Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.6% → 98.6%

Time: 10.9s

Alternatives: 14

Speedup: 2.1×

• could not determine a ground truth

  1. z = -56080536999961.74

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: 96.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.6× 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\\ t_3 := \frac{\pi}{\sin \left(\pi \cdot z\right)}\\ t_4 := \left(-z\right) + 7\\ t_5 := t\_4 + 0.5\\ t_6 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;t\_3 \cdot \left(\left(\left(t\_6 \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) \leq 5 \cdot 10^{+14}:\\ \;\;\;\;t\_3 \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {t\_5}^{\left(\left(-z\right) + 0.5\right)}\right) \cdot e^{-t\_5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(-z\right) + 1}\right) + \frac{-1259.1392167224028}{\left(-z\right) + 2}\right) + \frac{771.3234287776531}{\left(-z\right) + 3}\right) + \frac{-176.6150291621406}{\left(-z\right) + 4}\right) + \frac{12.507343278686905}{\left(-z\right) + 5}\right) + \frac{-0.13857109526572012}{\left(-z\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_4}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(-z\right) + 8}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_3 \cdot \left(\left(t\_6 \cdot {7.5}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-7.5}\right)\right) \cdot \left(\left(\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) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\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))
        (t_3 (/ PI (sin (* PI z))))
        (t_4 (+ (- z) 7.0))
        (t_5 (+ t_4 0.5))
        (t_6 (sqrt (* PI 2.0))))
   (if (<=
        (*
         t_3
         (*
          (* (* t_6 (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)))))
        5e+14)
     (*
      t_3
      (*
       (* (* (* (sqrt PI) (sqrt 2.0)) (pow t_5 (+ (- z) 0.5))) (exp (- t_5)))
       (+
        (+
         (+
          (+
           (+
            (+
             (+
              (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- z) 1.0)))
              (/ -1259.1392167224028 (+ (- z) 2.0)))
             (/ 771.3234287776531 (+ (- z) 3.0)))
            (/ -176.6150291621406 (+ (- z) 4.0)))
           (/ 12.507343278686905 (+ (- z) 5.0)))
          (/ -0.13857109526572012 (+ (- z) 6.0)))
         (/ 9.984369578019572e-6 t_4))
        (/ 1.5056327351493116e-7 (+ (- z) 8.0)))))
     (*
      (* t_3 (* (* t_6 (pow 7.5 (- (- 1.0 z) 0.5))) (exp (- 7.5))))
      (+
       (+
        (+
         (+
          (+
           (+
            (+ 0.9999999999998099 (/ 676.5203681218851 (- (- 1.0 z) 0.0)))
            (/ -1259.1392167224028 (- (- 1.0 z) -1.0)))
           (/ 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) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = t_0 + 7.0;
	double t_2 = t_1 + 0.5;
	double t_3 = ((double) M_PI) / sin((((double) M_PI) * z));
	double t_4 = -z + 7.0;
	double t_5 = t_4 + 0.5;
	double t_6 = sqrt((((double) M_PI) * 2.0));
	double tmp;
	if ((t_3 * (((t_6 * 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))))) <= 5e+14) {
		tmp = t_3 * ((((sqrt(((double) M_PI)) * sqrt(2.0)) * pow(t_5, (-z + 0.5))) * exp(-t_5)) * ((((((((0.9999999999998099 + (676.5203681218851 / (-z + 1.0))) + (-1259.1392167224028 / (-z + 2.0))) + (771.3234287776531 / (-z + 3.0))) + (-176.6150291621406 / (-z + 4.0))) + (12.507343278686905 / (-z + 5.0))) + (-0.13857109526572012 / (-z + 6.0))) + (9.984369578019572e-6 / t_4)) + (1.5056327351493116e-7 / (-z + 8.0))));
	} else {
		tmp = (t_3 * ((t_6 * pow(7.5, ((1.0 - z) - 0.5))) * exp(-7.5))) * (((((((0.9999999999998099 + (676.5203681218851 / ((1.0 - z) - 0.0))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + (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)));
	}
	return tmp;
}

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;
	double t_3 = Math.PI / Math.sin((Math.PI * z));
	double t_4 = -z + 7.0;
	double t_5 = t_4 + 0.5;
	double t_6 = Math.sqrt((Math.PI * 2.0));
	double tmp;
	if ((t_3 * (((t_6 * 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))))) <= 5e+14) {
		tmp = t_3 * ((((Math.sqrt(Math.PI) * Math.sqrt(2.0)) * Math.pow(t_5, (-z + 0.5))) * Math.exp(-t_5)) * ((((((((0.9999999999998099 + (676.5203681218851 / (-z + 1.0))) + (-1259.1392167224028 / (-z + 2.0))) + (771.3234287776531 / (-z + 3.0))) + (-176.6150291621406 / (-z + 4.0))) + (12.507343278686905 / (-z + 5.0))) + (-0.13857109526572012 / (-z + 6.0))) + (9.984369578019572e-6 / t_4)) + (1.5056327351493116e-7 / (-z + 8.0))));
	} else {
		tmp = (t_3 * ((t_6 * Math.pow(7.5, ((1.0 - z) - 0.5))) * Math.exp(-7.5))) * (((((((0.9999999999998099 + (676.5203681218851 / ((1.0 - z) - 0.0))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + (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)));
	}
	return tmp;
}

Python

def code(z):
	t_0 = (1.0 - z) - 1.0
	t_1 = t_0 + 7.0
	t_2 = t_1 + 0.5
	t_3 = math.pi / math.sin((math.pi * z))
	t_4 = -z + 7.0
	t_5 = t_4 + 0.5
	t_6 = math.sqrt((math.pi * 2.0))
	tmp = 0
	if (t_3 * (((t_6 * 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))))) <= 5e+14:
		tmp = t_3 * ((((math.sqrt(math.pi) * math.sqrt(2.0)) * math.pow(t_5, (-z + 0.5))) * math.exp(-t_5)) * ((((((((0.9999999999998099 + (676.5203681218851 / (-z + 1.0))) + (-1259.1392167224028 / (-z + 2.0))) + (771.3234287776531 / (-z + 3.0))) + (-176.6150291621406 / (-z + 4.0))) + (12.507343278686905 / (-z + 5.0))) + (-0.13857109526572012 / (-z + 6.0))) + (9.984369578019572e-6 / t_4)) + (1.5056327351493116e-7 / (-z + 8.0))))
	else:
		tmp = (t_3 * ((t_6 * math.pow(7.5, ((1.0 - z) - 0.5))) * math.exp(-7.5))) * (((((((0.9999999999998099 + (676.5203681218851 / ((1.0 - z) - 0.0))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + (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)))
	return tmp

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)
	t_3 = Float64(pi / sin(Float64(pi * z)))
	t_4 = Float64(Float64(-z) + 7.0)
	t_5 = Float64(t_4 + 0.5)
	t_6 = sqrt(Float64(pi * 2.0))
	tmp = 0.0
	if (Float64(t_3 * Float64(Float64(Float64(t_6 * (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))))) <= 5e+14)
		tmp = Float64(t_3 * Float64(Float64(Float64(Float64(sqrt(pi) * sqrt(2.0)) * (t_5 ^ Float64(Float64(-z) + 0.5))) * exp(Float64(-t_5))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(Float64(-z) + 1.0))) + Float64(-1259.1392167224028 / Float64(Float64(-z) + 2.0))) + Float64(771.3234287776531 / Float64(Float64(-z) + 3.0))) + Float64(-176.6150291621406 / Float64(Float64(-z) + 4.0))) + Float64(12.507343278686905 / Float64(Float64(-z) + 5.0))) + Float64(-0.13857109526572012 / Float64(Float64(-z) + 6.0))) + Float64(9.984369578019572e-6 / t_4)) + Float64(1.5056327351493116e-7 / Float64(Float64(-z) + 8.0)))));
	else
		tmp = Float64(Float64(t_3 * Float64(Float64(t_6 * (7.5 ^ Float64(Float64(1.0 - z) - 0.5))) * exp(Float64(-7.5)))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(Float64(1.0 - z) - 0.0))) + Float64(-1259.1392167224028 / Float64(Float64(1.0 - z) - -1.0))) + Float64(771.3234287776531 / Float64(Float64(1.0 - z) - -2.0))) + Float64(-176.6150291621406 / Float64(Float64(1.0 - z) - -3.0))) + Float64(12.507343278686905 / Float64(Float64(1.0 - z) - -4.0))) + Float64(Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) - -5.0)) + Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0)))) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = (1.0 - z) - 1.0;
	t_1 = t_0 + 7.0;
	t_2 = t_1 + 0.5;
	t_3 = pi / sin((pi * z));
	t_4 = -z + 7.0;
	t_5 = t_4 + 0.5;
	t_6 = sqrt((pi * 2.0));
	tmp = 0.0;
	if ((t_3 * (((t_6 * (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))))) <= 5e+14)
		tmp = t_3 * ((((sqrt(pi) * sqrt(2.0)) * (t_5 ^ (-z + 0.5))) * exp(-t_5)) * ((((((((0.9999999999998099 + (676.5203681218851 / (-z + 1.0))) + (-1259.1392167224028 / (-z + 2.0))) + (771.3234287776531 / (-z + 3.0))) + (-176.6150291621406 / (-z + 4.0))) + (12.507343278686905 / (-z + 5.0))) + (-0.13857109526572012 / (-z + 6.0))) + (9.984369578019572e-6 / t_4)) + (1.5056327351493116e-7 / (-z + 8.0))));
	else
		tmp = (t_3 * ((t_6 * (7.5 ^ ((1.0 - z) - 0.5))) * exp(-7.5))) * (((((((0.9999999999998099 + (676.5203681218851 / ((1.0 - z) - 0.0))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + (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)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 7.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + 0.5), $MachinePrecision]}, Block[{t$95$3 = N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[((-z) + 7.0), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 + 0.5), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$3 * N[(N[(N[(t$95$6 * 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], 5e+14], N[(t$95$3 * N[(N[(N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$5, N[((-z) + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$5)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[((-z) + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[((-z) + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[((-z) + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[((-z) + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[((-z) + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[((-z) + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$4), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[((-z) + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$3 * N[(N[(t$95$6 * N[Power[7.5, N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-7.5)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(N[(1.0 - z), $MachinePrecision] - 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(N[(1.0 - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(N[(1.0 - z), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] - -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] - -5.0), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (PI.f64) (sin.f64 (*.f64 (PI.f64) z))) (*.f64 (*.f64 (*.f64 (sqrt.f64 (*.f64 (PI.f64) #s(literal 2 binary64))) (pow.f64 (+.f64 (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)) #s(literal 1/2 binary64)) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 1/2 binary64)))) (exp.f64 (neg.f64 (+.f64 (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)) #s(literal 1/2 binary64))))) (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 #s(literal 9999999999998099/10000000000000000 binary64) (/.f64 #s(literal 6765203681218851/10000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 1 binary64)))) (/.f64 #s(literal -3147848041806007/2500000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 2 binary64)))) (/.f64 #s(literal 7713234287776531/10000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 3 binary64)))) (/.f64 #s(literal -883075145810703/5000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 4 binary64)))) (/.f64 #s(literal 2501468655737381/200000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 5 binary64)))) (/.f64 #s(literal -3464277381643003/25000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 6 binary64)))) (/.f64 #s(literal 2496092394504893/250000000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)))) (/.f64 #s(literal 3764081837873279/25000000000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 8 binary64)))))) < 5e14

   1. Initial program 96.6%

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

   2. Taylor expanded in z around 0

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\color{blue}{-1 \cdot z} + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\mathsf{neg}\left(z\right)\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      2. lower-neg.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in z around 0

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\color{blue}{-1 \cdot z} + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\mathsf{neg}\left(z\right)\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      2. lower-neg.f64 96.6

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

   7. Applied rewrites 96.6%

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

   8. Taylor expanded in z around 0

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\color{blue}{-1 \cdot z} + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\mathsf{neg}\left(z\right)\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      2. lower-neg.f64 96.6

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

   10. Applied rewrites 96.6%

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

   11. Taylor expanded in z around 0

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\color{blue}{-1 \cdot z} + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\mathsf{neg}\left(z\right)\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

       2. lower-neg.f64 96.6

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

   13. Applied rewrites 96.6%

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

   14. Taylor expanded in z around 0

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\color{blue}{-1 \cdot z} + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   15. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\mathsf{neg}\left(z\right)\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

       2. lower-neg.f64 96.6

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

   16. Applied rewrites 96.6%

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

   17. Taylor expanded in z around 0

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\color{blue}{-1 \cdot z} + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   18. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\mathsf{neg}\left(z\right)\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

       2. lower-neg.f64 96.6

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

   19. Applied rewrites 96.6%

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

   20. Taylor expanded in z around 0

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(-z\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\color{blue}{-1 \cdot z} + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   21. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(-z\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\mathsf{neg}\left(z\right)\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

       2. lower-neg.f64 96.6

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

   22. Applied rewrites 96.6%

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

   23. Taylor expanded in z around 0

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(-z\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(-z\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\color{blue}{-1 \cdot z} + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   24. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(-z\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(-z\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\mathsf{neg}\left(z\right)\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

       2. lower-neg.f64 96.6

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

   25. Applied rewrites 96.6%

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

   26. Taylor expanded in z around 0

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(-z\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(-z\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(-z\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\color{blue}{-1 \cdot z} + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   27. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(-z\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(-z\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(-z\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\mathsf{neg}\left(z\right)\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

       2. lower-neg.f64 96.6

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

   28. Applied rewrites 96.6%

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

   29. Taylor expanded in z around 0

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(-z\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(-z\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(-z\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(-z\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\color{blue}{-1 \cdot z} + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   30. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(-z\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(-z\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(-z\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(-z\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\mathsf{neg}\left(z\right)\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

       2. lower-neg.f64 96.6

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

   31. Applied rewrites 96.6%

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

   32. Taylor expanded in z around 0

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(-z\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(-z\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(-z\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(-z\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(-z\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\color{blue}{-1 \cdot z} + 8}\right)\right) \]
   33. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(-z\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(-z\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(-z\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(-z\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(-z\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\mathsf{neg}\left(z\right)\right) + 8}\right)\right) \]

       2. lower-neg.f64 96.6

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

   34. Applied rewrites 96.6%

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

       1. lift-sqrt.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\sqrt{\pi \cdot 2}} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(-z\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(-z\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(-z\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(-z\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(-z\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(-z\right) + 8}\right)\right) \]

       2. lift-PI.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)} \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(-z\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(-z\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(-z\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(-z\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(-z\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(-z\right) + 8}\right)\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right) \cdot 2}} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(-z\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(-z\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(-z\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(-z\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(-z\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(-z\right) + 8}\right)\right) \]

       4. sqrt-prod N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(-z\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(-z\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(-z\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(-z\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(-z\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(-z\right) + 8}\right)\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(-z\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(-z\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(-z\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(-z\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(-z\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(-z\right) + 8}\right)\right) \]

       6. lower-sqrt.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(-z\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(-z\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(-z\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(-z\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(-z\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(-z\right) + 8}\right)\right) \]

       7. lift-PI.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\color{blue}{\pi}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(-z\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(-z\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(-z\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(-z\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(-z\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(-z\right) + 8}\right)\right) \]

       8. lower-sqrt.f64 96.8

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

   36. Applied rewrites 96.8%

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

   if 5e14 < (*.f64 (/.f64 (PI.f64) (sin.f64 (*.f64 (PI.f64) z))) (*.f64 (*.f64 (*.f64 (sqrt.f64 (*.f64 (PI.f64) #s(literal 2 binary64))) (pow.f64 (+.f64 (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)) #s(literal 1/2 binary64)) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 1/2 binary64)))) (exp.f64 (neg.f64 (+.f64 (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)) #s(literal 1/2 binary64))))) (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 #s(literal 9999999999998099/10000000000000000 binary64) (/.f64 #s(literal 6765203681218851/10000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 1 binary64)))) (/.f64 #s(literal -3147848041806007/2500000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 2 binary64)))) (/.f64 #s(literal 7713234287776531/10000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 3 binary64)))) (/.f64 #s(literal -883075145810703/5000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 4 binary64)))) (/.f64 #s(literal 2501468655737381/200000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 5 binary64)))) (/.f64 #s(literal -3464277381643003/25000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 6 binary64)))) (/.f64 #s(literal 2496092394504893/250000000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)))) (/.f64 #s(literal 3764081837873279/25000000000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 8 binary64))))))

   1. Initial program 96.6%

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

   3. Applied rewrites 98.0%

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

   4. Taylor expanded in z around 0

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\color{blue}{\frac{15}{2}}}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 96.0%

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

   7. Taylor expanded in z around 0

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\frac{15}{2}}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\color{blue}{\frac{15}{2}}}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 96.5%

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

Alternative 2: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - -6\\ t_1 := t\_0 + 0.5\\ \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {t\_1}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-t\_1}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{t\_0}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) -6.0)) (t_1 (+ t_0 0.5)))
   (*
    (*
     (/ PI (sin (* PI z)))
     (*
      (* (* (sqrt PI) (sqrt 2.0)) (pow t_1 (- (- 1.0 z) 0.5)))
      (exp (- t_1))))
    (+
     (+
      (+
       (+
        (+
         (+
          (+ 0.9999999999998099 (/ 676.5203681218851 (- (- 1.0 z) 0.0)))
          (/ -1259.1392167224028 (- (- 1.0 z) -1.0)))
         (/ 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 t_0)))
     (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0))))))

C

double code(double z) {
	double t_0 = (1.0 - z) - -6.0;
	double t_1 = t_0 + 0.5;
	return ((((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt(((double) M_PI)) * sqrt(2.0)) * pow(t_1, ((1.0 - z) - 0.5))) * exp(-t_1))) * (((((((0.9999999999998099 + (676.5203681218851 / ((1.0 - z) - 0.0))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + (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 / t_0))) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - -6.0;
	double t_1 = t_0 + 0.5;
	return ((Math.PI / Math.sin((Math.PI * z))) * (((Math.sqrt(Math.PI) * Math.sqrt(2.0)) * Math.pow(t_1, ((1.0 - z) - 0.5))) * Math.exp(-t_1))) * (((((((0.9999999999998099 + (676.5203681218851 / ((1.0 - z) - 0.0))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + (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 / t_0))) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
}

Python

def code(z):
	t_0 = (1.0 - z) - -6.0
	t_1 = t_0 + 0.5
	return ((math.pi / math.sin((math.pi * z))) * (((math.sqrt(math.pi) * math.sqrt(2.0)) * math.pow(t_1, ((1.0 - z) - 0.5))) * math.exp(-t_1))) * (((((((0.9999999999998099 + (676.5203681218851 / ((1.0 - z) - 0.0))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + (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 / t_0))) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - -6.0)
	t_1 = Float64(t_0 + 0.5)
	return Float64(Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(sqrt(pi) * sqrt(2.0)) * (t_1 ^ Float64(Float64(1.0 - z) - 0.5))) * exp(Float64(-t_1)))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(Float64(1.0 - z) - 0.0))) + Float64(-1259.1392167224028 / Float64(Float64(1.0 - z) - -1.0))) + Float64(771.3234287776531 / Float64(Float64(1.0 - z) - -2.0))) + Float64(-176.6150291621406 / Float64(Float64(1.0 - z) - -3.0))) + Float64(12.507343278686905 / Float64(Float64(1.0 - z) - -4.0))) + Float64(Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) - -5.0)) + Float64(9.984369578019572e-6 / t_0))) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - -6.0;
	t_1 = t_0 + 0.5;
	tmp = ((pi / sin((pi * z))) * (((sqrt(pi) * sqrt(2.0)) * (t_1 ^ ((1.0 - z) - 0.5))) * exp(-t_1))) * (((((((0.9999999999998099 + (676.5203681218851 / ((1.0 - z) - 0.0))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + (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 / t_0))) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 0.5), $MachinePrecision]}, N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$1, N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$1)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(N[(1.0 - z), $MachinePrecision] - 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(N[(1.0 - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(N[(1.0 - z), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] - -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] - -5.0), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 96.6%

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

3. Applied rewrites 98.0%

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

   1. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\sqrt{\pi \cdot 2}} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lift-PI.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)} \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right) \cdot 2}} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   4. sqrt-prod N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   7. lift-PI.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\pi}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   8. lower-sqrt.f64 98.6

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

5. Applied rewrites 98.6%

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

Alternative 3: 97.9% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (/ PI (sin (* PI z)))
   (* (pow (- 7.5 z) (- 0.5 z)) (* (exp (- z 7.5)) (sqrt (* 2.0 PI)))))
  (+
   (+
    (+
     (+
      (+
       (+
        (+ 0.9999999999998099 (/ 676.5203681218851 (- (- 1.0 z) 0.0)))
        (/ -1259.1392167224028 (- (- 1.0 z) -1.0)))
       (/ 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))) * (pow((7.5 - z), (0.5 - z)) * (exp((z - 7.5)) * sqrt((2.0 * ((double) M_PI)))))) * (((((((0.9999999999998099 + (676.5203681218851 / ((1.0 - z) - 0.0))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + (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)));
}

Java

public static double code(double z) {
	return ((Math.PI / Math.sin((Math.PI * z))) * (Math.pow((7.5 - z), (0.5 - z)) * (Math.exp((z - 7.5)) * Math.sqrt((2.0 * Math.PI))))) * (((((((0.9999999999998099 + (676.5203681218851 / ((1.0 - z) - 0.0))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + (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)));
}

Python

def code(z):
	return ((math.pi / math.sin((math.pi * z))) * (math.pow((7.5 - z), (0.5 - z)) * (math.exp((z - 7.5)) * math.sqrt((2.0 * math.pi))))) * (((((((0.9999999999998099 + (676.5203681218851 / ((1.0 - z) - 0.0))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + (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))) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * Float64(exp(Float64(z - 7.5)) * sqrt(Float64(2.0 * pi))))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(Float64(1.0 - z) - 0.0))) + Float64(-1259.1392167224028 / Float64(Float64(1.0 - z) - -1.0))) + Float64(771.3234287776531 / Float64(Float64(1.0 - z) - -2.0))) + Float64(-176.6150291621406 / Float64(Float64(1.0 - z) - -3.0))) + Float64(12.507343278686905 / Float64(Float64(1.0 - z) - -4.0))) + Float64(Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) - -5.0)) + Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0)))) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0))))
end

MATLAB

function tmp = code(z)
	tmp = ((pi / sin((pi * z))) * (((7.5 - z) ^ (0.5 - z)) * (exp((z - 7.5)) * sqrt((2.0 * pi))))) * (((((((0.9999999999998099 + (676.5203681218851 / ((1.0 - z) - 0.0))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + (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)));
end

Wolfram

code[z_] := N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(z - 7.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(N[(1.0 - z), $MachinePrecision] - 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(N[(1.0 - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(N[(1.0 - z), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] - -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] - -5.0), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

3. Applied rewrites 98.0%

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

   1. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\sqrt{\pi \cdot 2}} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lift-PI.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)} \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right) \cdot 2}} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   4. sqrt-prod N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   7. lift-PI.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\pi}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   8. lower-sqrt.f64 98.6

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in z around inf

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]
7. Step-by-step derivation

   1. sqrt-prod N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \color{blue}{\left(e^{z - \frac{15}{2}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   3. exp-to-pow N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(\color{blue}{e^{z - \frac{15}{2}}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   4. lower-pow.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(\color{blue}{e^{z - \frac{15}{2}}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   5. lower--.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{\color{blue}{z - \frac{15}{2}}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   6. lower--.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \color{blue}{\sqrt{2 \cdot \mathsf{PI}\left(\right)}}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   8. lower-exp.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{\color{blue}{2 \cdot \mathsf{PI}\left(\right)}}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   9. lower--.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{\color{blue}{2} \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   12. lift-PI.f64 97.9

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

8. Applied rewrites 97.9%

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

Alternative 4: 97.8% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/
   (* PI (* (pow (- 7.5 z) (- 0.5 z)) (* (exp (- z 7.5)) (sqrt (* 2.0 PI)))))
   (sin (* z PI)))
  (+
   (+
    (+
     (+
      (+
       (+
        (+ 0.9999999999998099 (/ 676.5203681218851 (- (- 1.0 z) 0.0)))
        (/ -1259.1392167224028 (- (- 1.0 z) -1.0)))
       (/ 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) * (pow((7.5 - z), (0.5 - z)) * (exp((z - 7.5)) * sqrt((2.0 * ((double) M_PI)))))) / sin((z * ((double) M_PI)))) * (((((((0.9999999999998099 + (676.5203681218851 / ((1.0 - z) - 0.0))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + (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)));
}

Java

public static double code(double z) {
	return ((Math.PI * (Math.pow((7.5 - z), (0.5 - z)) * (Math.exp((z - 7.5)) * Math.sqrt((2.0 * Math.PI))))) / Math.sin((z * Math.PI))) * (((((((0.9999999999998099 + (676.5203681218851 / ((1.0 - z) - 0.0))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + (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)));
}

Python

def code(z):
	return ((math.pi * (math.pow((7.5 - z), (0.5 - z)) * (math.exp((z - 7.5)) * math.sqrt((2.0 * math.pi))))) / math.sin((z * math.pi))) * (((((((0.9999999999998099 + (676.5203681218851 / ((1.0 - z) - 0.0))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + (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 * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * Float64(exp(Float64(z - 7.5)) * sqrt(Float64(2.0 * pi))))) / sin(Float64(z * pi))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(Float64(1.0 - z) - 0.0))) + Float64(-1259.1392167224028 / Float64(Float64(1.0 - z) - -1.0))) + Float64(771.3234287776531 / Float64(Float64(1.0 - z) - -2.0))) + Float64(-176.6150291621406 / Float64(Float64(1.0 - z) - -3.0))) + Float64(12.507343278686905 / Float64(Float64(1.0 - z) - -4.0))) + Float64(Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) - -5.0)) + Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0)))) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0))))
end

MATLAB

function tmp = code(z)
	tmp = ((pi * (((7.5 - z) ^ (0.5 - z)) * (exp((z - 7.5)) * sqrt((2.0 * pi))))) / sin((z * pi))) * (((((((0.9999999999998099 + (676.5203681218851 / ((1.0 - z) - 0.0))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + (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)));
end

Wolfram

code[z_] := N[(N[(N[(Pi * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(z - 7.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(N[(1.0 - z), $MachinePrecision] - 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(N[(1.0 - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(N[(1.0 - z), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] - -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] - -5.0), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

3. Applied rewrites 98.0%

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

4. Taylor expanded in z around inf

   \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right) \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)}} \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right) \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)}{\color{blue}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)}} \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

6. Applied rewrites 97.8%

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

Alternative 5: 97.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - z\right) - -6\right) + 0.5\\ \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {t\_0}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-t\_0}\right)\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot \left(545.0353078425886 + 606.676680916724 \cdot z\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ (- (- 1.0 z) -6.0) 0.5)))
   (*
    (*
     (/ PI (sin (* PI z)))
     (*
      (* (* (sqrt PI) (sqrt 2.0)) (pow t_0 (- (- 1.0 z) 0.5)))
      (exp (- t_0))))
    (+
     (+
      263.383186962231
      (*
       z
       (+
        436.896172553987
        (* z (+ 545.0353078425886 (* 606.676680916724 z))))))
     (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0))))))

C

double code(double z) {
	double t_0 = ((1.0 - z) - -6.0) + 0.5;
	return ((((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt(((double) M_PI)) * sqrt(2.0)) * pow(t_0, ((1.0 - z) - 0.5))) * exp(-t_0))) * ((263.383186962231 + (z * (436.896172553987 + (z * (545.0353078425886 + (606.676680916724 * z)))))) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
}

Java

public static double code(double z) {
	double t_0 = ((1.0 - z) - -6.0) + 0.5;
	return ((Math.PI / Math.sin((Math.PI * z))) * (((Math.sqrt(Math.PI) * Math.sqrt(2.0)) * Math.pow(t_0, ((1.0 - z) - 0.5))) * Math.exp(-t_0))) * ((263.383186962231 + (z * (436.896172553987 + (z * (545.0353078425886 + (606.676680916724 * z)))))) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
}

Python

def code(z):
	t_0 = ((1.0 - z) - -6.0) + 0.5
	return ((math.pi / math.sin((math.pi * z))) * (((math.sqrt(math.pi) * math.sqrt(2.0)) * math.pow(t_0, ((1.0 - z) - 0.5))) * math.exp(-t_0))) * ((263.383186962231 + (z * (436.896172553987 + (z * (545.0353078425886 + (606.676680916724 * z)))))) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)))

Julia

function code(z)
	t_0 = Float64(Float64(Float64(1.0 - z) - -6.0) + 0.5)
	return Float64(Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(sqrt(pi) * sqrt(2.0)) * (t_0 ^ Float64(Float64(1.0 - z) - 0.5))) * exp(Float64(-t_0)))) * Float64(Float64(263.383186962231 + Float64(z * Float64(436.896172553987 + Float64(z * Float64(545.0353078425886 + Float64(606.676680916724 * z)))))) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0))))
end

MATLAB

function tmp = code(z)
	t_0 = ((1.0 - z) - -6.0) + 0.5;
	tmp = ((pi / sin((pi * z))) * (((sqrt(pi) * sqrt(2.0)) * (t_0 ^ ((1.0 - z) - 0.5))) * exp(-t_0))) * ((263.383186962231 + (z * (436.896172553987 + (z * (545.0353078425886 + (606.676680916724 * z)))))) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$0, N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$0)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(263.383186962231 + N[(z * N[(436.896172553987 + N[(z * N[(545.0353078425886 + N[(606.676680916724 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 96.6%

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

3. Applied rewrites 98.0%

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

   1. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\sqrt{\pi \cdot 2}} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lift-PI.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)} \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right) \cdot 2}} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   4. sqrt-prod N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   7. lift-PI.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\pi}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   8. lower-sqrt.f64 98.6

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in z around 0

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\color{blue}{\left(\frac{1382761731551712743134679}{5250000000000000000000} + z \cdot \left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]
7. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\frac{1382761731551712743134679}{5250000000000000000000} + \color{blue}{z \cdot \left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\frac{1382761731551712743134679}{5250000000000000000000} + z \cdot \color{blue}{\left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\frac{1382761731551712743134679}{5250000000000000000000} + z \cdot \left(\frac{16055934341359023345617179}{36750000000000000000000} + \color{blue}{z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\frac{1382761731551712743134679}{5250000000000000000000} + z \cdot \left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \color{blue}{\left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\frac{1382761731551712743134679}{5250000000000000000000} + z \cdot \left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \color{blue}{\frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z}\right)\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   6. lower-*.f64 97.7

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot \left(545.0353078425886 + 606.676680916724 \cdot \color{blue}{z}\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

8. Applied rewrites 97.7%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\color{blue}{\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot \left(545.0353078425886 + 606.676680916724 \cdot z\right)\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]
9. Add Preprocessing

Alternative 6: 97.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - z\right) - -6\right) + 0.5\\ \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {t\_0}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-t\_0}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ (- (- 1.0 z) -6.0) 0.5)))
   (*
    (*
     (/ PI (sin (* PI z)))
     (*
      (* (* (sqrt PI) (sqrt 2.0)) (pow t_0 (- (- 1.0 z) 0.5)))
      (exp (- t_0))))
    (+
     263.3831869810514
     (*
      z
      (+
       436.8961725563396
       (* z (+ 545.0353078428827 (* 606.6766809167608 z)))))))))

C

double code(double z) {
	double t_0 = ((1.0 - z) - -6.0) + 0.5;
	return ((((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt(((double) M_PI)) * sqrt(2.0)) * pow(t_0, ((1.0 - z) - 0.5))) * exp(-t_0))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))));
}

Java

public static double code(double z) {
	double t_0 = ((1.0 - z) - -6.0) + 0.5;
	return ((Math.PI / Math.sin((Math.PI * z))) * (((Math.sqrt(Math.PI) * Math.sqrt(2.0)) * Math.pow(t_0, ((1.0 - z) - 0.5))) * Math.exp(-t_0))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))));
}

Python

def code(z):
	t_0 = ((1.0 - z) - -6.0) + 0.5
	return ((math.pi / math.sin((math.pi * z))) * (((math.sqrt(math.pi) * math.sqrt(2.0)) * math.pow(t_0, ((1.0 - z) - 0.5))) * math.exp(-t_0))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))))

Julia

function code(z)
	t_0 = Float64(Float64(Float64(1.0 - z) - -6.0) + 0.5)
	return Float64(Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(sqrt(pi) * sqrt(2.0)) * (t_0 ^ Float64(Float64(1.0 - z) - 0.5))) * exp(Float64(-t_0)))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * Float64(545.0353078428827 + Float64(606.6766809167608 * z)))))))
end

MATLAB

function tmp = code(z)
	t_0 = ((1.0 - z) - -6.0) + 0.5;
	tmp = ((pi / sin((pi * z))) * (((sqrt(pi) * sqrt(2.0)) * (t_0 ^ ((1.0 - z) - 0.5))) * exp(-t_0))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$0, N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$0)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * N[(545.0353078428827 + N[(606.6766809167608 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 96.6%

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

3. Applied rewrites 98.0%

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

   1. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\sqrt{\pi \cdot 2}} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lift-PI.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)} \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right) \cdot 2}} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   4. sqrt-prod N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   7. lift-PI.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\pi}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   8. lower-sqrt.f64 98.6

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in z around 0

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)} \]
7. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + \color{blue}{z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \color{blue}{\left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)}\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \color{blue}{z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \color{blue}{\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)}\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \color{blue}{\frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z}\right)\right)\right) \]

   6. lower-*.f64 97.7

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot \color{blue}{z}\right)\right)\right) \]

8. Applied rewrites 97.7%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)} \]
9. Add Preprocessing

Alternative 7: 97.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - z\right) - -6\right) + 0.5\\ \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {t\_0}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-t\_0}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ (- (- 1.0 z) -6.0) 0.5)))
   (*
    (*
     (/ PI (sin (* PI z)))
     (*
      (* (* (sqrt PI) (sqrt 2.0)) (pow t_0 (- (- 1.0 z) 0.5)))
      (exp (- t_0))))
    (+
     263.3831869810514
     (* z (+ 436.8961725563396 (* 545.0353078428827 z)))))))

C

double code(double z) {
	double t_0 = ((1.0 - z) - -6.0) + 0.5;
	return ((((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt(((double) M_PI)) * sqrt(2.0)) * pow(t_0, ((1.0 - z) - 0.5))) * exp(-t_0))) * (263.3831869810514 + (z * (436.8961725563396 + (545.0353078428827 * z))));
}

Java

public static double code(double z) {
	double t_0 = ((1.0 - z) - -6.0) + 0.5;
	return ((Math.PI / Math.sin((Math.PI * z))) * (((Math.sqrt(Math.PI) * Math.sqrt(2.0)) * Math.pow(t_0, ((1.0 - z) - 0.5))) * Math.exp(-t_0))) * (263.3831869810514 + (z * (436.8961725563396 + (545.0353078428827 * z))));
}

Python

def code(z):
	t_0 = ((1.0 - z) - -6.0) + 0.5
	return ((math.pi / math.sin((math.pi * z))) * (((math.sqrt(math.pi) * math.sqrt(2.0)) * math.pow(t_0, ((1.0 - z) - 0.5))) * math.exp(-t_0))) * (263.3831869810514 + (z * (436.8961725563396 + (545.0353078428827 * z))))

Julia

function code(z)
	t_0 = Float64(Float64(Float64(1.0 - z) - -6.0) + 0.5)
	return Float64(Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(sqrt(pi) * sqrt(2.0)) * (t_0 ^ Float64(Float64(1.0 - z) - 0.5))) * exp(Float64(-t_0)))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(545.0353078428827 * z)))))
end

MATLAB

function tmp = code(z)
	t_0 = ((1.0 - z) - -6.0) + 0.5;
	tmp = ((pi / sin((pi * z))) * (((sqrt(pi) * sqrt(2.0)) * (t_0 ^ ((1.0 - z) - 0.5))) * exp(-t_0))) * (263.3831869810514 + (z * (436.8961725563396 + (545.0353078428827 * z))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$0, N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$0)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(545.0353078428827 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 96.6%

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

3. Applied rewrites 98.0%

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

   1. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\sqrt{\pi \cdot 2}} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lift-PI.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)} \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right) \cdot 2}} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   4. sqrt-prod N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   7. lift-PI.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\pi}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   8. lower-sqrt.f64 98.6

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in z around 0

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)} \]
7. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + \color{blue}{z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \color{blue}{\left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)}\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \color{blue}{\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z}\right)\right) \]

   4. lower-*.f64 97.6

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

8. Applied rewrites 97.6%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right)} \]
9. Add Preprocessing

Alternative 8: 97.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-7.5} \cdot \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot \sqrt{7.5}\right)\right)\\ \frac{\mathsf{fma}\left(263.3831869810514, t\_0, z \cdot \mathsf{fma}\left(263.3831869810514, t\_0, e^{-7.5} \cdot \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot \mathsf{fma}\left(263.3831869810514, \left(\log 0.13333333333333333 - 0.06666666666666667\right) \cdot \sqrt{7.5}, 436.8961725563396 \cdot \sqrt{7.5}\right)\right)\right)\right)\right)}{z} \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (* (exp -7.5) (* (sqrt 2.0) (* (sqrt PI) (sqrt 7.5))))))
   (/
    (fma
     263.3831869810514
     t_0
     (*
      z
      (fma
       263.3831869810514
       t_0
       (*
        (exp -7.5)
        (*
         (sqrt 2.0)
         (*
          (sqrt PI)
          (fma
           263.3831869810514
           (* (- (log 0.13333333333333333) 0.06666666666666667) (sqrt 7.5))
           (* 436.8961725563396 (sqrt 7.5)))))))))
    z)))

C

double code(double z) {
	double t_0 = exp(-7.5) * (sqrt(2.0) * (sqrt(((double) M_PI)) * sqrt(7.5)));
	return fma(263.3831869810514, t_0, (z * fma(263.3831869810514, t_0, (exp(-7.5) * (sqrt(2.0) * (sqrt(((double) M_PI)) * fma(263.3831869810514, ((log(0.13333333333333333) - 0.06666666666666667) * sqrt(7.5)), (436.8961725563396 * sqrt(7.5))))))))) / z;
}

Julia

function code(z)
	t_0 = Float64(exp(-7.5) * Float64(sqrt(2.0) * Float64(sqrt(pi) * sqrt(7.5))))
	return Float64(fma(263.3831869810514, t_0, Float64(z * fma(263.3831869810514, t_0, Float64(exp(-7.5) * Float64(sqrt(2.0) * Float64(sqrt(pi) * fma(263.3831869810514, Float64(Float64(log(0.13333333333333333) - 0.06666666666666667) * sqrt(7.5)), Float64(436.8961725563396 * sqrt(7.5))))))))) / z)
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[Exp[-7.5], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(263.3831869810514 * t$95$0 + N[(z * N[(263.3831869810514 * t$95$0 + N[(N[Exp[-7.5], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[Pi], $MachinePrecision] * N[(263.3831869810514 * N[(N[(N[Log[0.13333333333333333], $MachinePrecision] - 0.06666666666666667), $MachinePrecision] * N[Sqrt[7.5], $MachinePrecision]), $MachinePrecision] + N[(436.8961725563396 * N[Sqrt[7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Initial program 96.6%

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

3. Applied rewrites 98.0%

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

   1. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\sqrt{\pi \cdot 2}} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lift-PI.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)} \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right) \cdot 2}} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   4. sqrt-prod N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   7. lift-PI.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\pi}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   8. lower-sqrt.f64 98.6

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)\right) + z \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)\right) + e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right) \cdot {\frac{15}{2}}^{\frac{1}{2}}\right) + \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)\right)\right)}{z}} \]

8. Applied rewrites 97.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(263.3831869810514, e^{-7.5} \cdot \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot \sqrt{7.5}\right)\right), z \cdot \mathsf{fma}\left(263.3831869810514, e^{-7.5} \cdot \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot \sqrt{7.5}\right)\right), e^{-7.5} \cdot \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot \mathsf{fma}\left(263.3831869810514, \left(\log 0.13333333333333333 - 0.06666666666666667\right) \cdot \sqrt{7.5}, 436.8961725563396 \cdot \sqrt{7.5}\right)\right)\right)\right)\right)}{z}} \]
9. Add Preprocessing

Alternative 9: 97.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - z\right) - -6\right) + 0.5\\ \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {t\_0}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-t\_0}\right)\right) \cdot \left(263.3831869810514 + 436.8961725563396 \cdot z\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ (- (- 1.0 z) -6.0) 0.5)))
   (*
    (*
     (/ PI (sin (* PI z)))
     (*
      (* (* (sqrt PI) (sqrt 2.0)) (pow t_0 (- (- 1.0 z) 0.5)))
      (exp (- t_0))))
    (+ 263.3831869810514 (* 436.8961725563396 z)))))

C

double code(double z) {
	double t_0 = ((1.0 - z) - -6.0) + 0.5;
	return ((((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt(((double) M_PI)) * sqrt(2.0)) * pow(t_0, ((1.0 - z) - 0.5))) * exp(-t_0))) * (263.3831869810514 + (436.8961725563396 * z));
}

Java

public static double code(double z) {
	double t_0 = ((1.0 - z) - -6.0) + 0.5;
	return ((Math.PI / Math.sin((Math.PI * z))) * (((Math.sqrt(Math.PI) * Math.sqrt(2.0)) * Math.pow(t_0, ((1.0 - z) - 0.5))) * Math.exp(-t_0))) * (263.3831869810514 + (436.8961725563396 * z));
}

Python

def code(z):
	t_0 = ((1.0 - z) - -6.0) + 0.5
	return ((math.pi / math.sin((math.pi * z))) * (((math.sqrt(math.pi) * math.sqrt(2.0)) * math.pow(t_0, ((1.0 - z) - 0.5))) * math.exp(-t_0))) * (263.3831869810514 + (436.8961725563396 * z))

Julia

function code(z)
	t_0 = Float64(Float64(Float64(1.0 - z) - -6.0) + 0.5)
	return Float64(Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(sqrt(pi) * sqrt(2.0)) * (t_0 ^ Float64(Float64(1.0 - z) - 0.5))) * exp(Float64(-t_0)))) * Float64(263.3831869810514 + Float64(436.8961725563396 * z)))
end

MATLAB

function tmp = code(z)
	t_0 = ((1.0 - z) - -6.0) + 0.5;
	tmp = ((pi / sin((pi * z))) * (((sqrt(pi) * sqrt(2.0)) * (t_0 ^ ((1.0 - z) - 0.5))) * exp(-t_0))) * (263.3831869810514 + (436.8961725563396 * z));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$0, N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$0)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(436.8961725563396 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 96.6%

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

3. Applied rewrites 98.0%

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

   1. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\sqrt{\pi \cdot 2}} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lift-PI.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)} \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right) \cdot 2}} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   4. sqrt-prod N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   7. lift-PI.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\pi}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   8. lower-sqrt.f64 98.6

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in z around 0

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot z\right)} \]
7. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + \color{blue}{\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot z}\right) \]

   2. lower-*.f64 97.2

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

8. Applied rewrites 97.2%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \color{blue}{\left(263.3831869810514 + 436.8961725563396 \cdot z\right)} \]
9. Add Preprocessing

Alternative 10: 96.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := \left(t\_0 + 7\right) + 0.5\\ \frac{1 + 0.16666666666666666 \cdot \left(\left(z \cdot z\right) \cdot \left(\pi \cdot \pi\right)\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {t\_1}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{-t\_1}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 - -606.6766809167608 \cdot z\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)) (t_1 (+ (+ t_0 7.0) 0.5)))
   (*
    (/ (+ 1.0 (* 0.16666666666666666 (* (* z z) (* PI PI)))) z)
    (*
     (* (* (sqrt (* PI 2.0)) (pow t_1 (+ t_0 0.5))) (exp (- t_1)))
     (+
      263.3831869810514
      (*
       z
       (+
        436.8961725563396
        (* z (- 545.0353078428827 (* -606.6766809167608 z))))))))))

C

double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = (t_0 + 7.0) + 0.5;
	return ((1.0 + (0.16666666666666666 * ((z * z) * (((double) M_PI) * ((double) M_PI))))) / z) * (((sqrt((((double) M_PI) * 2.0)) * pow(t_1, (t_0 + 0.5))) * exp(-t_1)) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 - (-606.6766809167608 * z)))))));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = (t_0 + 7.0) + 0.5;
	return ((1.0 + (0.16666666666666666 * ((z * z) * (Math.PI * Math.PI)))) / z) * (((Math.sqrt((Math.PI * 2.0)) * Math.pow(t_1, (t_0 + 0.5))) * Math.exp(-t_1)) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 - (-606.6766809167608 * z)))))));
}

Python

def code(z):
	t_0 = (1.0 - z) - 1.0
	t_1 = (t_0 + 7.0) + 0.5
	return ((1.0 + (0.16666666666666666 * ((z * z) * (math.pi * math.pi)))) / z) * (((math.sqrt((math.pi * 2.0)) * math.pow(t_1, (t_0 + 0.5))) * math.exp(-t_1)) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 - (-606.6766809167608 * z)))))))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - 1.0)
	t_1 = Float64(Float64(t_0 + 7.0) + 0.5)
	return Float64(Float64(Float64(1.0 + Float64(0.16666666666666666 * Float64(Float64(z * z) * Float64(pi * pi)))) / z) * Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (t_1 ^ Float64(t_0 + 0.5))) * exp(Float64(-t_1))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * Float64(545.0353078428827 - Float64(-606.6766809167608 * z))))))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - 1.0;
	t_1 = (t_0 + 7.0) + 0.5;
	tmp = ((1.0 + (0.16666666666666666 * ((z * z) * (pi * pi)))) / z) * (((sqrt((pi * 2.0)) * (t_1 ^ (t_0 + 0.5))) * exp(-t_1)) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 - (-606.6766809167608 * z)))))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + 7.0), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(N[(N[(1.0 + N[(0.16666666666666666 * N[(N[(z * z), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$1, N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$1)], $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * N[(545.0353078428827 - N[(-606.6766809167608 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 96.6%

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

2. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)}\right) \]
3. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + \color{blue}{z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \color{blue}{\left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \color{blue}{z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)}\right)\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \color{blue}{\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)}\right)\right)\right) \]

   5. fp-cancel-sign-sub-inv N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}\right)\right) \cdot z}\right)\right)\right)\right) \]

   6. lower--.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}\right)\right) \cdot z}\right)\right)\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} - \left(\mathsf{neg}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}\right)\right) \cdot \color{blue}{z}\right)\right)\right)\right) \]

   8. metadata-eval 96.8

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 - -606.6766809167608 \cdot z\right)\right)\right)\right) \]

4. Applied rewrites 96.8%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 - -606.6766809167608 \cdot z\right)\right)\right)}\right) \]

5. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1 + \frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} - \frac{-4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1 + \frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{\color{blue}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} - \frac{-4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{1 + \frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} - \frac{-4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1 + \frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} - \frac{-4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{1 + \frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} - \frac{-4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \frac{1 + \frac{1}{6} \cdot \left(\left(z \cdot z\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} - \frac{-4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{1 + \frac{1}{6} \cdot \left(\left(z \cdot z\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} - \frac{-4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \frac{1 + \frac{1}{6} \cdot \left(\left(z \cdot z\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} - \frac{-4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{1 + \frac{1}{6} \cdot \left(\left(z \cdot z\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} - \frac{-4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]

   9. lift-PI.f64 N/A

      \[\leadsto \frac{1 + \frac{1}{6} \cdot \left(\left(z \cdot z\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} - \frac{-4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]

   10. lift-PI.f64 96.8

       \[\leadsto \frac{1 + 0.16666666666666666 \cdot \left(\left(z \cdot z\right) \cdot \left(\pi \cdot \pi\right)\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 - -606.6766809167608 \cdot z\right)\right)\right)\right) \]

7. Applied rewrites 96.8%

   \[\leadsto \color{blue}{\frac{1 + 0.16666666666666666 \cdot \left(\left(z \cdot z\right) \cdot \left(\pi \cdot \pi\right)\right)}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 - -606.6766809167608 \cdot z\right)\right)\right)\right) \]
8. Add Preprocessing

Alternative 11: 96.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) + 7\\ t_1 := t\_0 + 0.5\\ \frac{1}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {t\_1}^{\left(\left(-z\right) + 0.5\right)}\right) \cdot e^{-t\_1}\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + 545.0353078134797 \cdot z\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_0}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(-z\right) + 8}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ (- z) 7.0)) (t_1 (+ t_0 0.5)))
   (*
    (/ 1.0 z)
    (*
     (* (* (sqrt (* PI 2.0)) (pow t_1 (+ (- z) 0.5))) (exp (- t_1)))
     (+
      (+
       (+
        263.3831855358925
        (* z (+ 436.8961723502244 (* 545.0353078134797 z))))
       (/ 9.984369578019572e-6 t_0))
      (/ 1.5056327351493116e-7 (+ (- z) 8.0)))))))

C

double code(double z) {
	double t_0 = -z + 7.0;
	double t_1 = t_0 + 0.5;
	return (1.0 / z) * (((sqrt((((double) M_PI) * 2.0)) * pow(t_1, (-z + 0.5))) * exp(-t_1)) * (((263.3831855358925 + (z * (436.8961723502244 + (545.0353078134797 * z)))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / (-z + 8.0))));
}

Java

public static double code(double z) {
	double t_0 = -z + 7.0;
	double t_1 = t_0 + 0.5;
	return (1.0 / z) * (((Math.sqrt((Math.PI * 2.0)) * Math.pow(t_1, (-z + 0.5))) * Math.exp(-t_1)) * (((263.3831855358925 + (z * (436.8961723502244 + (545.0353078134797 * z)))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / (-z + 8.0))));
}

Python

def code(z):
	t_0 = -z + 7.0
	t_1 = t_0 + 0.5
	return (1.0 / z) * (((math.sqrt((math.pi * 2.0)) * math.pow(t_1, (-z + 0.5))) * math.exp(-t_1)) * (((263.3831855358925 + (z * (436.8961723502244 + (545.0353078134797 * z)))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / (-z + 8.0))))

Julia

function code(z)
	t_0 = Float64(Float64(-z) + 7.0)
	t_1 = Float64(t_0 + 0.5)
	return Float64(Float64(1.0 / z) * Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (t_1 ^ Float64(Float64(-z) + 0.5))) * exp(Float64(-t_1))) * Float64(Float64(Float64(263.3831855358925 + Float64(z * Float64(436.8961723502244 + Float64(545.0353078134797 * z)))) + Float64(9.984369578019572e-6 / t_0)) + Float64(1.5056327351493116e-7 / Float64(Float64(-z) + 8.0)))))
end

MATLAB

function tmp = code(z)
	t_0 = -z + 7.0;
	t_1 = t_0 + 0.5;
	tmp = (1.0 / z) * (((sqrt((pi * 2.0)) * (t_1 ^ (-z + 0.5))) * exp(-t_1)) * (((263.3831855358925 + (z * (436.8961723502244 + (545.0353078134797 * z)))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / (-z + 8.0))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[((-z) + 7.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 0.5), $MachinePrecision]}, N[(N[(1.0 / z), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$1, N[((-z) + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$1)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(263.3831855358925 + N[(z * N[(436.8961723502244 + N[(545.0353078134797 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[((-z) + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 96.6%

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

2. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\color{blue}{-1 \cdot z} + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\mathsf{neg}\left(z\right)\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-neg.f64 96.6

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

4. Applied rewrites 96.6%

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

5. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\color{blue}{-1 \cdot z} + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\mathsf{neg}\left(z\right)\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-neg.f64 96.6

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

7. Applied rewrites 96.6%

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

8. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\color{blue}{-1 \cdot z} + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
9. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\mathsf{neg}\left(z\right)\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-neg.f64 96.6

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

10. Applied rewrites 96.6%

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

11. Taylor expanded in z around 0

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\color{blue}{-1 \cdot z} + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
12. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\mathsf{neg}\left(z\right)\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

    2. lower-neg.f64 96.6

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

13. Applied rewrites 96.6%

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

14. Taylor expanded in z around 0

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\color{blue}{-1 \cdot z} + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
15. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\mathsf{neg}\left(z\right)\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

    2. lower-neg.f64 96.6

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

16. Applied rewrites 96.6%

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

17. Taylor expanded in z around 0

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\color{blue}{-1 \cdot z} + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
18. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\mathsf{neg}\left(z\right)\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

    2. lower-neg.f64 96.6

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

19. Applied rewrites 96.6%

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

20. Taylor expanded in z around 0

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(-z\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\color{blue}{-1 \cdot z} + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
21. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(-z\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\mathsf{neg}\left(z\right)\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

    2. lower-neg.f64 96.6

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

22. Applied rewrites 96.6%

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

23. Taylor expanded in z around 0

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(-z\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(-z\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\color{blue}{-1 \cdot z} + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
24. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(-z\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(-z\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\mathsf{neg}\left(z\right)\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

    2. lower-neg.f64 96.6

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

25. Applied rewrites 96.6%

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

26. Taylor expanded in z around 0

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(-z\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(-z\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(-z\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\color{blue}{-1 \cdot z} + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
27. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(-z\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(-z\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(-z\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\mathsf{neg}\left(z\right)\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

    2. lower-neg.f64 96.6

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

28. Applied rewrites 96.6%

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

29. Taylor expanded in z around 0

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(-z\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(-z\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(-z\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(-z\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\color{blue}{-1 \cdot z} + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
30. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(-z\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(-z\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(-z\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(-z\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\mathsf{neg}\left(z\right)\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

    2. lower-neg.f64 96.6

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

31. Applied rewrites 96.6%

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

32. Taylor expanded in z around 0

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(-z\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(-z\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(-z\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(-z\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(-z\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\color{blue}{-1 \cdot z} + 8}\right)\right) \]
33. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(-z\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(-z\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(-z\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(-z\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(-z\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(-z\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(-z\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\mathsf{neg}\left(z\right)\right) + 8}\right)\right) \]

    2. lower-neg.f64 96.6

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

34. Applied rewrites 96.6%

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

35. Taylor expanded in z around 0

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\color{blue}{\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + \frac{367898832774098786021}{675000000000000000} \cdot z\right)\right)} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(-z\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(-z\right) + 8}\right)\right) \]
36. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + \color{blue}{z \cdot \left(\frac{131068851705067315609}{300000000000000000} + \frac{367898832774098786021}{675000000000000000} \cdot z\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(-z\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(-z\right) + 8}\right)\right) \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \color{blue}{\left(\frac{131068851705067315609}{300000000000000000} + \frac{367898832774098786021}{675000000000000000} \cdot z\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(-z\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(-z\right) + 8}\right)\right) \]

    3. lower-+.f64 N/A

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + \color{blue}{\frac{367898832774098786021}{675000000000000000} \cdot z}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(-z\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(-z\right) + 8}\right)\right) \]

    4. lower-*.f64 96.8

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + 0.5\right)}^{\left(\left(-z\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + 545.0353078134797 \cdot \color{blue}{z}\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(-z\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(-z\right) + 8}\right)\right) \]

37. Applied rewrites 96.8%

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + 0.5\right)}^{\left(\left(-z\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + 545.0353078134797 \cdot z\right)\right)} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(-z\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(-z\right) + 8}\right)\right) \]

38. Taylor expanded in z around 0

    \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + \frac{367898832774098786021}{675000000000000000} \cdot z\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(-z\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(-z\right) + 8}\right)\right) \]
39. Step-by-step derivation

    1. lower-/.f64 96.3

       \[\leadsto \frac{1}{\color{blue}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + 0.5\right)}^{\left(\left(-z\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + 545.0353078134797 \cdot z\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(-z\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(-z\right) + 8}\right)\right) \]

40. Applied rewrites 96.3%

    \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + 0.5\right)}^{\left(\left(-z\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + 545.0353078134797 \cdot z\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(-z\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(-z\right) + 8}\right)\right) \]
41. Add Preprocessing

Alternative 12: 96.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := \left(t\_0 + 7\right) + 0.5\\ \frac{1}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {t\_1}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{-t\_1}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 - -606.6766809167608 \cdot z\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)) (t_1 (+ (+ t_0 7.0) 0.5)))
   (*
    (/ 1.0 z)
    (*
     (* (* (sqrt (* PI 2.0)) (pow t_1 (+ t_0 0.5))) (exp (- t_1)))
     (+
      263.3831869810514
      (*
       z
       (+
        436.8961725563396
        (* z (- 545.0353078428827 (* -606.6766809167608 z))))))))))

C

double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = (t_0 + 7.0) + 0.5;
	return (1.0 / z) * (((sqrt((((double) M_PI) * 2.0)) * pow(t_1, (t_0 + 0.5))) * exp(-t_1)) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 - (-606.6766809167608 * z)))))));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = (t_0 + 7.0) + 0.5;
	return (1.0 / z) * (((Math.sqrt((Math.PI * 2.0)) * Math.pow(t_1, (t_0 + 0.5))) * Math.exp(-t_1)) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 - (-606.6766809167608 * z)))))));
}

Python

def code(z):
	t_0 = (1.0 - z) - 1.0
	t_1 = (t_0 + 7.0) + 0.5
	return (1.0 / z) * (((math.sqrt((math.pi * 2.0)) * math.pow(t_1, (t_0 + 0.5))) * math.exp(-t_1)) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 - (-606.6766809167608 * z)))))))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - 1.0)
	t_1 = Float64(Float64(t_0 + 7.0) + 0.5)
	return Float64(Float64(1.0 / z) * Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (t_1 ^ Float64(t_0 + 0.5))) * exp(Float64(-t_1))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * Float64(545.0353078428827 - Float64(-606.6766809167608 * z))))))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - 1.0;
	t_1 = (t_0 + 7.0) + 0.5;
	tmp = (1.0 / z) * (((sqrt((pi * 2.0)) * (t_1 ^ (t_0 + 0.5))) * exp(-t_1)) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 - (-606.6766809167608 * z)))))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + 7.0), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(N[(1.0 / z), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$1, N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$1)], $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * N[(545.0353078428827 - N[(-606.6766809167608 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 96.6%

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

2. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)}\right) \]
3. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + \color{blue}{z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \color{blue}{\left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \color{blue}{z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)}\right)\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \color{blue}{\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)}\right)\right)\right) \]

   5. fp-cancel-sign-sub-inv N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}\right)\right) \cdot z}\right)\right)\right)\right) \]

   6. lower--.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}\right)\right) \cdot z}\right)\right)\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} - \left(\mathsf{neg}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}\right)\right) \cdot \color{blue}{z}\right)\right)\right)\right) \]

   8. metadata-eval 96.8

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 - -606.6766809167608 \cdot z\right)\right)\right)\right) \]

4. Applied rewrites 96.8%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 - -606.6766809167608 \cdot z\right)\right)\right)}\right) \]

5. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} - \frac{-4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. lower-/.f64 96.3

      \[\leadsto \frac{1}{\color{blue}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 - -606.6766809167608 \cdot z\right)\right)\right)\right) \]

7. Applied rewrites 96.3%

   \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 - -606.6766809167608 \cdot z\right)\right)\right)\right) \]
8. Add Preprocessing

Alternative 13: 96.3% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

3. Applied rewrites 98.0%

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

   1. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\sqrt{\pi \cdot 2}} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lift-PI.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)} \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right) \cdot 2}} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   4. sqrt-prod N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   7. lift-PI.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\pi}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   8. lower-sqrt.f64 98.6

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)}{z}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \color{blue}{\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)}{z}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)}{\color{blue}{z}} \]

8. Applied rewrites 96.5%

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

Alternative 14: 95.8% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

2. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{z}} \]
3. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)}{\color{blue}{z}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)}{\color{blue}{z}} \]

4. Applied rewrites 95.4%

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

   1. lift-*.f64 N/A

      \[\leadsto \frac{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot e^{\frac{-15}{2}}\right) \cdot \left(\sqrt{\pi + \pi} \cdot \sqrt{\frac{15}{2}}\right)}{z} \]

   2. lift-sqrt.f64 N/A

      \[\leadsto \frac{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot e^{\frac{-15}{2}}\right) \cdot \left(\sqrt{\pi + \pi} \cdot \sqrt{\frac{15}{2}}\right)}{z} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \frac{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot e^{\frac{-15}{2}}\right) \cdot \left(\sqrt{\pi + \pi} \cdot \sqrt{\frac{15}{2}}\right)}{z} \]

   4. sqrt-unprod N/A

      \[\leadsto \frac{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot e^{\frac{-15}{2}}\right) \cdot \sqrt{\left(\pi + \pi\right) \cdot \frac{15}{2}}}{z} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot e^{\frac{-15}{2}}\right) \cdot \sqrt{\left(\pi + \pi\right) \cdot \frac{15}{2}}}{z} \]

   6. lower-*.f64 95.4

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

6. Applied rewrites 95.4%

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

7. Taylor expanded in z around 0

   \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \color{blue}{\frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}{z}} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}{\color{blue}{z}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}{z} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}{z} \]

   4. lift-exp.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}{z} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}{z} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}{z} \]

   7. lift-PI.f64 95.8

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

9. Applied rewrites 95.8%

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

Reproduce

herbie shell --seed 2025140 
(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))))))