5 7.5)))))
     (*
      t_3
      (*
       t_2
       (exp
        (+
         (log (* (sqrt 2.0) (* (pow (- 7.5 z) (- 0.5 z)) (sqrt PI))))
         (- z 7.5))))))))

C

double code(double z) {
	double t_0 = -1.0 + (1.0 - z);
	double t_1 = t_0 + 7.0;
	double t_2 = ((double) M_PI) / sin((((double) M_PI) * z));
	double t_3 = ((((-1259.1392167224028 / ((1.0 - z) - -1.0)) + (0.9999999999998099 + (676.5203681218851 / (1.0 - z)))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) + (-176.6150291621406 / ((1.0 - z) - -3.0)))) + ((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0)))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
	double t_4 = sqrt((((double) M_PI) * 2.0));
	double t_5 = (z + -1.0) - -1.0;
	double tmp;
	if ((t_2 * (((t_4 * pow((0.5 + t_1), (0.5 + t_0))) * exp(((t_5 - 7.0) - 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / (1.0 + t_0))) + (-1259.1392167224028 / (2.0 + t_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))))) <= 2e+307) {
		tmp = t_3 * (t_2 * ((t_4 * pow((7.5 + expm1(log1p(-z))), ((1.0 - z) - 0.5))) * exp((t_5 - 7.5))));
	} else {
		tmp = t_3 * (t_2 * exp((log((sqrt(2.0) * (pow((7.5 - z), (0.5 - z)) * sqrt(((double) M_PI))))) + (z - 7.5))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = -1.0 + (1.0 - z);
	double t_1 = t_0 + 7.0;
	double t_2 = Math.PI / Math.sin((Math.PI * z));
	double t_3 = ((((-1259.1392167224028 / ((1.0 - z) - -1.0)) + (0.9999999999998099 + (676.5203681218851 / (1.0 - z)))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) + (-176.6150291621406 / ((1.0 - z) - -3.0)))) + ((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0)))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
	double t_4 = Math.sqrt((Math.PI * 2.0));
	double t_5 = (z + -1.0) - -1.0;
	double tmp;
	if ((t_2 * (((t_4 * Math.pow((0.5 + t_1), (0.5 + t_0))) * Math.exp(((t_5 - 7.0) - 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / (1.0 + t_0))) + (-1259.1392167224028 / (2.0 + t_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))))) <= 2e+307) {
		tmp = t_3 * (t_2 * ((t_4 * Math.pow((7.5 + Math.expm1(Math.log1p(-z))), ((1.0 - z) - 0.5))) * Math.exp((t_5 - 7.5))));
	} else {
		tmp = t_3 * (t_2 * Math.exp((Math.log((Math.sqrt(2.0) * (Math.pow((7.5 - z), (0.5 - z)) * Math.sqrt(Math.PI)))) + (z - 7.5))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = -1.0 + (1.0 - z)
	t_1 = t_0 + 7.0
	t_2 = math.pi / math.sin((math.pi * z))
	t_3 = ((((-1259.1392167224028 / ((1.0 - z) - -1.0)) + (0.9999999999998099 + (676.5203681218851 / (1.0 - z)))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) + (-176.6150291621406 / ((1.0 - z) - -3.0)))) + ((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0)))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)))
	t_4 = math.sqrt((math.pi * 2.0))
	t_5 = (z + -1.0) - -1.0
	tmp = 0
	if (t_2 * (((t_4 * math.pow((0.5 + t_1), (0.5 + t_0))) * math.exp(((t_5 - 7.0) - 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / (1.0 + t_0))) + (-1259.1392167224028 / (2.0 + t_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))))) <= 2e+307:
		tmp = t_3 * (t_2 * ((t_4 * math.pow((7.5 + math.expm1(math.log1p(-z))), ((1.0 - z) - 0.5))) * math.exp((t_5 - 7.5))))
	else:
		tmp = t_3 * (t_2 * math.exp((math.log((math.sqrt(2.0) * (math.pow((7.5 - z), (0.5 - z)) * math.sqrt(math.pi)))) + (z - 7.5))))
	return tmp

Julia

function code(z)
	t_0 = Float64(-1.0 + Float64(1.0 - z))
	t_1 = Float64(t_0 + 7.0)
	t_2 = Float64(pi / sin(Float64(pi * z)))
	t_3 = Float64(Float64(Float64(Float64(Float64(-1259.1392167224028 / Float64(Float64(1.0 - z) - -1.0)) + Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 - z)))) + Float64(Float64(771.3234287776531 / Float64(Float64(1.0 - z) - -2.0)) + Float64(-176.6150291621406 / Float64(Float64(1.0 - z) - -3.0)))) + Float64(Float64(12.507343278686905 / Float64(Float64(1.0 - z) - -4.0)) + Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) - -5.0)))) + Float64(Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0)) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0))))
	t_4 = sqrt(Float64(pi * 2.0))
	t_5 = Float64(Float64(z + -1.0) - -1.0)
	tmp = 0.0
	if (Float64(t_2 * Float64(Float64(Float64(t_4 * (Float64(0.5 + t_1) ^ Float64(0.5 + t_0))) * exp(Float64(Float64(t_5 - 7.0) - 0.5))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 + t_0))) + Float64(-1259.1392167224028 / Float64(2.0 + t_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))))) <= 2e+307)
		tmp = Float64(t_3 * Float64(t_2 * Float64(Float64(t_4 * (Float64(7.5 + expm1(log1p(Float64(-z)))) ^ Float64(Float64(1.0 - z) - 0.5))) * exp(Float64(t_5 - 7.5)))));
	else
		tmp = Float64(t_3 * Float64(t_2 * exp(Float64(log(Float64(sqrt(2.0) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * sqrt(pi)))) + Float64(z - 7.5)))));
	end
	return tmp
end

Wolfram

code[z_] := Block[{t$95$0 = N[(-1.0 + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 7.0), $MachinePrecision]}, Block[{t$95$2 = N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(-1259.1392167224028 / N[(N[(1.0 - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(N[(1.0 - z), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] - -4.0), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] - -5.0), $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]}, Block[{t$95$4 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(z + -1.0), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[N[(t$95$2 * N[(N[(N[(t$95$4 * N[Power[N[(0.5 + t$95$1), $MachinePrecision], N[(0.5 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t$95$5 - 7.0), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 + t$95$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], 2e+307], N[(t$95$3 * N[(t$95$2 * N[(N[(t$95$4 * N[Power[N[(7.5 + N[(Exp[N[Log[1 + (-z)], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(t$95$5 - 7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * N[(t$95$2 * N[Exp[N[(N[Log[N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(z - 7.5), $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) 2)) (pow.f64 (+.f64 (+.f64 (-.f64 (-.f64 1 z) 1) 7) 1/2) (+.f64 (-.f64 (-.f64 1 z) 1) 1/2))) (exp.f64 (neg.f64 (+.f64 (+.f64 (-.f64 (-.f64 1 z) 1) 7) 1/2)))) (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 9999999999998099/10000000000000000 (/.f64 6765203681218851/10000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 1))) (/.f64 -3147848041806007/2500000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 2))) (/.f64 7713234287776531/10000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 3))) (/.f64 -883075145810703/5000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 4))) (/.f64 2501468655737381/200000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 5))) (/.f64 -3464277381643003/25000000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 6))) (/.f64 2496092394504893/250000000000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 7))) (/.f64 3764081837873279/25000000000000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 8))))) < 1.99999999999999997e307

   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 98.9%

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

      1. expm1-log1p-u 98.9%

         \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - z\right) + -1\right)\right)} + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

      2. metadata-eval 98.9%

         \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - z\right) + \color{blue}{\left(-1\right)}\right)\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

      3. sub-neg 98.9%

         \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(1 - z\right) - 1}\right)\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

      4. add-exp-log 98.9%

         \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(1 - z\right)}} - 1\right)\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

      5. expm1-def 98.9%

         \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(1 - z\right)\right)}\right)\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

      6. log1p-expm1 98.9%

         \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\mathsf{expm1}\left(\color{blue}{\log \left(1 - z\right)}\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

      7. sub-neg 98.9%

         \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\mathsf{expm1}\left(\log \color{blue}{\left(1 + \left(-z\right)\right)}\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

      8. log1p-udef 98.9%

         \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(-z\right)}\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   5. Applied egg-rr 98.9%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-z\right)\right)} + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   if 1.99999999999999997e307 < (*.f64 (/.f64 (PI.f64) (sin.f64 (*.f64 (PI.f64) z))) (*.f64 (*.f64 (*.f64 (sqrt.f64 (*.f64 (PI.f64) 2)) (pow.f64 (+.f64 (+.f64 (-.f64 (-.f64 1 z) 1) 7) 1/2) (+.f64 (-.f64 (-.f64 1 z) 1) 1/2))) (exp.f64 (neg.f64 (+.f64 (+.f64 (-.f64 (-.f64 1 z) 1) 7) 1/2)))) (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 9999999999998099/10000000000000000 (/.f64 6765203681218851/10000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 1))) (/.f64 -3147848041806007/2500000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 2))) (/.f64 7713234287776531/10000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 3))) (/.f64 -883075145810703/5000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 4))) (/.f64 2501468655737381/200000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 5))) (/.f64 -3464277381643003/25000000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 6))) (/.f64 2496092394504893/250000000000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 7))) (/.f64 3764081837873279/25000000000000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 8)))))

   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%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in z around inf 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(0.5 + \left(\left(-1 + \left(1 - z\right)\right) + 7\right)\right)}^{\left(0.5 + \left(-1 + \left(1 - z\right)\right)\right)}\right) \cdot e^{\left(\left(\left(z + -1\right) - -1\right) - 7\right) - 0.5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 + \left(-1 + \left(1 - z\right)\right)}\right) + \frac{-1259.1392167224028}{2 + \left(-1 + \left(1 - z\right)\right)}\right) + \frac{771.3234287776531}{\left(-1 + \left(1 - z\right)\right) + 3}\right) + \frac{-176.6150291621406}{\left(-1 + \left(1 - z\right)\right) + 4}\right) + \frac{12.507343278686905}{\left(-1 + \left(1 - z\right)\right) + 5}\right) + \frac{-0.13857109526572012}{\left(-1 + \left(1 - z\right)\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(-1 + \left(1 - z\right)\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(-1 + \left(1 - z\right)\right) + 8}\right)\right) \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\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) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \mathsf{expm1}\left(\mathsf{log1p}\left(-z\right)\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{\left(\left(z + -1\right) - -1\right) - 7.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\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) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\log \left(\sqrt{2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi}\right)\right) + \left(z - 7.5\right)}\right)\\ \end{array} \]

Alternative 2: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \left(1 - z\right)\\ t_1 := t_0 + 7\\ t_2 := \frac{\pi}{\sin \left(\pi \cdot z\right)}\\ t_3 := \frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\\ t_4 := \frac{676.5203681218851}{1 - z}\\ t_5 := \sqrt{\pi \cdot 2}\\ t_6 := \left(z + -1\right) - -1\\ t_7 := \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\\ t_8 := \frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\\ \mathbf{if}\;t_2 \cdot \left(\left(\left(t_5 \cdot {\left(0.5 + t_1\right)}^{\left(0.5 + t_0\right)}\right) \cdot e^{\left(t_6 - 7\right) - 0.5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 + t_0}\right) + \frac{-1259.1392167224028}{2 + t_0}\right) + \frac{771.3234287776531}{t_0 + 3}\right) + \frac{-176.6150291621406}{t_0 + 4}\right) + \frac{12.507343278686905}{t_0 + 5}\right) + \frac{-0.13857109526572012}{t_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t_1}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t_0 + 8}\right)\right) \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\left(t_2 \cdot \left(e^{t_6 - 7.5} \cdot \left(t_5 \cdot {\left(7.5 + t_0\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right)\right)\right) \cdot \left(t_7 + \left(t_3 + \left(t_8 + \left(0.9999999999998099 + \left(t_4 + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \left(0.9999999999998099 + t_4\right)\right) + t_8\right) + t_3\right) + t_7\right) \cdot \left(t_2 \cdot e^{\log \left(\sqrt{2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi}\right)\right) + \left(z - 7.5\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ -1.0 (- 1.0 z)))
        (t_1 (+ t_0 7.0))
        (t_2 (/ PI (sin (* PI z))))
        (t_3
         (+
          (/ 12.507343278686905 (- (- 1.0 z) -4.0))
          (/ -0.13857109526572012 (- (- 1.0 z) -5.0))))
        (t_4 (/ 676.5203681218851 (- 1.0 z)))
        (t_5 (sqrt (* PI 2.0)))
        (t_6 (- (+ z -1.0) -1.0))
        (t_7
         (+
          (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))
          (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0))))
        (t_8
         (+
          (/ 771.3234287776531 (- (- 1.0 z) -2.0))
          (/ -176.6150291621406 (- (- 1.0 z) -3.0)))))
   (if (<=
        (*
         t_2
         (*
          (* (* t_5 (pow (+ 0.5 t_1) (+ 0.5 t_0))) (exp (- (- t_6 7.0) 0.5)))
          (+
           (+
            (+
             (+
              (+
               (+
                (+
                 (+ 0.9999999999998099 (/ 676.5203681218851 (+ 1.0 t_0)))
                 (/ -1259.1392167224028 (+ 2.0 t_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)))))
        2e+307)
     (*
      (* t_2 (* (exp (- t_6 7.5)) (* t_5 (pow (+ 7.5 t_0) (- (- 1.0 z) 0.5)))))
      (+
       t_7
       (+
        t_3
        (+
         t_8
         (+ 0.9999999999998099 (+ t_4 (/ -1259.1392167224028 (- 2.0 z))))))))
     (*
      (+
       (+
        (+
         (+
          (/ -1259.1392167224028 (- (- 1.0 z) -1.0))
          (+ 0.9999999999998099 t_4))
         t_8)
        t_3)
       t_7)
      (*
       t_2
       (exp
        (+
         (log (* (sqrt 2.0) (* (pow (- 7.5 z) (- 0.5 z)) (sqrt PI))))
         (- z 7.5))))))))

C

double code(double z) {
	double t_0 = -1.0 + (1.0 - z);
	double t_1 = t_0 + 7.0;
	double t_2 = ((double) M_PI) / sin((((double) M_PI) * z));
	double t_3 = (12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0));
	double t_4 = 676.5203681218851 / (1.0 - z);
	double t_5 = sqrt((((double) M_PI) * 2.0));
	double t_6 = (z + -1.0) - -1.0;
	double t_7 = (9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0));
	double t_8 = (771.3234287776531 / ((1.0 - z) - -2.0)) + (-176.6150291621406 / ((1.0 - z) - -3.0));
	double tmp;
	if ((t_2 * (((t_5 * pow((0.5 + t_1), (0.5 + t_0))) * exp(((t_6 - 7.0) - 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / (1.0 + t_0))) + (-1259.1392167224028 / (2.0 + t_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))))) <= 2e+307) {
		tmp = (t_2 * (exp((t_6 - 7.5)) * (t_5 * pow((7.5 + t_0), ((1.0 - z) - 0.5))))) * (t_7 + (t_3 + (t_8 + (0.9999999999998099 + (t_4 + (-1259.1392167224028 / (2.0 - z)))))));
	} else {
		tmp = (((((-1259.1392167224028 / ((1.0 - z) - -1.0)) + (0.9999999999998099 + t_4)) + t_8) + t_3) + t_7) * (t_2 * exp((log((sqrt(2.0) * (pow((7.5 - z), (0.5 - z)) * sqrt(((double) M_PI))))) + (z - 7.5))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = -1.0 + (1.0 - z);
	double t_1 = t_0 + 7.0;
	double t_2 = Math.PI / Math.sin((Math.PI * z));
	double t_3 = (12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0));
	double t_4 = 676.5203681218851 / (1.0 - z);
	double t_5 = Math.sqrt((Math.PI * 2.0));
	double t_6 = (z + -1.0) - -1.0;
	double t_7 = (9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0));
	double t_8 = (771.3234287776531 / ((1.0 - z) - -2.0)) + (-176.6150291621406 / ((1.0 - z) - -3.0));
	double tmp;
	if ((t_2 * (((t_5 * Math.pow((0.5 + t_1), (0.5 + t_0))) * Math.exp(((t_6 - 7.0) - 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / (1.0 + t_0))) + (-1259.1392167224028 / (2.0 + t_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))))) <= 2e+307) {
		tmp = (t_2 * (Math.exp((t_6 - 7.5)) * (t_5 * Math.pow((7.5 + t_0), ((1.0 - z) - 0.5))))) * (t_7 + (t_3 + (t_8 + (0.9999999999998099 + (t_4 + (-1259.1392167224028 / (2.0 - z)))))));
	} else {
		tmp = (((((-1259.1392167224028 / ((1.0 - z) - -1.0)) + (0.9999999999998099 + t_4)) + t_8) + t_3) + t_7) * (t_2 * Math.exp((Math.log((Math.sqrt(2.0) * (Math.pow((7.5 - z), (0.5 - z)) * Math.sqrt(Math.PI)))) + (z - 7.5))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = -1.0 + (1.0 - z)
	t_1 = t_0 + 7.0
	t_2 = math.pi / math.sin((math.pi * z))
	t_3 = (12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0))
	t_4 = 676.5203681218851 / (1.0 - z)
	t_5 = math.sqrt((math.pi * 2.0))
	t_6 = (z + -1.0) - -1.0
	t_7 = (9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))
	t_8 = (771.3234287776531 / ((1.0 - z) - -2.0)) + (-176.6150291621406 / ((1.0 - z) - -3.0))
	tmp = 0
	if (t_2 * (((t_5 * math.pow((0.5 + t_1), (0.5 + t_0))) * math.exp(((t_6 - 7.0) - 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / (1.0 + t_0))) + (-1259.1392167224028 / (2.0 + t_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))))) <= 2e+307:
		tmp = (t_2 * (math.exp((t_6 - 7.5)) * (t_5 * math.pow((7.5 + t_0), ((1.0 - z) - 0.5))))) * (t_7 + (t_3 + (t_8 + (0.9999999999998099 + (t_4 + (-1259.1392167224028 / (2.0 - z)))))))
	else:
		tmp = (((((-1259.1392167224028 / ((1.0 - z) - -1.0)) + (0.9999999999998099 + t_4)) + t_8) + t_3) + t_7) * (t_2 * math.exp((math.log((math.sqrt(2.0) * (math.pow((7.5 - z), (0.5 - z)) * math.sqrt(math.pi)))) + (z - 7.5))))
	return tmp

Julia

function code(z)
	t_0 = Float64(-1.0 + Float64(1.0 - z))
	t_1 = Float64(t_0 + 7.0)
	t_2 = Float64(pi / sin(Float64(pi * z)))
	t_3 = Float64(Float64(12.507343278686905 / Float64(Float64(1.0 - z) - -4.0)) + Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) - -5.0)))
	t_4 = Float64(676.5203681218851 / Float64(1.0 - z))
	t_5 = sqrt(Float64(pi * 2.0))
	t_6 = Float64(Float64(z + -1.0) - -1.0)
	t_7 = Float64(Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0)) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0)))
	t_8 = Float64(Float64(771.3234287776531 / Float64(Float64(1.0 - z) - -2.0)) + Float64(-176.6150291621406 / Float64(Float64(1.0 - z) - -3.0)))
	tmp = 0.0
	if (Float64(t_2 * Float64(Float64(Float64(t_5 * (Float64(0.5 + t_1) ^ Float64(0.5 + t_0))) * exp(Float64(Float64(t_6 - 7.0) - 0.5))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 + t_0))) + Float64(-1259.1392167224028 / Float64(2.0 + t_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))))) <= 2e+307)
		tmp = Float64(Float64(t_2 * Float64(exp(Float64(t_6 - 7.5)) * Float64(t_5 * (Float64(7.5 + t_0) ^ Float64(Float64(1.0 - z) - 0.5))))) * Float64(t_7 + Float64(t_3 + Float64(t_8 + Float64(0.9999999999998099 + Float64(t_4 + Float64(-1259.1392167224028 / Float64(2.0 - z))))))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(-1259.1392167224028 / Float64(Float64(1.0 - z) - -1.0)) + Float64(0.9999999999998099 + t_4)) + t_8) + t_3) + t_7) * Float64(t_2 * exp(Float64(log(Float64(sqrt(2.0) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * sqrt(pi)))) + Float64(z - 7.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = -1.0 + (1.0 - z);
	t_1 = t_0 + 7.0;
	t_2 = pi / sin((pi * z));
	t_3 = (12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0));
	t_4 = 676.5203681218851 / (1.0 - z);
	t_5 = sqrt((pi * 2.0));
	t_6 = (z + -1.0) - -1.0;
	t_7 = (9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0));
	t_8 = (771.3234287776531 / ((1.0 - z) - -2.0)) + (-176.6150291621406 / ((1.0 - z) - -3.0));
	tmp = 0.0;
	if ((t_2 * (((t_5 * ((0.5 + t_1) ^ (0.5 + t_0))) * exp(((t_6 - 7.0) - 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / (1.0 + t_0))) + (-1259.1392167224028 / (2.0 + t_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))))) <= 2e+307)
		tmp = (t_2 * (exp((t_6 - 7.5)) * (t_5 * ((7.5 + t_0) ^ ((1.0 - z) - 0.5))))) * (t_7 + (t_3 + (t_8 + (0.9999999999998099 + (t_4 + (-1259.1392167224028 / (2.0 - z)))))));
	else
		tmp = (((((-1259.1392167224028 / ((1.0 - z) - -1.0)) + (0.9999999999998099 + t_4)) + t_8) + t_3) + t_7) * (t_2 * exp((log((sqrt(2.0) * (((7.5 - z) ^ (0.5 - z)) * sqrt(pi)))) + (z - 7.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[(-1.0 + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 7.0), $MachinePrecision]}, Block[{t$95$2 = N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] - -4.0), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] - -5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[(z + -1.0), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$7 = 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]}, Block[{t$95$8 = N[(N[(771.3234287776531 / N[(N[(1.0 - z), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$2 * N[(N[(N[(t$95$5 * N[Power[N[(0.5 + t$95$1), $MachinePrecision], N[(0.5 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t$95$6 - 7.0), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 + t$95$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], 2e+307], N[(N[(t$95$2 * N[(N[Exp[N[(t$95$6 - 7.5), $MachinePrecision]], $MachinePrecision] * N[(t$95$5 * N[Power[N[(7.5 + t$95$0), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$7 + N[(t$95$3 + N[(t$95$8 + N[(0.9999999999998099 + N[(t$95$4 + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(-1259.1392167224028 / N[(N[(1.0 - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + t$95$4), $MachinePrecision]), $MachinePrecision] + t$95$8), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$7), $MachinePrecision] * N[(t$95$2 * N[Exp[N[(N[Log[N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(z - 7.5), $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) 2)) (pow.f64 (+.f64 (+.f64 (-.f64 (-.f64 1 z) 1) 7) 1/2) (+.f64 (-.f64 (-.f64 1 z) 1) 1/2))) (exp.f64 (neg.f64 (+.f64 (+.f64 (-.f64 (-.f64 1 z) 1) 7) 1/2)))) (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 9999999999998099/10000000000000000 (/.f64 6765203681218851/10000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 1))) (/.f64 -3147848041806007/2500000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 2))) (/.f64 7713234287776531/10000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 3))) (/.f64 -883075145810703/5000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 4))) (/.f64 2501468655737381/200000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 5))) (/.f64 -3464277381643003/25000000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 6))) (/.f64 2496092394504893/250000000000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 7))) (/.f64 3764081837873279/25000000000000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 8))))) < 1.99999999999999997e307

   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 98.9%

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

      1. *-un-lft-identity 98.9%

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

      2. associate-+l+ 98.9%

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

      3. --rgt-identity 98.9%

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

      4. metadata-eval 98.9%

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

      5. associate-+l- 98.9%

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

      6. +-commutative 98.9%

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

      7. expm1-log1p-u 98.9%

         \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(1 \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - z\right) - 1\right)\right)}}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

      8. add-exp-log 98.9%

         \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(1 \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(1 - z\right)}} - 1\right)\right)}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

      9. expm1-def 98.9%

         \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(1 \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(1 - z\right)\right)}\right)\right)}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

      10. log1p-expm1 98.9%

          \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(1 \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \mathsf{expm1}\left(\color{blue}{\log \left(1 - z\right)}\right)}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

      11. sub-neg 98.9%

          \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(1 \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \mathsf{expm1}\left(\log \color{blue}{\left(1 + \left(-z\right)\right)}\right)}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

      12. log1p-udef 98.9%

          \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(1 \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(-z\right)}\right)}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

      13. expm1-log1p-u 98.9%

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

      14. sub-neg 98.9%

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

   5. Applied egg-rr 98.9%

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

   if 1.99999999999999997e307 < (*.f64 (/.f64 (PI.f64) (sin.f64 (*.f64 (PI.f64) z))) (*.f64 (*.f64 (*.f64 (sqrt.f64 (*.f64 (PI.f64) 2)) (pow.f64 (+.f64 (+.f64 (-.f64 (-.f64 1 z) 1) 7) 1/2) (+.f64 (-.f64 (-.f64 1 z) 1) 1/2))) (exp.f64 (neg.f64 (+.f64 (+.f64 (-.f64 (-.f64 1 z) 1) 7) 1/2)))) (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 9999999999998099/10000000000000000 (/.f64 6765203681218851/10000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 1))) (/.f64 -3147848041806007/2500000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 2))) (/.f64 7713234287776531/10000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 3))) (/.f64 -883075145810703/5000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 4))) (/.f64 2501468655737381/200000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 5))) (/.f64 -3464277381643003/25000000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 6))) (/.f64 2496092394504893/250000000000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 7))) (/.f64 3764081837873279/25000000000000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 8)))))

   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%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in z around inf 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(0.5 + \left(\left(-1 + \left(1 - z\right)\right) + 7\right)\right)}^{\left(0.5 + \left(-1 + \left(1 - z\right)\right)\right)}\right) \cdot e^{\left(\left(\left(z + -1\right) - -1\right) - 7\right) - 0.5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 + \left(-1 + \left(1 - z\right)\right)}\right) + \frac{-1259.1392167224028}{2 + \left(-1 + \left(1 - z\right)\right)}\right) + \frac{771.3234287776531}{\left(-1 + \left(1 - z\right)\right) + 3}\right) + \frac{-176.6150291621406}{\left(-1 + \left(1 - z\right)\right) + 4}\right) + \frac{12.507343278686905}{\left(-1 + \left(1 - z\right)\right) + 5}\right) + \frac{-0.13857109526572012}{\left(-1 + \left(1 - z\right)\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(-1 + \left(1 - z\right)\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(-1 + \left(1 - z\right)\right) + 8}\right)\right) \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(e^{\left(\left(z + -1\right) - -1\right) - 7.5} \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \left(-1 + \left(1 - z\right)\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right)\right)\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right) + \left(\left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\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) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\log \left(\sqrt{2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi}\right)\right) + \left(z - 7.5\right)}\right)\\ \end{array} \]

Alternative 3: 98.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[z_] := N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Log[N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(-1.0 * z + 7.5), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(-1.0 * z + 7.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(-1259.1392167224028 / N[(N[(1.0 - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(N[(1.0 - z), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] - -4.0), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] - -5.0), $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. Initial program 95.4%

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

3. Simplified 97.0%

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

5. Applied egg-rr 98.0%

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

6. Final simplification 98.0%

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

Alternative 4: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(z)
	return Float64(Float64(Float64(pi / sin(Float64(pi * z))) * Float64(exp(Float64(Float64(Float64(z + -1.0) - -1.0) - 7.5)) * Float64(sqrt(Float64(pi * 2.0)) * (Float64(7.5 + Float64(-1.0 + Float64(1.0 - z))) ^ Float64(Float64(1.0 - z) - 0.5))))) * Float64(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(Float64(12.507343278686905 / Float64(Float64(1.0 - z) - -4.0)) + Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) - -5.0))) + Float64(Float64(Float64(771.3234287776531 / Float64(Float64(1.0 - z) - -2.0)) + Float64(-176.6150291621406 / Float64(Float64(1.0 - z) - -3.0))) + Float64(0.9999999999998099 + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(-1259.1392167224028 / Float64(2.0 - z))))))))
end

MATLAB

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

Wolfram

code[z_] := N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(N[(N[(z + -1.0), $MachinePrecision] - -1.0), $MachinePrecision] - 7.5), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 + N[(-1.0 + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(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] + N[(N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] - -4.0), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] - -5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(771.3234287776531 / N[(N[(1.0 - z), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.4%

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

3. Simplified 97.0%

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

   1. *-un-lft-identity 97.0%

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

   2. associate-+l+ 97.0%

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

   3. --rgt-identity 97.0%

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

   4. metadata-eval 97.0%

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

   5. associate-+l- 97.0%

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

   6. +-commutative 97.0%

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

   7. expm1-log1p-u 97.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(1 \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - z\right) - 1\right)\right)}}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   8. add-exp-log 97.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(1 \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(1 - z\right)}} - 1\right)\right)}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   9. expm1-def 97.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(1 \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(1 - z\right)\right)}\right)\right)}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   10. log1p-expm1 97.0%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(1 \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \mathsf{expm1}\left(\color{blue}{\log \left(1 - z\right)}\right)}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   11. sub-neg 97.0%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(1 \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \mathsf{expm1}\left(\log \color{blue}{\left(1 + \left(-z\right)\right)}\right)}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   12. log1p-udef 97.0%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(1 \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(-z\right)}\right)}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   13. expm1-log1p-u 97.0%

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

   14. sub-neg 97.0%

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

5. Applied egg-rr 97.0%

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

6. Final simplification 97.0%

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

Alternative 5: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(z)
	return Float64(Float64(Float64(Float64(Float64(Float64(-1259.1392167224028 / Float64(Float64(1.0 - z) - -1.0)) + Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 - z)))) + Float64(Float64(771.3234287776531 / Float64(Float64(1.0 - z) - -2.0)) + Float64(-176.6150291621406 / Float64(Float64(1.0 - z) - -3.0)))) + Float64(Float64(12.507343278686905 / Float64(Float64(1.0 - z) - -4.0)) + Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) - -5.0)))) + 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(pi / sin(Float64(pi * z))) * Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5))))))
end

MATLAB

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

Wolfram

code[z_] := N[(N[(N[(N[(N[(N[(-1259.1392167224028 / N[(N[(1.0 - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(N[(1.0 - z), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] - -4.0), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] - -5.0), $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[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.4%

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

3. Simplified 97.0%

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

5. Applied egg-rr 88.1%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)} \cdot e^{-\mathsf{fma}\left(-1, z, 7.5\right)}\right)\right)} - 1\right)}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
6. Step-by-step derivation

   1. expm1-def 97.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)} \cdot e^{-\mathsf{fma}\left(-1, z, 7.5\right)}\right)\right)\right)}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   2. expm1-log1p 97.0%

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

   3. *-commutative 97.0%

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

7. Simplified 97.0%

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

8. Final simplification 97.0%

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

Alternative 6: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 95.1%

   \[\leadsto \color{blue}{\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)} \]

5. Applied egg-rr 42.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot e^{-\mathsf{fma}\left(-1, z, 7.5\right)}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{3 - \left(1 + z\right)}\right) + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \left(\frac{-0.13857109526572012}{1 - \left(z + -5\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(1 + z\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(1 + z\right)}\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right)\right)\right)} - 1} \]

7. Simplified 96.8%

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

8. Final simplification 96.8%

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

Alternative 7: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ {\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{\pi} \cdot \left(263.3831869810514 \cdot \frac{\sqrt{2}}{\frac{z}{e^{-7.5}}} + \sqrt{2} \cdot \left(e^{-7.5} \cdot 700.279359537391\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (pow (- 7.5 z) (- 0.5 z))
  (*
   (sqrt PI)
   (+
    (* 263.3831869810514 (/ (sqrt 2.0) (/ z (exp -7.5))))
    (* (sqrt 2.0) (* (exp -7.5) 700.279359537391))))))

C

double code(double z) {
	return pow((7.5 - z), (0.5 - z)) * (sqrt(((double) M_PI)) * ((263.3831869810514 * (sqrt(2.0) / (z / exp(-7.5)))) + (sqrt(2.0) * (exp(-7.5) * 700.279359537391))));
}

Java

public static double code(double z) {
	return Math.pow((7.5 - z), (0.5 - z)) * (Math.sqrt(Math.PI) * ((263.3831869810514 * (Math.sqrt(2.0) / (z / Math.exp(-7.5)))) + (Math.sqrt(2.0) * (Math.exp(-7.5) * 700.279359537391))));
}

Python

def code(z):
	return math.pow((7.5 - z), (0.5 - z)) * (math.sqrt(math.pi) * ((263.3831869810514 * (math.sqrt(2.0) / (z / math.exp(-7.5)))) + (math.sqrt(2.0) * (math.exp(-7.5) * 700.279359537391))))

Julia

function code(z)
	return Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * Float64(sqrt(pi) * Float64(Float64(263.3831869810514 * Float64(sqrt(2.0) / Float64(z / exp(-7.5)))) + Float64(sqrt(2.0) * Float64(exp(-7.5) * 700.279359537391)))))
end

MATLAB

function tmp = code(z)
	tmp = ((7.5 - z) ^ (0.5 - z)) * (sqrt(pi) * ((263.3831869810514 * (sqrt(2.0) / (z / exp(-7.5)))) + (sqrt(2.0) * (exp(-7.5) * 700.279359537391))));
end

Wolfram

code[z_] := N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[(263.3831869810514 * N[(N[Sqrt[2.0], $MachinePrecision] / N[(z / N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] * 700.279359537391), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.4%

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

3. Simplified 95.1%

   \[\leadsto \color{blue}{\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)} \]

5. Applied egg-rr 42.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot e^{-\mathsf{fma}\left(-1, z, 7.5\right)}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{3 - \left(1 + z\right)}\right) + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \left(\frac{-0.13857109526572012}{1 - \left(z + -5\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(1 + z\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(1 + z\right)}\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right)\right)\right)} - 1} \]

7. Simplified 94.9%

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

8. Taylor expanded in z around 0 96.8%

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

   1. associate-*r* 96.8%

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

   2. distribute-rgt-out 96.8%

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

   3. associate-/l* 96.8%

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

   4. distribute-rgt-out 96.8%

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

   5. metadata-eval 96.8%

      \[\leadsto {\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{\pi} \cdot \left(263.3831869810514 \cdot \frac{\sqrt{2}}{\frac{z}{e^{-7.5}}} + \sqrt{2} \cdot \left(e^{-7.5} \cdot \color{blue}{700.279359537391}\right)\right)\right) \]

10. Simplified 96.8%

    \[\leadsto {\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \color{blue}{\left(\sqrt{\pi} \cdot \left(263.3831869810514 \cdot \frac{\sqrt{2}}{\frac{z}{e^{-7.5}}} + \sqrt{2} \cdot \left(e^{-7.5} \cdot 700.279359537391\right)\right)\right)} \]

11. Final simplification 96.8%

    \[\leadsto {\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{\pi} \cdot \left(263.3831869810514 \cdot \frac{\sqrt{2}}{\frac{z}{e^{-7.5}}} + \sqrt{2} \cdot \left(e^{-7.5} \cdot 700.279359537391\right)\right)\right) \]

Alternative 8: 96.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 95.1%

   \[\leadsto \color{blue}{\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)} \]

5. Applied egg-rr 42.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot e^{-\mathsf{fma}\left(-1, z, 7.5\right)}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{3 - \left(1 + z\right)}\right) + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \left(\frac{-0.13857109526572012}{1 - \left(z + -5\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(1 + z\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(1 + z\right)}\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right)\right)\right)} - 1} \]

7. Simplified 94.9%

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

8. Taylor expanded in z around 0 95.3%

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

9. Final simplification 95.3%

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

Alternative 9: 96.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 95.1%

   \[\leadsto \color{blue}{\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)} \]

4. Taylor expanded in z around 0 94.5%

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

   1. *-commutative 94.5%

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

   2. associate-/l* 94.8%

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

6. Simplified 94.8%

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

7. Final simplification 94.8%

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

Alternative 10: 96.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ 263.3831869810514 \cdot \frac{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot 15\right)\right)}}{z \cdot e^{7.5}} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (* 263.3831869810514 (/ (sqrt (expm1 (log1p (* PI 15.0)))) (* z (exp 7.5)))))

C

double code(double z) {
	return 263.3831869810514 * (sqrt(expm1(log1p((((double) M_PI) * 15.0)))) / (z * exp(7.5)));
}

Java

public static double code(double z) {
	return 263.3831869810514 * (Math.sqrt(Math.expm1(Math.log1p((Math.PI * 15.0)))) / (z * Math.exp(7.5)));
}

Python

def code(z):
	return 263.3831869810514 * (math.sqrt(math.expm1(math.log1p((math.pi * 15.0)))) / (z * math.exp(7.5)))

Julia

function code(z)
	return Float64(263.3831869810514 * Float64(sqrt(expm1(log1p(Float64(pi * 15.0)))) / Float64(z * exp(7.5))))
end

Wolfram

code[z_] := N[(263.3831869810514 * N[(N[Sqrt[N[(Exp[N[Log[1 + N[(Pi * 15.0), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] / N[(z * N[Exp[7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.4%

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

3. Simplified 95.1%

   \[\leadsto \color{blue}{\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)} \]

4. Taylor expanded in z around 0 94.5%

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

   1. *-commutative 94.5%

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

   2. associate-/l* 94.8%

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

6. Simplified 94.8%

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

   1. expm1-log1p-u 41.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(263.3831869810514 \cdot \left(\sqrt{\pi} \cdot \frac{\sqrt{2}}{\frac{z}{e^{-7.5} \cdot \sqrt{7.5}}}\right)\right)\right)} \]

   2. expm1-udef 41.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(263.3831869810514 \cdot \left(\sqrt{\pi} \cdot \frac{\sqrt{2}}{\frac{z}{e^{-7.5} \cdot \sqrt{7.5}}}\right)\right)} - 1} \]

   3. associate-*r/ 41.9%

      \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \color{blue}{\frac{\sqrt{\pi} \cdot \sqrt{2}}{\frac{z}{e^{-7.5} \cdot \sqrt{7.5}}}}\right)} - 1 \]

   4. sqrt-prod 41.9%

      \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{\color{blue}{\sqrt{\pi \cdot 2}}}{\frac{z}{e^{-7.5} \cdot \sqrt{7.5}}}\right)} - 1 \]

   5. associate-/r* 41.9%

      \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{\sqrt{\pi \cdot 2}}{\color{blue}{\frac{\frac{z}{e^{-7.5}}}{\sqrt{7.5}}}}\right)} - 1 \]

8. Applied egg-rr 41.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{\sqrt{\pi \cdot 2}}{\frac{\frac{z}{e^{-7.5}}}{\sqrt{7.5}}}\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 41.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(263.3831869810514 \cdot \frac{\sqrt{\pi \cdot 2}}{\frac{\frac{z}{e^{-7.5}}}{\sqrt{7.5}}}\right)\right)} \]

   2. expm1-log1p 94.4%

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

   3. associate-/r/ 94.4%

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

   4. *-commutative 94.4%

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

10. Simplified 94.4%

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

    1. expm1-log1p-u 41.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(263.3831869810514 \cdot \left(\frac{\sqrt{2 \cdot \pi}}{\frac{z}{e^{-7.5}}} \cdot \sqrt{7.5}\right)\right)\right)} \]

    2. expm1-udef 41.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(263.3831869810514 \cdot \left(\frac{\sqrt{2 \cdot \pi}}{\frac{z}{e^{-7.5}}} \cdot \sqrt{7.5}\right)\right)} - 1} \]

    3. associate-*l/ 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \color{blue}{\frac{\sqrt{2 \cdot \pi} \cdot \sqrt{7.5}}{\frac{z}{e^{-7.5}}}}\right)} - 1 \]

    4. pow1/2 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{\color{blue}{{\left(2 \cdot \pi\right)}^{0.5}} \cdot \sqrt{7.5}}{\frac{z}{e^{-7.5}}}\right)} - 1 \]

    5. pow1/2 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{{\left(2 \cdot \pi\right)}^{0.5} \cdot \color{blue}{{7.5}^{0.5}}}{\frac{z}{e^{-7.5}}}\right)} - 1 \]

    6. pow-prod-down 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{\color{blue}{{\left(\left(2 \cdot \pi\right) \cdot 7.5\right)}^{0.5}}}{\frac{z}{e^{-7.5}}}\right)} - 1 \]

    7. *-commutative 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{{\left(\color{blue}{\left(\pi \cdot 2\right)} \cdot 7.5\right)}^{0.5}}{\frac{z}{e^{-7.5}}}\right)} - 1 \]

    8. div-inv 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}}{\color{blue}{z \cdot \frac{1}{e^{-7.5}}}}\right)} - 1 \]

    9. rec-exp 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}}{z \cdot \color{blue}{e^{--7.5}}}\right)} - 1 \]

    10. metadata-eval 41.9%

        \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}}{z \cdot e^{\color{blue}{7.5}}}\right)} - 1 \]

12. Applied egg-rr 41.9%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}}{z \cdot e^{7.5}}\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 41.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(263.3831869810514 \cdot \frac{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}}{z \cdot e^{7.5}}\right)\right)} \]

    2. expm1-log1p 93.9%

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

    3. unpow1/2 93.9%

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

    4. associate-*l* 93.9%

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

    5. metadata-eval 93.9%

       \[\leadsto 263.3831869810514 \cdot \frac{\sqrt{\pi \cdot \color{blue}{15}}}{z \cdot e^{7.5}} \]

14. Simplified 93.9%

    \[\leadsto \color{blue}{263.3831869810514 \cdot \frac{\sqrt{\pi \cdot 15}}{z \cdot e^{7.5}}} \]
15. Step-by-step derivation

    1. expm1-log1p-u 94.8%

       \[\leadsto 263.3831869810514 \cdot \frac{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot 15\right)\right)}}}{z \cdot e^{7.5}} \]

16. Applied egg-rr 94.8%

    \[\leadsto 263.3831869810514 \cdot \frac{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot 15\right)\right)}}}{z \cdot e^{7.5}} \]

17. Final simplification 94.8%

    \[\leadsto 263.3831869810514 \cdot \frac{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot 15\right)\right)}}{z \cdot e^{7.5}} \]

Alternative 11: 96.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 95.1%

   \[\leadsto \color{blue}{\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)} \]

4. Taylor expanded in z around 0 94.5%

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

   1. *-commutative 94.5%

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

   2. associate-/l* 94.8%

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

6. Simplified 94.8%

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

   1. expm1-log1p-u 41.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(263.3831869810514 \cdot \left(\sqrt{\pi} \cdot \frac{\sqrt{2}}{\frac{z}{e^{-7.5} \cdot \sqrt{7.5}}}\right)\right)\right)} \]

   2. expm1-udef 41.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(263.3831869810514 \cdot \left(\sqrt{\pi} \cdot \frac{\sqrt{2}}{\frac{z}{e^{-7.5} \cdot \sqrt{7.5}}}\right)\right)} - 1} \]

   3. associate-*r/ 41.9%

      \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \color{blue}{\frac{\sqrt{\pi} \cdot \sqrt{2}}{\frac{z}{e^{-7.5} \cdot \sqrt{7.5}}}}\right)} - 1 \]

   4. sqrt-prod 41.9%

      \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{\color{blue}{\sqrt{\pi \cdot 2}}}{\frac{z}{e^{-7.5} \cdot \sqrt{7.5}}}\right)} - 1 \]

   5. associate-/r* 41.9%

      \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{\sqrt{\pi \cdot 2}}{\color{blue}{\frac{\frac{z}{e^{-7.5}}}{\sqrt{7.5}}}}\right)} - 1 \]

8. Applied egg-rr 41.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{\sqrt{\pi \cdot 2}}{\frac{\frac{z}{e^{-7.5}}}{\sqrt{7.5}}}\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 41.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(263.3831869810514 \cdot \frac{\sqrt{\pi \cdot 2}}{\frac{\frac{z}{e^{-7.5}}}{\sqrt{7.5}}}\right)\right)} \]

   2. expm1-log1p 94.4%

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

   3. associate-/r/ 94.4%

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

   4. *-commutative 94.4%

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

10. Simplified 94.4%

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

    1. expm1-log1p-u 41.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(263.3831869810514 \cdot \left(\frac{\sqrt{2 \cdot \pi}}{\frac{z}{e^{-7.5}}} \cdot \sqrt{7.5}\right)\right)\right)} \]

    2. expm1-udef 41.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(263.3831869810514 \cdot \left(\frac{\sqrt{2 \cdot \pi}}{\frac{z}{e^{-7.5}}} \cdot \sqrt{7.5}\right)\right)} - 1} \]

    3. associate-*l/ 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \color{blue}{\frac{\sqrt{2 \cdot \pi} \cdot \sqrt{7.5}}{\frac{z}{e^{-7.5}}}}\right)} - 1 \]

    4. pow1/2 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{\color{blue}{{\left(2 \cdot \pi\right)}^{0.5}} \cdot \sqrt{7.5}}{\frac{z}{e^{-7.5}}}\right)} - 1 \]

    5. pow1/2 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{{\left(2 \cdot \pi\right)}^{0.5} \cdot \color{blue}{{7.5}^{0.5}}}{\frac{z}{e^{-7.5}}}\right)} - 1 \]

    6. pow-prod-down 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{\color{blue}{{\left(\left(2 \cdot \pi\right) \cdot 7.5\right)}^{0.5}}}{\frac{z}{e^{-7.5}}}\right)} - 1 \]

    7. *-commutative 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{{\left(\color{blue}{\left(\pi \cdot 2\right)} \cdot 7.5\right)}^{0.5}}{\frac{z}{e^{-7.5}}}\right)} - 1 \]

    8. div-inv 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}}{\color{blue}{z \cdot \frac{1}{e^{-7.5}}}}\right)} - 1 \]

    9. rec-exp 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}}{z \cdot \color{blue}{e^{--7.5}}}\right)} - 1 \]

    10. metadata-eval 41.9%

        \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}}{z \cdot e^{\color{blue}{7.5}}}\right)} - 1 \]

12. Applied egg-rr 41.9%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}}{z \cdot e^{7.5}}\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 41.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(263.3831869810514 \cdot \frac{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}}{z \cdot e^{7.5}}\right)\right)} \]

    2. expm1-log1p 93.9%

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

    3. unpow1/2 93.9%

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

    4. associate-*l* 93.9%

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

    5. metadata-eval 93.9%

       \[\leadsto 263.3831869810514 \cdot \frac{\sqrt{\pi \cdot \color{blue}{15}}}{z \cdot e^{7.5}} \]

14. Simplified 93.9%

    \[\leadsto \color{blue}{263.3831869810514 \cdot \frac{\sqrt{\pi \cdot 15}}{z \cdot e^{7.5}}} \]

15. Taylor expanded in z around 0 94.3%

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

    1. associate-/r* 94.5%

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

17. Simplified 94.5%

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

18. Final simplification 94.5%

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

Alternative 12: 96.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 95.1%

   \[\leadsto \color{blue}{\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)} \]

4. Taylor expanded in z around 0 94.5%

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

   1. *-commutative 94.5%

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

   2. associate-/l* 94.8%

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

6. Simplified 94.8%

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

   1. expm1-log1p-u 41.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(263.3831869810514 \cdot \left(\sqrt{\pi} \cdot \frac{\sqrt{2}}{\frac{z}{e^{-7.5} \cdot \sqrt{7.5}}}\right)\right)\right)} \]

   2. expm1-udef 41.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(263.3831869810514 \cdot \left(\sqrt{\pi} \cdot \frac{\sqrt{2}}{\frac{z}{e^{-7.5} \cdot \sqrt{7.5}}}\right)\right)} - 1} \]

   3. associate-*r/ 41.9%

      \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \color{blue}{\frac{\sqrt{\pi} \cdot \sqrt{2}}{\frac{z}{e^{-7.5} \cdot \sqrt{7.5}}}}\right)} - 1 \]

   4. sqrt-prod 41.9%

      \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{\color{blue}{\sqrt{\pi \cdot 2}}}{\frac{z}{e^{-7.5} \cdot \sqrt{7.5}}}\right)} - 1 \]

   5. associate-/r* 41.9%

      \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{\sqrt{\pi \cdot 2}}{\color{blue}{\frac{\frac{z}{e^{-7.5}}}{\sqrt{7.5}}}}\right)} - 1 \]

8. Applied egg-rr 41.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{\sqrt{\pi \cdot 2}}{\frac{\frac{z}{e^{-7.5}}}{\sqrt{7.5}}}\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 41.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(263.3831869810514 \cdot \frac{\sqrt{\pi \cdot 2}}{\frac{\frac{z}{e^{-7.5}}}{\sqrt{7.5}}}\right)\right)} \]

   2. expm1-log1p 94.4%

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

   3. associate-/r/ 94.4%

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

   4. *-commutative 94.4%

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

10. Simplified 94.4%

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

    1. expm1-log1p-u 41.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(263.3831869810514 \cdot \left(\frac{\sqrt{2 \cdot \pi}}{\frac{z}{e^{-7.5}}} \cdot \sqrt{7.5}\right)\right)\right)} \]

    2. expm1-udef 41.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(263.3831869810514 \cdot \left(\frac{\sqrt{2 \cdot \pi}}{\frac{z}{e^{-7.5}}} \cdot \sqrt{7.5}\right)\right)} - 1} \]

    3. associate-*l/ 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \color{blue}{\frac{\sqrt{2 \cdot \pi} \cdot \sqrt{7.5}}{\frac{z}{e^{-7.5}}}}\right)} - 1 \]

    4. pow1/2 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{\color{blue}{{\left(2 \cdot \pi\right)}^{0.5}} \cdot \sqrt{7.5}}{\frac{z}{e^{-7.5}}}\right)} - 1 \]

    5. pow1/2 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{{\left(2 \cdot \pi\right)}^{0.5} \cdot \color{blue}{{7.5}^{0.5}}}{\frac{z}{e^{-7.5}}}\right)} - 1 \]

    6. pow-prod-down 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{\color{blue}{{\left(\left(2 \cdot \pi\right) \cdot 7.5\right)}^{0.5}}}{\frac{z}{e^{-7.5}}}\right)} - 1 \]

    7. *-commutative 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{{\left(\color{blue}{\left(\pi \cdot 2\right)} \cdot 7.5\right)}^{0.5}}{\frac{z}{e^{-7.5}}}\right)} - 1 \]

    8. div-inv 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}}{\color{blue}{z \cdot \frac{1}{e^{-7.5}}}}\right)} - 1 \]

    9. rec-exp 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}}{z \cdot \color{blue}{e^{--7.5}}}\right)} - 1 \]

    10. metadata-eval 41.9%

        \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}}{z \cdot e^{\color{blue}{7.5}}}\right)} - 1 \]

12. Applied egg-rr 41.9%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}}{z \cdot e^{7.5}}\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 41.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(263.3831869810514 \cdot \frac{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}}{z \cdot e^{7.5}}\right)\right)} \]

    2. expm1-log1p 93.9%

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

    3. unpow1/2 93.9%

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

    4. associate-*l* 93.9%

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

    5. metadata-eval 93.9%

       \[\leadsto 263.3831869810514 \cdot \frac{\sqrt{\pi \cdot \color{blue}{15}}}{z \cdot e^{7.5}} \]

14. Simplified 93.9%

    \[\leadsto \color{blue}{263.3831869810514 \cdot \frac{\sqrt{\pi \cdot 15}}{z \cdot e^{7.5}}} \]

15. Taylor expanded in z around 0 94.3%

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

    1. associate-*l/ 93.9%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{\frac{\sqrt{15} \cdot \sqrt{\pi}}{e^{7.5} \cdot z}} \]

    2. *-commutative 93.9%

       \[\leadsto 263.3831869810514 \cdot \frac{\sqrt{15} \cdot \sqrt{\pi}}{\color{blue}{z \cdot e^{7.5}}} \]

    3. associate-/l* 94.4%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{\frac{\sqrt{15}}{\frac{z \cdot e^{7.5}}{\sqrt{\pi}}}} \]

    4. *-commutative 94.4%

       \[\leadsto 263.3831869810514 \cdot \frac{\sqrt{15}}{\frac{\color{blue}{e^{7.5} \cdot z}}{\sqrt{\pi}}} \]

    5. metadata-eval 94.4%

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

    6. rec-exp 94.4%

       \[\leadsto 263.3831869810514 \cdot \frac{\sqrt{15}}{\frac{\color{blue}{\frac{1}{e^{-7.5}}} \cdot z}{\sqrt{\pi}}} \]

    7. associate-*l/ 94.7%

       \[\leadsto 263.3831869810514 \cdot \frac{\sqrt{15}}{\frac{\color{blue}{\frac{1 \cdot z}{e^{-7.5}}}}{\sqrt{\pi}}} \]

    8. metadata-eval 94.7%

       \[\leadsto 263.3831869810514 \cdot \frac{\sqrt{15}}{\frac{\frac{\color{blue}{\left(--1\right)} \cdot z}{e^{-7.5}}}{\sqrt{\pi}}} \]

    9. distribute-lft-neg-in 94.7%

       \[\leadsto 263.3831869810514 \cdot \frac{\sqrt{15}}{\frac{\frac{\color{blue}{--1 \cdot z}}{e^{-7.5}}}{\sqrt{\pi}}} \]

    10. neg-mul-1 94.7%

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

    11. remove-double-neg 94.7%

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

17. Simplified 94.7%

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

18. Final simplification 94.7%

    \[\leadsto 263.3831869810514 \cdot \frac{\sqrt{15}}{\frac{\frac{z}{e^{-7.5}}}{\sqrt{\pi}}} \]

Alternative 13: 96.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 95.1%

   \[\leadsto \color{blue}{\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)} \]

4. Taylor expanded in z around 0 94.5%

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

   1. *-commutative 94.5%

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

   2. associate-/l* 94.8%

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

6. Simplified 94.8%

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

7. Taylor expanded in z around 0 94.5%

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

   1. *-commutative 94.5%

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

   2. associate-*l/ 94.8%

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

   3. *-commutative 94.8%

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

   4. associate-*l* 94.8%

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

   5. *-commutative 94.8%

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

   6. associate-*l* 94.8%

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

9. Simplified 94.8%

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

    1. expm1-log1p-u 41.8%

       \[\leadsto \sqrt{\pi} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(e^{-7.5} \cdot \left(\sqrt{7.5} \cdot \frac{\sqrt{2}}{z}\right)\right) \cdot 263.3831869810514\right)\right)} \]

    2. expm1-udef 41.8%

       \[\leadsto \sqrt{\pi} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(e^{-7.5} \cdot \left(\sqrt{7.5} \cdot \frac{\sqrt{2}}{z}\right)\right) \cdot 263.3831869810514\right)} - 1\right)} \]

    3. *-commutative 41.8%

       \[\leadsto \sqrt{\pi} \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{263.3831869810514 \cdot \left(e^{-7.5} \cdot \left(\sqrt{7.5} \cdot \frac{\sqrt{2}}{z}\right)\right)}\right)} - 1\right) \]

    4. associate-*r/ 41.8%

       \[\leadsto \sqrt{\pi} \cdot \left(e^{\mathsf{log1p}\left(263.3831869810514 \cdot \left(e^{-7.5} \cdot \color{blue}{\frac{\sqrt{7.5} \cdot \sqrt{2}}{z}}\right)\right)} - 1\right) \]

    5. sqrt-unprod 41.8%

       \[\leadsto \sqrt{\pi} \cdot \left(e^{\mathsf{log1p}\left(263.3831869810514 \cdot \left(e^{-7.5} \cdot \frac{\color{blue}{\sqrt{7.5 \cdot 2}}}{z}\right)\right)} - 1\right) \]

    6. metadata-eval 41.8%

       \[\leadsto \sqrt{\pi} \cdot \left(e^{\mathsf{log1p}\left(263.3831869810514 \cdot \left(e^{-7.5} \cdot \frac{\sqrt{\color{blue}{15}}}{z}\right)\right)} - 1\right) \]

11. Applied egg-rr 41.8%

    \[\leadsto \sqrt{\pi} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(263.3831869810514 \cdot \left(e^{-7.5} \cdot \frac{\sqrt{15}}{z}\right)\right)} - 1\right)} \]
12. Step-by-step derivation

    1. expm1-def 41.8%

       \[\leadsto \sqrt{\pi} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(263.3831869810514 \cdot \left(e^{-7.5} \cdot \frac{\sqrt{15}}{z}\right)\right)\right)} \]

    2. expm1-log1p 94.7%

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

    3. associate-*r* 94.7%

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

13. Simplified 94.7%

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

14. Final simplification 94.7%

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

Alternative 14: 95.7% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 95.1%

   \[\leadsto \color{blue}{\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)} \]

4. Taylor expanded in z around 0 94.5%

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

   1. *-commutative 94.5%

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

   2. associate-/l* 94.8%

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

6. Simplified 94.8%

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

   1. expm1-log1p-u 41.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(263.3831869810514 \cdot \left(\sqrt{\pi} \cdot \frac{\sqrt{2}}{\frac{z}{e^{-7.5} \cdot \sqrt{7.5}}}\right)\right)\right)} \]

   2. expm1-udef 41.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(263.3831869810514 \cdot \left(\sqrt{\pi} \cdot \frac{\sqrt{2}}{\frac{z}{e^{-7.5} \cdot \sqrt{7.5}}}\right)\right)} - 1} \]

   3. associate-*r/ 41.9%

      \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \color{blue}{\frac{\sqrt{\pi} \cdot \sqrt{2}}{\frac{z}{e^{-7.5} \cdot \sqrt{7.5}}}}\right)} - 1 \]

   4. sqrt-prod 41.9%

      \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{\color{blue}{\sqrt{\pi \cdot 2}}}{\frac{z}{e^{-7.5} \cdot \sqrt{7.5}}}\right)} - 1 \]

   5. associate-/r* 41.9%

      \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{\sqrt{\pi \cdot 2}}{\color{blue}{\frac{\frac{z}{e^{-7.5}}}{\sqrt{7.5}}}}\right)} - 1 \]

8. Applied egg-rr 41.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{\sqrt{\pi \cdot 2}}{\frac{\frac{z}{e^{-7.5}}}{\sqrt{7.5}}}\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 41.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(263.3831869810514 \cdot \frac{\sqrt{\pi \cdot 2}}{\frac{\frac{z}{e^{-7.5}}}{\sqrt{7.5}}}\right)\right)} \]

   2. expm1-log1p 94.4%

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

   3. associate-/r/ 94.4%

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

   4. *-commutative 94.4%

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

10. Simplified 94.4%

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

    1. expm1-log1p-u 41.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(263.3831869810514 \cdot \left(\frac{\sqrt{2 \cdot \pi}}{\frac{z}{e^{-7.5}}} \cdot \sqrt{7.5}\right)\right)\right)} \]

    2. expm1-udef 41.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(263.3831869810514 \cdot \left(\frac{\sqrt{2 \cdot \pi}}{\frac{z}{e^{-7.5}}} \cdot \sqrt{7.5}\right)\right)} - 1} \]

    3. associate-*l/ 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \color{blue}{\frac{\sqrt{2 \cdot \pi} \cdot \sqrt{7.5}}{\frac{z}{e^{-7.5}}}}\right)} - 1 \]

    4. pow1/2 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{\color{blue}{{\left(2 \cdot \pi\right)}^{0.5}} \cdot \sqrt{7.5}}{\frac{z}{e^{-7.5}}}\right)} - 1 \]

    5. pow1/2 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{{\left(2 \cdot \pi\right)}^{0.5} \cdot \color{blue}{{7.5}^{0.5}}}{\frac{z}{e^{-7.5}}}\right)} - 1 \]

    6. pow-prod-down 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{\color{blue}{{\left(\left(2 \cdot \pi\right) \cdot 7.5\right)}^{0.5}}}{\frac{z}{e^{-7.5}}}\right)} - 1 \]

    7. *-commutative 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{{\left(\color{blue}{\left(\pi \cdot 2\right)} \cdot 7.5\right)}^{0.5}}{\frac{z}{e^{-7.5}}}\right)} - 1 \]

    8. div-inv 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}}{\color{blue}{z \cdot \frac{1}{e^{-7.5}}}}\right)} - 1 \]

    9. rec-exp 41.9%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}}{z \cdot \color{blue}{e^{--7.5}}}\right)} - 1 \]

    10. metadata-eval 41.9%

        \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}}{z \cdot e^{\color{blue}{7.5}}}\right)} - 1 \]

12. Applied egg-rr 41.9%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(263.3831869810514 \cdot \frac{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}}{z \cdot e^{7.5}}\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 41.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(263.3831869810514 \cdot \frac{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}}{z \cdot e^{7.5}}\right)\right)} \]

    2. expm1-log1p 93.9%

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

    3. unpow1/2 93.9%

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

    4. associate-*l* 93.9%

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

    5. metadata-eval 93.9%

       \[\leadsto 263.3831869810514 \cdot \frac{\sqrt{\pi \cdot \color{blue}{15}}}{z \cdot e^{7.5}} \]

14. Simplified 93.9%

    \[\leadsto \color{blue}{263.3831869810514 \cdot \frac{\sqrt{\pi \cdot 15}}{z \cdot e^{7.5}}} \]
15. Step-by-step derivation

    1. expm1-log1p-u 41.9%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{\pi \cdot 15}}{z \cdot e^{7.5}}\right)\right)} \]

    2. expm1-udef 41.9%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sqrt{\pi \cdot 15}}{z \cdot e^{7.5}}\right)} - 1\right)} \]

16. Applied egg-rr 41.9%

    \[\leadsto 263.3831869810514 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sqrt{\pi \cdot 15}}{z \cdot e^{7.5}}\right)} - 1\right)} \]
17. Step-by-step derivation

    1. expm1-def 41.9%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{\pi \cdot 15}}{z \cdot e^{7.5}}\right)\right)} \]

    2. expm1-log1p 93.9%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{\frac{\sqrt{\pi \cdot 15}}{z \cdot e^{7.5}}} \]

    3. associate-/r* 93.9%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{\frac{\frac{\sqrt{\pi \cdot 15}}{z}}{e^{7.5}}} \]

    4. rem-exp-log 41.7%

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

    5. exp-diff 41.7%

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

    6. sub-neg 41.7%

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

    7. metadata-eval 41.7%

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

    8. exp-sum 41.7%

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

    9. rem-exp-log 94.3%

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

    10. metadata-eval 94.3%

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

    11. associate-*l* 94.3%

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

    12. associate-*l* 94.3%

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

    13. metadata-eval 94.3%

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

    14. *-commutative 94.3%

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

18. Simplified 94.3%

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

19. Final simplification 94.3%

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

Reproduce

herbie shell --seed 2023268 
(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))))))