Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.4% → 98.3%

Time: 16.2s

Alternatives: 12

Speedup: 1.8×

• could not determine a ground truth

  1. z = -2488937797818419.5

Specification

\[z \leq \frac{1}{2}\]

Math

\[\begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := t\_0 + 7\\ t_2 := t\_1 + \frac{1}{2}\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {t\_2}^{\left(t\_0 + \frac{1}{2}\right)}\right) \cdot e^{-t\_2}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{t\_0 + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{t\_0 + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{t\_0 + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{t\_0 + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{t\_0 + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{t\_0 + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{t\_1}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{t\_0 + 8}\right)\right) \end{array} \]

FPCore

(FPCore (z)
  :precision binary64
  (let* ((t_0 (- (- 1 z) 1)) (t_1 (+ t_0 7)) (t_2 (+ t_1 1/2)))
  (*
   (/ PI (sin (* PI z)))
   (*
    (* (* (sqrt (* PI 2)) (pow t_2 (+ t_0 1/2))) (exp (- t_2)))
    (+
     (+
      (+
       (+
        (+
         (+
          (+
           (+
            9999999999998099/10000000000000000
            (/ 6765203681218851/10000000000000 (+ t_0 1)))
           (/ -3147848041806007/2500000000000 (+ t_0 2)))
          (/ 7713234287776531/10000000000000 (+ t_0 3)))
         (/ -883075145810703/5000000000000 (+ t_0 4)))
        (/ 2501468655737381/200000000000000 (+ t_0 5)))
       (/ -3464277381643003/25000000000000000 (+ t_0 6)))
      (/ 2496092394504893/250000000000000000000 t_1))
     (/ 3764081837873279/25000000000000000000000 (+ t_0 8)))))))

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 - z), $MachinePrecision] - 1), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 7), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + 1/2), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(Pi * 2), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$2, N[(t$95$0 + 1/2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$2)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(9999999999998099/10000000000000000 + N[(6765203681218851/10000000000000 / N[(t$95$0 + 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-3147848041806007/2500000000000 / N[(t$95$0 + 2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(7713234287776531/10000000000000 / N[(t$95$0 + 3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-883075145810703/5000000000000 / N[(t$95$0 + 4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2501468655737381/200000000000000 / N[(t$95$0 + 5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-3464277381643003/25000000000000000 / N[(t$95$0 + 6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2496092394504893/250000000000000000000 / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(3764081837873279/25000000000000000000000 / N[(t$95$0 + 8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Initial Program: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := t\_0 + 7\\ t_2 := t\_1 + \frac{1}{2}\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {t\_2}^{\left(t\_0 + \frac{1}{2}\right)}\right) \cdot e^{-t\_2}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{t\_0 + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{t\_0 + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{t\_0 + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{t\_0 + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{t\_0 + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{t\_0 + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{t\_1}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{t\_0 + 8}\right)\right) \end{array} \]

FPCore

(FPCore (z)
  :precision binary64
  (let* ((t_0 (- (- 1 z) 1)) (t_1 (+ t_0 7)) (t_2 (+ t_1 1/2)))
  (*
   (/ PI (sin (* PI z)))
   (*
    (* (* (sqrt (* PI 2)) (pow t_2 (+ t_0 1/2))) (exp (- t_2)))
    (+
     (+
      (+
       (+
        (+
         (+
          (+
           (+
            9999999999998099/10000000000000000
            (/ 6765203681218851/10000000000000 (+ t_0 1)))
           (/ -3147848041806007/2500000000000 (+ t_0 2)))
          (/ 7713234287776531/10000000000000 (+ t_0 3)))
         (/ -883075145810703/5000000000000 (+ t_0 4)))
        (/ 2501468655737381/200000000000000 (+ t_0 5)))
       (/ -3464277381643003/25000000000000000 (+ t_0 6)))
      (/ 2496092394504893/250000000000000000000 t_1))
     (/ 3764081837873279/25000000000000000000000 (+ t_0 8)))))))

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 - z), $MachinePrecision] - 1), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 7), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + 1/2), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(Pi * 2), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$2, N[(t$95$0 + 1/2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$2)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(9999999999998099/10000000000000000 + N[(6765203681218851/10000000000000 / N[(t$95$0 + 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-3147848041806007/2500000000000 / N[(t$95$0 + 2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(7713234287776531/10000000000000 / N[(t$95$0 + 3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-883075145810703/5000000000000 / N[(t$95$0 + 4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2501468655737381/200000000000000 / N[(t$95$0 + 5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-3464277381643003/25000000000000000 / N[(t$95$0 + 6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2496092394504893/250000000000000000000 / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(3764081837873279/25000000000000000000000 / N[(t$95$0 + 8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Alternative 1: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := t\_0 + 7\\ t_2 := t\_1 + \frac{1}{2}\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {t\_2}^{\left(t\_0 + \frac{1}{2}\right)}\right) \cdot e^{-t\_2}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{1}{\frac{1 - z}{\frac{6765203681218851}{10000000000000}}} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{t\_0 + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{t\_0 + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{t\_1}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{t\_0 + 8}\right)\right) \end{array} \]

FPCore

(FPCore (z)
  :precision binary64
  (let* ((t_0 (- (- 1 z) 1)) (t_1 (+ t_0 7)) (t_2 (+ t_1 1/2)))
  (*
   (/ PI (sin (* PI z)))
   (*
    (* (* (sqrt (* PI 2)) (pow t_2 (+ t_0 1/2))) (exp (- t_2)))
    (+
     (+
      (+
       (+
        (+
         9999999999998099/10000000000000000
         (+
          (-
           (/ 1 (/ (- 1 z) 6765203681218851/10000000000000))
           (/ 3147848041806007/2500000000000 (- (- 1 z) -1)))
          (-
           (/ 7713234287776531/10000000000000 (- (- 1 z) -2))
           (/ 883075145810703/5000000000000 (- (- 1 z) -3)))))
        (/ 2501468655737381/200000000000000 (+ t_0 5)))
       (/ -3464277381643003/25000000000000000 (+ t_0 6)))
      (/ 2496092394504893/250000000000000000000 t_1))
     (/ 3764081837873279/25000000000000000000000 (+ t_0 8)))))))

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 + (((1.0 / ((1.0 - z) / 676.5203681218851)) - (1259.1392167224028 / ((1.0 - z) - -1.0))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) - (176.6150291621406 / ((1.0 - z) - -3.0))))) + (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 + (((1.0 / ((1.0 - z) / 676.5203681218851)) - (1259.1392167224028 / ((1.0 - z) - -1.0))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) - (176.6150291621406 / ((1.0 - z) - -3.0))))) + (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 + (((1.0 / ((1.0 - z) / 676.5203681218851)) - (1259.1392167224028 / ((1.0 - z) - -1.0))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) - (176.6150291621406 / ((1.0 - z) - -3.0))))) + (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(0.9999999999998099 + Float64(Float64(Float64(1.0 / Float64(Float64(1.0 - z) / 676.5203681218851)) - Float64(1259.1392167224028 / Float64(Float64(1.0 - z) - -1.0))) + Float64(Float64(771.3234287776531 / Float64(Float64(1.0 - z) - -2.0)) - Float64(176.6150291621406 / Float64(Float64(1.0 - z) - -3.0))))) + 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 + (((1.0 / ((1.0 - z) / 676.5203681218851)) - (1259.1392167224028 / ((1.0 - z) - -1.0))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) - (176.6150291621406 / ((1.0 - z) - -3.0))))) + (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 - z), $MachinePrecision] - 1), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 7), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + 1/2), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(Pi * 2), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$2, N[(t$95$0 + 1/2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$2)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(9999999999998099/10000000000000000 + N[(N[(N[(1 / N[(N[(1 - z), $MachinePrecision] / 6765203681218851/10000000000000), $MachinePrecision]), $MachinePrecision] - N[(3147848041806007/2500000000000 / N[(N[(1 - z), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(7713234287776531/10000000000000 / N[(N[(1 - z), $MachinePrecision] - -2), $MachinePrecision]), $MachinePrecision] - N[(883075145810703/5000000000000 / N[(N[(1 - z), $MachinePrecision] - -3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2501468655737381/200000000000000 / N[(t$95$0 + 5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-3464277381643003/25000000000000000 / N[(t$95$0 + 6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2496092394504893/250000000000000000000 / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(3764081837873279/25000000000000000000000 / N[(t$95$0 + 8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 96.4%

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

3. Applied rewrites 98.3%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

   3. lower-unsound-/.f64 N/A

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

   4. lower-unsound-/.f64 98.3%

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

5. Applied rewrites 98.3%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := 1 - \left(z + -6\right)\\ t_2 := t\_1 + \frac{1}{2}\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {t\_2}^{\left(t\_0 + \frac{1}{2}\right)}\right) \cdot e^{-t\_2}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{t\_0 + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{t\_0 + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{t\_1}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{t\_0 + 8}\right)\right) \end{array} \]

FPCore

(FPCore (z)
  :precision binary64
  (let* ((t_0 (- (- 1 z) 1)) (t_1 (- 1 (+ z -6))) (t_2 (+ t_1 1/2)))
  (*
   (/ PI (sin (* PI z)))
   (*
    (* (* (sqrt (* PI 2)) (pow t_2 (+ t_0 1/2))) (exp (- t_2)))
    (+
     (+
      (+
       (+
        (+
         9999999999998099/10000000000000000
         (+
          (-
           (/ 6765203681218851/10000000000000 (- 1 z))
           (/ 3147848041806007/2500000000000 (- (- 1 z) -1)))
          (-
           (/ 7713234287776531/10000000000000 (- (- 1 z) -2))
           (/ 883075145810703/5000000000000 (- (- 1 z) -3)))))
        (/ 2501468655737381/200000000000000 (+ t_0 5)))
       (/ -3464277381643003/25000000000000000 (+ t_0 6)))
      (/ 2496092394504893/250000000000000000000 t_1))
     (/ 3764081837873279/25000000000000000000000 (+ t_0 8)))))))

C

double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = 1.0 - (z + -6.0);
	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 / (1.0 - z)) - (1259.1392167224028 / ((1.0 - z) - -1.0))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) - (176.6150291621406 / ((1.0 - z) - -3.0))))) + (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 = 1.0 - (z + -6.0);
	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 / (1.0 - z)) - (1259.1392167224028 / ((1.0 - z) - -1.0))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) - (176.6150291621406 / ((1.0 - z) - -3.0))))) + (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 = 1.0 - (z + -6.0)
	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 / (1.0 - z)) - (1259.1392167224028 / ((1.0 - z) - -1.0))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) - (176.6150291621406 / ((1.0 - z) - -3.0))))) + (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(1.0 - Float64(z + -6.0))
	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(0.9999999999998099 + Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) - Float64(1259.1392167224028 / Float64(Float64(1.0 - z) - -1.0))) + Float64(Float64(771.3234287776531 / Float64(Float64(1.0 - z) - -2.0)) - Float64(176.6150291621406 / Float64(Float64(1.0 - z) - -3.0))))) + 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 = 1.0 - (z + -6.0);
	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 / (1.0 - z)) - (1259.1392167224028 / ((1.0 - z) - -1.0))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) - (176.6150291621406 / ((1.0 - z) - -3.0))))) + (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 - z), $MachinePrecision] - 1), $MachinePrecision]}, Block[{t$95$1 = N[(1 - N[(z + -6), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + 1/2), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(Pi * 2), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$2, N[(t$95$0 + 1/2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$2)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(9999999999998099/10000000000000000 + N[(N[(N[(6765203681218851/10000000000000 / N[(1 - z), $MachinePrecision]), $MachinePrecision] - N[(3147848041806007/2500000000000 / N[(N[(1 - z), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(7713234287776531/10000000000000 / N[(N[(1 - z), $MachinePrecision] - -2), $MachinePrecision]), $MachinePrecision] - N[(883075145810703/5000000000000 / N[(N[(1 - z), $MachinePrecision] - -3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2501468655737381/200000000000000 / N[(t$95$0 + 5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-3464277381643003/25000000000000000 / N[(t$95$0 + 6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2496092394504893/250000000000000000000 / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(3764081837873279/25000000000000000000000 / N[(t$95$0 + 8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 96.4%

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

3. Applied rewrites 98.3%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+l- N/A

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

   4. lift--.f64 N/A

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

   5. metadata-eval N/A

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

   6. associate--l- N/A

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

   7. lower--.f64 N/A

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

   8. lower-+.f64 98.3%

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

5. Applied rewrites 98.3%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+l- N/A

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

   4. lift--.f64 N/A

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

   5. metadata-eval N/A

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

   6. associate--l- N/A

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

   7. lower--.f64 N/A

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

   8. lower-+.f64 98.3%

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

7. Applied rewrites 98.3%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+l- N/A

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

   4. lift--.f64 N/A

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

   5. metadata-eval N/A

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

   6. associate--l- N/A

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

   7. lower--.f64 N/A

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

   8. lower-+.f64 98.3%

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

9. Applied rewrites 98.3%

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

Alternative 3: 98.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \left(1 - z\right) - \frac{-13}{2}\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \frac{\frac{-3764081837873279}{25000000000000000000000}}{-7 - \left(1 - z\right)}\right)\right) \cdot \left(e^{-t\_0} \cdot \left({t\_0}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{\pi + \pi}\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
  :precision binary64
  (let* ((t_0 (- (- 1 z) -13/2)))
  (*
   (/ PI (sin (* PI z)))
   (*
    (+
     (+
      (+
       (+
        9999999999998099/10000000000000000
        (-
         (/ 6765203681218851/10000000000000 (- 1 z))
         (/ 3147848041806007/2500000000000 (- (- 1 z) -1))))
       (-
        (/ 7713234287776531/10000000000000 (- (- 1 z) -2))
        (/ 883075145810703/5000000000000 (- (- 1 z) -3))))
      (+
       (/ 2501468655737381/200000000000000 (- (- 1 z) -4))
       (/ -3464277381643003/25000000000000000 (- (- 1 z) -5))))
     (+
      (/ 2496092394504893/250000000000000000000 (- (- 1 z) -6))
      (/ -3764081837873279/25000000000000000000000 (- -7 (- 1 z)))))
    (*
     (exp (- t_0))
     (* (pow t_0 (- (- 1 z) 1/2)) (sqrt (+ PI PI))))))))

C

double code(double z) {
	double t_0 = (1.0 - z) - -6.5;
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (((((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) - (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 / (-7.0 - (1.0 - z))))) * (exp(-t_0) * (pow(t_0, ((1.0 - z) - 0.5)) * sqrt((((double) M_PI) + ((double) M_PI))))));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - -6.5;
	return (Math.PI / Math.sin((Math.PI * z))) * (((((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) - (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 / (-7.0 - (1.0 - z))))) * (Math.exp(-t_0) * (Math.pow(t_0, ((1.0 - z) - 0.5)) * Math.sqrt((Math.PI + Math.PI)))));
}

Python

def code(z):
	t_0 = (1.0 - z) - -6.5
	return (math.pi / math.sin((math.pi * z))) * (((((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) - (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 / (-7.0 - (1.0 - z))))) * (math.exp(-t_0) * (math.pow(t_0, ((1.0 - z) - 0.5)) * math.sqrt((math.pi + math.pi)))))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - -6.5)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) - Float64(1259.1392167224028 / Float64(Float64(1.0 - z) - -1.0)))) + Float64(Float64(771.3234287776531 / Float64(Float64(1.0 - z) - -2.0)) - Float64(176.6150291621406 / Float64(Float64(1.0 - z) - -3.0)))) + Float64(Float64(12.507343278686905 / Float64(Float64(1.0 - z) - -4.0)) + Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) - -5.0)))) + Float64(Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0)) + Float64(-1.5056327351493116e-7 / Float64(-7.0 - Float64(1.0 - z))))) * Float64(exp(Float64(-t_0)) * Float64((t_0 ^ Float64(Float64(1.0 - z) - 0.5)) * sqrt(Float64(pi + pi))))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - -6.5;
	tmp = (pi / sin((pi * z))) * (((((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) - (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 / (-7.0 - (1.0 - z))))) * (exp(-t_0) * ((t_0 ^ ((1.0 - z) - 0.5)) * sqrt((pi + pi)))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1 - z), $MachinePrecision] - -13/2), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(9999999999998099/10000000000000000 + N[(N[(6765203681218851/10000000000000 / N[(1 - z), $MachinePrecision]), $MachinePrecision] - N[(3147848041806007/2500000000000 / N[(N[(1 - z), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(7713234287776531/10000000000000 / N[(N[(1 - z), $MachinePrecision] - -2), $MachinePrecision]), $MachinePrecision] - N[(883075145810703/5000000000000 / N[(N[(1 - z), $MachinePrecision] - -3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2501468655737381/200000000000000 / N[(N[(1 - z), $MachinePrecision] - -4), $MachinePrecision]), $MachinePrecision] + N[(-3464277381643003/25000000000000000 / N[(N[(1 - z), $MachinePrecision] - -5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2496092394504893/250000000000000000000 / N[(N[(1 - z), $MachinePrecision] - -6), $MachinePrecision]), $MachinePrecision] + N[(-3764081837873279/25000000000000000000000 / N[(-7 - N[(1 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-t$95$0)], $MachinePrecision] * N[(N[Power[t$95$0, N[(N[(1 - z), $MachinePrecision] - 1/2), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi + Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 96.4%

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

3. Applied rewrites 98.3%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+l- N/A

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

   4. lift--.f64 N/A

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

   5. metadata-eval N/A

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

   6. associate--l- N/A

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

   7. lower--.f64 N/A

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

   8. lower-+.f64 98.3%

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

5. Applied rewrites 98.3%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+l- N/A

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

   4. lift--.f64 N/A

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

   5. metadata-eval N/A

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

   6. associate--l- N/A

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

   7. lower--.f64 N/A

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

   8. lower-+.f64 98.3%

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

7. Applied rewrites 98.3%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+l- N/A

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

   4. lift--.f64 N/A

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

   5. metadata-eval N/A

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

   6. associate--l- N/A

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

   7. lower--.f64 N/A

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

   8. lower-+.f64 98.3%

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

9. Applied rewrites 98.3%

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

11. Applied rewrites 98.3%

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \frac{\frac{-3764081837873279}{25000000000000000000000}}{-7 - \left(1 - z\right)}\right)\right) \cdot \left(e^{-\left(\left(1 - z\right) - \frac{-13}{2}\right)} \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{\pi + \pi}\right)\right)\right)} \]
12. Add Preprocessing

Alternative 4: 97.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := t\_0 + 7\\ t_2 := t\_1 + \frac{1}{2}\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {t\_2}^{\left(t\_0 + \frac{1}{2}\right)}\right) \cdot e^{-t\_2}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{t\_1}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{t\_0 + 8}\right)\right) \end{array} \]

FPCore

(FPCore (z)
  :precision binary64
  (let* ((t_0 (- (- 1 z) 1)) (t_1 (+ t_0 7)) (t_2 (+ t_1 1/2)))
  (*
   (/ PI (sin (* PI z)))
   (*
    (* (* (* (sqrt PI) (sqrt 2)) (pow t_2 (+ t_0 1/2))) (exp (- t_2)))
    (+
     (+
      (+
       9876869457595968283/37500000000000000
       (*
        z
        (+
         131068851705067315609/300000000000000000
         (*
          z
          (+
           367898832774098786021/675000000000000000
           (* 65521081538557082921549/108000000000000000000 z))))))
      (/ 2496092394504893/250000000000000000000 t_1))
     (/ 3764081837873279/25000000000000000000000 (+ t_0 8)))))))

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)) * sqrt(2.0)) * pow(t_2, (t_0 + 0.5))) * exp(-t_2)) * (((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (606.6766809125655 * z)))))) + (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) * Math.sqrt(2.0)) * Math.pow(t_2, (t_0 + 0.5))) * Math.exp(-t_2)) * (((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (606.6766809125655 * z)))))) + (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) * math.sqrt(2.0)) * math.pow(t_2, (t_0 + 0.5))) * math.exp(-t_2)) * (((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (606.6766809125655 * z)))))) + (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(Float64(sqrt(pi) * sqrt(2.0)) * (t_2 ^ Float64(t_0 + 0.5))) * exp(Float64(-t_2))) * Float64(Float64(Float64(263.3831855358925 + Float64(z * Float64(436.8961723502244 + Float64(z * Float64(545.0353078134797 + Float64(606.6766809125655 * z)))))) + 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) * sqrt(2.0)) * (t_2 ^ (t_0 + 0.5))) * exp(-t_2)) * (((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (606.6766809125655 * z)))))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1 - z), $MachinePrecision] - 1), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 7), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + 1/2), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$2, N[(t$95$0 + 1/2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$2)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(9876869457595968283/37500000000000000 + N[(z * N[(131068851705067315609/300000000000000000 + N[(z * N[(367898832774098786021/675000000000000000 + N[(65521081538557082921549/108000000000000000000 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2496092394504893/250000000000000000000 / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(3764081837873279/25000000000000000000000 / N[(t$95$0 + 8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 96.4%

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

2. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\color{blue}{\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + \color{blue}{z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \color{blue}{\left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + \color{blue}{z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \color{blue}{\left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \color{blue}{\frac{65521081538557082921549}{108000000000000000000} \cdot z}\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower-*.f64 96.6%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot \color{blue}{z}\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 96.6%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\color{blue}{\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\sqrt{\pi \cdot 2}} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\pi \cdot 2}} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. sqrt-prod N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lift-PI.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-unsound-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lift-PI.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\color{blue}{\pi}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lower-unsound-sqrt.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\color{blue}{\sqrt{\pi}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lower-unsound-sqrt.f64 97.4%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \color{blue}{\sqrt{2}}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Applied rewrites 97.4%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
7. Add Preprocessing

Alternative 5: 97.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \left(1 - z\right) - -6\\ t_1 := t\_0 - \frac{-1}{2}\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left({t\_1}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{\pi + \pi}\right) \cdot \left(e^{-t\_1} \cdot \left(\left(\left(\left(\frac{65521081538557082921549}{108000000000000000000} \cdot z - \frac{-367898832774098786021}{675000000000000000}\right) \cdot z - \frac{-131068851705067315609}{300000000000000000}\right) \cdot z - \frac{-9876869457595968283}{37500000000000000}\right) + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{t\_0} + \frac{\frac{-3764081837873279}{25000000000000000000000}}{-7 - \left(1 - z\right)}\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
  :precision binary64
  (let* ((t_0 (- (- 1 z) -6)) (t_1 (- t_0 -1/2)))
  (*
   (/ PI (sin (* PI z)))
   (*
    (* (pow t_1 (- (- 1 z) 1/2)) (sqrt (+ PI PI)))
    (*
     (exp (- t_1))
     (+
      (-
       (*
        (-
         (*
          (-
           (* 65521081538557082921549/108000000000000000000 z)
           -367898832774098786021/675000000000000000)
          z)
         -131068851705067315609/300000000000000000)
        z)
       -9876869457595968283/37500000000000000)
      (+
       (/ 2496092394504893/250000000000000000000 t_0)
       (/
        -3764081837873279/25000000000000000000000
        (- -7 (- 1 z))))))))))

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))) * ((pow(t_1, ((1.0 - z) - 0.5)) * sqrt((((double) M_PI) + ((double) M_PI)))) * (exp(-t_1) * (((((((606.6766809125655 * z) - -545.0353078134797) * z) - -436.8961723502244) * z) - -263.3831855358925) + ((9.984369578019572e-6 / t_0) + (-1.5056327351493116e-7 / (-7.0 - (1.0 - z)))))));
}

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.pow(t_1, ((1.0 - z) - 0.5)) * Math.sqrt((Math.PI + Math.PI))) * (Math.exp(-t_1) * (((((((606.6766809125655 * z) - -545.0353078134797) * z) - -436.8961723502244) * z) - -263.3831855358925) + ((9.984369578019572e-6 / t_0) + (-1.5056327351493116e-7 / (-7.0 - (1.0 - z)))))));
}

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.pow(t_1, ((1.0 - z) - 0.5)) * math.sqrt((math.pi + math.pi))) * (math.exp(-t_1) * (((((((606.6766809125655 * z) - -545.0353078134797) * z) - -436.8961723502244) * z) - -263.3831855358925) + ((9.984369578019572e-6 / t_0) + (-1.5056327351493116e-7 / (-7.0 - (1.0 - z)))))))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - -6.0)
	t_1 = Float64(t_0 - -0.5)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64((t_1 ^ Float64(Float64(1.0 - z) - 0.5)) * sqrt(Float64(pi + pi))) * Float64(exp(Float64(-t_1)) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(606.6766809125655 * z) - -545.0353078134797) * z) - -436.8961723502244) * z) - -263.3831855358925) + Float64(Float64(9.984369578019572e-6 / t_0) + Float64(-1.5056327351493116e-7 / Float64(-7.0 - Float64(1.0 - z))))))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - -6.0;
	t_1 = t_0 - -0.5;
	tmp = (pi / sin((pi * z))) * (((t_1 ^ ((1.0 - z) - 0.5)) * sqrt((pi + pi))) * (exp(-t_1) * (((((((606.6766809125655 * z) - -545.0353078134797) * z) - -436.8961723502244) * z) - -263.3831855358925) + ((9.984369578019572e-6 / t_0) + (-1.5056327351493116e-7 / (-7.0 - (1.0 - z)))))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1 - z), $MachinePrecision] - -6), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - -1/2), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$1, N[(N[(1 - z), $MachinePrecision] - 1/2), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi + Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-t$95$1)], $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(65521081538557082921549/108000000000000000000 * z), $MachinePrecision] - -367898832774098786021/675000000000000000), $MachinePrecision] * z), $MachinePrecision] - -131068851705067315609/300000000000000000), $MachinePrecision] * z), $MachinePrecision] - -9876869457595968283/37500000000000000), $MachinePrecision] + N[(N[(2496092394504893/250000000000000000000 / t$95$0), $MachinePrecision] + N[(-3764081837873279/25000000000000000000000 / N[(-7 - N[(1 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 96.4%

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

2. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\color{blue}{\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + \color{blue}{z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \color{blue}{\left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + \color{blue}{z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \color{blue}{\left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \color{blue}{\frac{65521081538557082921549}{108000000000000000000} \cdot z}\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower-*.f64 96.6%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot \color{blue}{z}\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 96.6%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\color{blue}{\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\sqrt{\pi \cdot 2}} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\pi \cdot 2}} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. sqrt-prod N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lift-PI.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-unsound-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lift-PI.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\color{blue}{\pi}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lower-unsound-sqrt.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\color{blue}{\sqrt{\pi}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lower-unsound-sqrt.f64 97.4%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \color{blue}{\sqrt{2}}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Applied rewrites 97.4%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

8. Applied rewrites 96.6%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(\left({\left(\left(\left(1 - z\right) - -6\right) - \frac{-1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{\pi + \pi}\right) \cdot \left(e^{-\left(\left(\left(1 - z\right) - -6\right) - \frac{-1}{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{65521081538557082921549}{108000000000000000000} \cdot z + \frac{367898832774098786021}{675000000000000000}\right) \cdot z + \frac{131068851705067315609}{300000000000000000}\right) \cdot z + \frac{9876869457595968283}{37500000000000000}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right)\right)\right)} \]

10. Applied rewrites 97.4%

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left({\left(\left(\left(1 - z\right) - -6\right) - \frac{-1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{\pi + \pi}\right) \cdot \left(e^{-\left(\left(\left(1 - z\right) - -6\right) - \frac{-1}{2}\right)} \cdot \color{blue}{\left(\left(\left(\left(\frac{65521081538557082921549}{108000000000000000000} \cdot z - \frac{-367898832774098786021}{675000000000000000}\right) \cdot z - \frac{-131068851705067315609}{300000000000000000}\right) \cdot z - \frac{-9876869457595968283}{37500000000000000}\right) + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \frac{\frac{-3764081837873279}{25000000000000000000000}}{-7 - \left(1 - z\right)}\right)\right)}\right)\right) \]
11. Add Preprocessing

Alternative 6: 97.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \left(1 - z\right) - -6\\ t_1 := t\_0 - \frac{-1}{2}\\ \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(e^{-t\_1} \cdot \left({t\_1}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{\pi + \pi}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\frac{65521081538557082921549}{108000000000000000000} \cdot z + \frac{367898832774098786021}{675000000000000000}\right) \cdot z + \frac{131068851705067315609}{300000000000000000}\right) \cdot z + \frac{9876869457595968283}{37500000000000000}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{t\_0}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \end{array} \]

FPCore

(FPCore (z)
  :precision binary64
  (let* ((t_0 (- (- 1 z) -6)) (t_1 (- t_0 -1/2)))
  (*
   (*
    (/ PI (sin (* z PI)))
    (* (exp (- t_1)) (* (pow t_1 (- (- 1 z) 1/2)) (sqrt (+ PI PI)))))
   (+
    (+
     (+
      (*
       (+
        (*
         (+
          (* 65521081538557082921549/108000000000000000000 z)
          367898832774098786021/675000000000000000)
         z)
        131068851705067315609/300000000000000000)
       z)
      9876869457595968283/37500000000000000)
     (/ 2496092394504893/250000000000000000000 t_0))
    (/ 3764081837873279/25000000000000000000000 (- (- 1 z) -7))))))

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((z * ((double) M_PI)))) * (exp(-t_1) * (pow(t_1, ((1.0 - z) - 0.5)) * sqrt((((double) M_PI) + ((double) M_PI)))))) * ((((((((606.6766809125655 * z) + 545.0353078134797) * z) + 436.8961723502244) * z) + 263.3831855358925) + (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((z * Math.PI))) * (Math.exp(-t_1) * (Math.pow(t_1, ((1.0 - z) - 0.5)) * Math.sqrt((Math.PI + Math.PI))))) * ((((((((606.6766809125655 * z) + 545.0353078134797) * z) + 436.8961723502244) * z) + 263.3831855358925) + (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((z * math.pi))) * (math.exp(-t_1) * (math.pow(t_1, ((1.0 - z) - 0.5)) * math.sqrt((math.pi + math.pi))))) * ((((((((606.6766809125655 * z) + 545.0353078134797) * z) + 436.8961723502244) * z) + 263.3831855358925) + (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(z * pi))) * Float64(exp(Float64(-t_1)) * Float64((t_1 ^ Float64(Float64(1.0 - z) - 0.5)) * sqrt(Float64(pi + pi))))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(606.6766809125655 * z) + 545.0353078134797) * z) + 436.8961723502244) * z) + 263.3831855358925) + 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((z * pi))) * (exp(-t_1) * ((t_1 ^ ((1.0 - z) - 0.5)) * sqrt((pi + pi))))) * ((((((((606.6766809125655 * z) + 545.0353078134797) * z) + 436.8961723502244) * z) + 263.3831855358925) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1 - z), $MachinePrecision] - -6), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - -1/2), $MachinePrecision]}, N[(N[(N[(Pi / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-t$95$1)], $MachinePrecision] * N[(N[Power[t$95$1, N[(N[(1 - z), $MachinePrecision] - 1/2), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi + Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(65521081538557082921549/108000000000000000000 * z), $MachinePrecision] + 367898832774098786021/675000000000000000), $MachinePrecision] * z), $MachinePrecision] + 131068851705067315609/300000000000000000), $MachinePrecision] * z), $MachinePrecision] + 9876869457595968283/37500000000000000), $MachinePrecision] + N[(2496092394504893/250000000000000000000 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(3764081837873279/25000000000000000000000 / N[(N[(1 - z), $MachinePrecision] - -7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 96.4%

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

2. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\color{blue}{\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + \color{blue}{z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \color{blue}{\left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + \color{blue}{z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \color{blue}{\left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \color{blue}{\frac{65521081538557082921549}{108000000000000000000} \cdot z}\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower-*.f64 96.6%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot \color{blue}{z}\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 96.6%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\color{blue}{\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Applied rewrites 96.8%

   \[\leadsto \color{blue}{\left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(e^{-\left(\left(\left(1 - z\right) - -6\right) - \frac{-1}{2}\right)} \cdot \left({\left(\left(\left(1 - z\right) - -6\right) - \frac{-1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{\pi + \pi}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\frac{65521081538557082921549}{108000000000000000000} \cdot z + \frac{367898832774098786021}{675000000000000000}\right) \cdot z + \frac{131068851705067315609}{300000000000000000}\right) \cdot z + \frac{9876869457595968283}{37500000000000000}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right)} \]
7. Add Preprocessing

Alternative 7: 96.8% accurate, 1.4× speedup

Math

\[\left(\left(\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \pi - \frac{-4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \left(z \cdot \pi\right)\right) \cdot z + \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \pi\right) \cdot z + \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \pi\right) \cdot \left(\left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{\pi + \pi}\right)\right) \cdot \frac{1}{\sin \left(z \cdot \pi\right)}\right) \]

FPCore

(FPCore (z)
  :precision binary64
  (*
 (+
  (*
   (+
    (*
     (-
      (*
       64608921419941589693928044520019/118540800000000000000000000000
       PI)
      (*
       -4027292589444183035165374538123333/6638284800000000000000000000000
       (* z PI)))
     z)
    (* 102757979785251069442117317613/235200000000000000000000000 PI))
   z)
  (* 1106209385320415913103082059/4200000000000000000000000 PI))
 (*
  (*
   (pow (- (- 1 z) -13/2) (- (- 1 z) 1/2))
   (* (exp (- z 15/2)) (sqrt (+ PI PI))))
  (/ 1 (sin (* z PI))))))

C

double code(double z) {
	return ((((((545.0353078428827 * ((double) M_PI)) - (-606.6766809167608 * (z * ((double) M_PI)))) * z) + (436.8961725563396 * ((double) M_PI))) * z) + (263.3831869810514 * ((double) M_PI))) * ((pow(((1.0 - z) - -6.5), ((1.0 - z) - 0.5)) * (exp((z - 7.5)) * sqrt((((double) M_PI) + ((double) M_PI))))) * (1.0 / sin((z * ((double) M_PI)))));
}

Java

public static double code(double z) {
	return ((((((545.0353078428827 * Math.PI) - (-606.6766809167608 * (z * Math.PI))) * z) + (436.8961725563396 * Math.PI)) * z) + (263.3831869810514 * Math.PI)) * ((Math.pow(((1.0 - z) - -6.5), ((1.0 - z) - 0.5)) * (Math.exp((z - 7.5)) * Math.sqrt((Math.PI + Math.PI)))) * (1.0 / Math.sin((z * Math.PI))));
}

Python

def code(z):
	return ((((((545.0353078428827 * math.pi) - (-606.6766809167608 * (z * math.pi))) * z) + (436.8961725563396 * math.pi)) * z) + (263.3831869810514 * math.pi)) * ((math.pow(((1.0 - z) - -6.5), ((1.0 - z) - 0.5)) * (math.exp((z - 7.5)) * math.sqrt((math.pi + math.pi)))) * (1.0 / math.sin((z * math.pi))))

Julia

function code(z)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(545.0353078428827 * pi) - Float64(-606.6766809167608 * Float64(z * pi))) * z) + Float64(436.8961725563396 * pi)) * z) + Float64(263.3831869810514 * pi)) * Float64(Float64((Float64(Float64(1.0 - z) - -6.5) ^ Float64(Float64(1.0 - z) - 0.5)) * Float64(exp(Float64(z - 7.5)) * sqrt(Float64(pi + pi)))) * Float64(1.0 / sin(Float64(z * pi)))))
end

MATLAB

function tmp = code(z)
	tmp = ((((((545.0353078428827 * pi) - (-606.6766809167608 * (z * pi))) * z) + (436.8961725563396 * pi)) * z) + (263.3831869810514 * pi)) * (((((1.0 - z) - -6.5) ^ ((1.0 - z) - 0.5)) * (exp((z - 7.5)) * sqrt((pi + pi)))) * (1.0 / sin((z * pi))));
end

Wolfram

code[z_] := N[(N[(N[(N[(N[(N[(N[(64608921419941589693928044520019/118540800000000000000000000000 * Pi), $MachinePrecision] - N[(-4027292589444183035165374538123333/6638284800000000000000000000000 * N[(z * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] + N[(102757979785251069442117317613/235200000000000000000000000 * Pi), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] + N[(1106209385320415913103082059/4200000000000000000000000 * Pi), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(N[(1 - z), $MachinePrecision] - -13/2), $MachinePrecision], N[(N[(1 - z), $MachinePrecision] - 1/2), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(z - 15/2), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi + Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1 / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.4%

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

3. Applied rewrites 96.7%

   \[\leadsto \color{blue}{\left(\left(\pi \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) - \frac{\frac{-6765203681218851}{10000000000000}}{1 - z}\right) - \frac{\frac{-7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) - \frac{\frac{-2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) - \frac{\frac{3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5}\right) - \frac{\frac{-2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) - \frac{\frac{-3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(\left(e^{\left(0 + z\right) + \frac{-15}{2}} \cdot \sqrt{\pi + \pi}\right) \cdot {\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right)\right) \cdot \frac{1}{\sin \left(z \cdot \pi\right)}} \]

4. Taylor expanded in z around 0

   \[\leadsto \left(\color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \pi + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \pi + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \pi + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \left(z \cdot \pi\right)\right)\right)\right)} \cdot \left(\left(e^{\left(0 + z\right) + \frac{-15}{2}} \cdot \sqrt{\pi + \pi}\right) \cdot {\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right)\right) \cdot \frac{1}{\sin \left(z \cdot \pi\right)} \]
5. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \mathsf{PI}\left(\right) + \color{blue}{z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \mathsf{PI}\left(\right) + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \mathsf{PI}\left(\right) + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \left(z \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \cdot \left(\left(e^{\left(0 + z\right) + \frac{-15}{2}} \cdot \sqrt{\pi + \pi}\right) \cdot {\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right)\right) \cdot \frac{1}{\sin \left(z \cdot \pi\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \mathsf{PI}\left(\right) + \color{blue}{z} \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \mathsf{PI}\left(\right) + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \mathsf{PI}\left(\right) + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \left(z \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(\left(e^{\left(0 + z\right) + \frac{-15}{2}} \cdot \sqrt{\pi + \pi}\right) \cdot {\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right)\right) \cdot \frac{1}{\sin \left(z \cdot \pi\right)} \]

   3. lower-PI.f64 N/A

      \[\leadsto \left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \pi + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \mathsf{PI}\left(\right) + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \mathsf{PI}\left(\right) + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \left(z \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(\left(e^{\left(0 + z\right) + \frac{-15}{2}} \cdot \sqrt{\pi + \pi}\right) \cdot {\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right)\right) \cdot \frac{1}{\sin \left(z \cdot \pi\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \pi + z \cdot \color{blue}{\left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \mathsf{PI}\left(\right) + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \mathsf{PI}\left(\right) + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \left(z \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \cdot \left(\left(e^{\left(0 + z\right) + \frac{-15}{2}} \cdot \sqrt{\pi + \pi}\right) \cdot {\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right)\right) \cdot \frac{1}{\sin \left(z \cdot \pi\right)} \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \pi + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \mathsf{PI}\left(\right) + \color{blue}{z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \mathsf{PI}\left(\right) + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \left(z \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \cdot \left(\left(e^{\left(0 + z\right) + \frac{-15}{2}} \cdot \sqrt{\pi + \pi}\right) \cdot {\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right)\right) \cdot \frac{1}{\sin \left(z \cdot \pi\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \pi + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \mathsf{PI}\left(\right) + \color{blue}{z} \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \mathsf{PI}\left(\right) + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \left(z \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(\left(e^{\left(0 + z\right) + \frac{-15}{2}} \cdot \sqrt{\pi + \pi}\right) \cdot {\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right)\right) \cdot \frac{1}{\sin \left(z \cdot \pi\right)} \]

   7. lower-PI.f64 N/A

      \[\leadsto \left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \pi + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \pi + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \mathsf{PI}\left(\right) + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \left(z \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(\left(e^{\left(0 + z\right) + \frac{-15}{2}} \cdot \sqrt{\pi + \pi}\right) \cdot {\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right)\right) \cdot \frac{1}{\sin \left(z \cdot \pi\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \pi + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \pi + z \cdot \color{blue}{\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \mathsf{PI}\left(\right) + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \left(z \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \cdot \left(\left(e^{\left(0 + z\right) + \frac{-15}{2}} \cdot \sqrt{\pi + \pi}\right) \cdot {\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right)\right) \cdot \frac{1}{\sin \left(z \cdot \pi\right)} \]

6. Applied rewrites 96.6%

   \[\leadsto \left(\color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \pi + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \pi + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \pi + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \left(z \cdot \pi\right)\right)\right)\right)} \cdot \left(\left(e^{\left(0 + z\right) + \frac{-15}{2}} \cdot \sqrt{\pi + \pi}\right) \cdot {\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right)\right) \cdot \frac{1}{\sin \left(z \cdot \pi\right)} \]

8. Applied rewrites 96.8%

   \[\leadsto \color{blue}{\left(\left(\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \pi - \frac{-4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \left(z \cdot \pi\right)\right) \cdot z + \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \pi\right) \cdot z + \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \pi\right) \cdot \left(\left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{\pi + \pi}\right)\right) \cdot \frac{1}{\sin \left(z \cdot \pi\right)}\right)} \]
9. Add Preprocessing

Alternative 8: 96.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := t\_0 + 7\\ t_2 := t\_1 + \frac{1}{2}\\ \frac{1}{z} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {t\_2}^{\left(t\_0 + \frac{1}{2}\right)}\right) \cdot e^{-t\_2}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{t\_1}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{t\_0 + 8}\right)\right) \end{array} \]

FPCore

(FPCore (z)
  :precision binary64
  (let* ((t_0 (- (- 1 z) 1)) (t_1 (+ t_0 7)) (t_2 (+ t_1 1/2)))
  (*
   (/ 1 z)
   (*
    (* (* (* (sqrt PI) (sqrt 2)) (pow t_2 (+ t_0 1/2))) (exp (- t_2)))
    (+
     (+
      (+
       9876869457595968283/37500000000000000
       (*
        z
        (+
         131068851705067315609/300000000000000000
         (*
          z
          (+
           367898832774098786021/675000000000000000
           (* 65521081538557082921549/108000000000000000000 z))))))
      (/ 2496092394504893/250000000000000000000 t_1))
     (/ 3764081837873279/25000000000000000000000 (+ t_0 8)))))))

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 (1.0 / z) * ((((sqrt(((double) M_PI)) * sqrt(2.0)) * pow(t_2, (t_0 + 0.5))) * exp(-t_2)) * (((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (606.6766809125655 * z)))))) + (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 (1.0 / z) * ((((Math.sqrt(Math.PI) * Math.sqrt(2.0)) * Math.pow(t_2, (t_0 + 0.5))) * Math.exp(-t_2)) * (((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (606.6766809125655 * z)))))) + (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 (1.0 / z) * ((((math.sqrt(math.pi) * math.sqrt(2.0)) * math.pow(t_2, (t_0 + 0.5))) * math.exp(-t_2)) * (((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (606.6766809125655 * z)))))) + (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(1.0 / z) * Float64(Float64(Float64(Float64(sqrt(pi) * sqrt(2.0)) * (t_2 ^ Float64(t_0 + 0.5))) * exp(Float64(-t_2))) * Float64(Float64(Float64(263.3831855358925 + Float64(z * Float64(436.8961723502244 + Float64(z * Float64(545.0353078134797 + Float64(606.6766809125655 * z)))))) + 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 = (1.0 / z) * ((((sqrt(pi) * sqrt(2.0)) * (t_2 ^ (t_0 + 0.5))) * exp(-t_2)) * (((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (606.6766809125655 * z)))))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1 - z), $MachinePrecision] - 1), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 7), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + 1/2), $MachinePrecision]}, N[(N[(1 / z), $MachinePrecision] * N[(N[(N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$2, N[(t$95$0 + 1/2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$2)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(9876869457595968283/37500000000000000 + N[(z * N[(131068851705067315609/300000000000000000 + N[(z * N[(367898832774098786021/675000000000000000 + N[(65521081538557082921549/108000000000000000000 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2496092394504893/250000000000000000000 / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(3764081837873279/25000000000000000000000 / N[(t$95$0 + 8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 96.4%

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

2. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\color{blue}{\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + \color{blue}{z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \color{blue}{\left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + \color{blue}{z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \color{blue}{\left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \color{blue}{\frac{65521081538557082921549}{108000000000000000000} \cdot z}\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower-*.f64 96.6%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot \color{blue}{z}\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 96.6%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\color{blue}{\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\sqrt{\pi \cdot 2}} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\pi \cdot 2}} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. sqrt-prod N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lift-PI.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-unsound-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lift-PI.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\color{blue}{\pi}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lower-unsound-sqrt.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\color{blue}{\sqrt{\pi}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lower-unsound-sqrt.f64 97.4%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \color{blue}{\sqrt{2}}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Applied rewrites 97.4%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

7. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
8. Step-by-step derivation

   1. lower-/.f64 97.2%

      \[\leadsto \frac{1}{\color{blue}{z}} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

9. Applied rewrites 97.2%

   \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
10. Add Preprocessing

Alternative 9: 96.4% accurate, 1.8× speedup

Math

\[\left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \pi + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \pi + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \pi + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \left(z \cdot \pi\right)\right)\right)\right) \cdot \left(\left(e^{\left(0 + z\right) + \frac{-15}{2}} \cdot \sqrt{\pi + \pi}\right) \cdot {\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right)\right) \cdot \frac{1}{z \cdot \pi} \]

FPCore

(FPCore (z)
  :precision binary64
  (*
 (*
  (+
   (* 1106209385320415913103082059/4200000000000000000000000 PI)
   (*
    z
    (+
     (* 102757979785251069442117317613/235200000000000000000000000 PI)
     (*
      z
      (+
       (*
        64608921419941589693928044520019/118540800000000000000000000000
        PI)
       (*
        4027292589444183035165374538123333/6638284800000000000000000000000
        (* z PI)))))))
  (*
   (* (exp (+ (+ 0 z) -15/2)) (sqrt (+ PI PI)))
   (pow (- (- 1 z) -13/2) (- (- 1 z) 1/2))))
 (/ 1 (* z PI))))

C

double code(double z) {
	return (((263.3831869810514 * ((double) M_PI)) + (z * ((436.8961725563396 * ((double) M_PI)) + (z * ((545.0353078428827 * ((double) M_PI)) + (606.6766809167608 * (z * ((double) M_PI)))))))) * ((exp(((0.0 + z) + -7.5)) * sqrt((((double) M_PI) + ((double) M_PI)))) * pow(((1.0 - z) - -6.5), ((1.0 - z) - 0.5)))) * (1.0 / (z * ((double) M_PI)));
}

Java

public static double code(double z) {
	return (((263.3831869810514 * Math.PI) + (z * ((436.8961725563396 * Math.PI) + (z * ((545.0353078428827 * Math.PI) + (606.6766809167608 * (z * Math.PI))))))) * ((Math.exp(((0.0 + z) + -7.5)) * Math.sqrt((Math.PI + Math.PI))) * Math.pow(((1.0 - z) - -6.5), ((1.0 - z) - 0.5)))) * (1.0 / (z * Math.PI));
}

Python

def code(z):
	return (((263.3831869810514 * math.pi) + (z * ((436.8961725563396 * math.pi) + (z * ((545.0353078428827 * math.pi) + (606.6766809167608 * (z * math.pi))))))) * ((math.exp(((0.0 + z) + -7.5)) * math.sqrt((math.pi + math.pi))) * math.pow(((1.0 - z) - -6.5), ((1.0 - z) - 0.5)))) * (1.0 / (z * math.pi))

Julia

function code(z)
	return Float64(Float64(Float64(Float64(263.3831869810514 * pi) + Float64(z * Float64(Float64(436.8961725563396 * pi) + Float64(z * Float64(Float64(545.0353078428827 * pi) + Float64(606.6766809167608 * Float64(z * pi))))))) * Float64(Float64(exp(Float64(Float64(0.0 + z) + -7.5)) * sqrt(Float64(pi + pi))) * (Float64(Float64(1.0 - z) - -6.5) ^ Float64(Float64(1.0 - z) - 0.5)))) * Float64(1.0 / Float64(z * pi)))
end

MATLAB

function tmp = code(z)
	tmp = (((263.3831869810514 * pi) + (z * ((436.8961725563396 * pi) + (z * ((545.0353078428827 * pi) + (606.6766809167608 * (z * pi))))))) * ((exp(((0.0 + z) + -7.5)) * sqrt((pi + pi))) * (((1.0 - z) - -6.5) ^ ((1.0 - z) - 0.5)))) * (1.0 / (z * pi));
end

Wolfram

code[z_] := N[(N[(N[(N[(1106209385320415913103082059/4200000000000000000000000 * Pi), $MachinePrecision] + N[(z * N[(N[(102757979785251069442117317613/235200000000000000000000000 * Pi), $MachinePrecision] + N[(z * N[(N[(64608921419941589693928044520019/118540800000000000000000000000 * Pi), $MachinePrecision] + N[(4027292589444183035165374538123333/6638284800000000000000000000000 * N[(z * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Exp[N[(N[(0 + z), $MachinePrecision] + -15/2), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi + Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(1 - z), $MachinePrecision] - -13/2), $MachinePrecision], N[(N[(1 - z), $MachinePrecision] - 1/2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1 / N[(z * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.4%

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

3. Applied rewrites 96.7%

   \[\leadsto \color{blue}{\left(\left(\pi \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) - \frac{\frac{-6765203681218851}{10000000000000}}{1 - z}\right) - \frac{\frac{-7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) - \frac{\frac{-2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) - \frac{\frac{3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5}\right) - \frac{\frac{-2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) - \frac{\frac{-3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(\left(e^{\left(0 + z\right) + \frac{-15}{2}} \cdot \sqrt{\pi + \pi}\right) \cdot {\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right)\right) \cdot \frac{1}{\sin \left(z \cdot \pi\right)}} \]

4. Taylor expanded in z around 0

   \[\leadsto \left(\color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \pi + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \pi + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \pi + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \left(z \cdot \pi\right)\right)\right)\right)} \cdot \left(\left(e^{\left(0 + z\right) + \frac{-15}{2}} \cdot \sqrt{\pi + \pi}\right) \cdot {\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right)\right) \cdot \frac{1}{\sin \left(z \cdot \pi\right)} \]
5. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \mathsf{PI}\left(\right) + \color{blue}{z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \mathsf{PI}\left(\right) + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \mathsf{PI}\left(\right) + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \left(z \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \cdot \left(\left(e^{\left(0 + z\right) + \frac{-15}{2}} \cdot \sqrt{\pi + \pi}\right) \cdot {\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right)\right) \cdot \frac{1}{\sin \left(z \cdot \pi\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \mathsf{PI}\left(\right) + \color{blue}{z} \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \mathsf{PI}\left(\right) + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \mathsf{PI}\left(\right) + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \left(z \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(\left(e^{\left(0 + z\right) + \frac{-15}{2}} \cdot \sqrt{\pi + \pi}\right) \cdot {\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right)\right) \cdot \frac{1}{\sin \left(z \cdot \pi\right)} \]

   3. lower-PI.f64 N/A

      \[\leadsto \left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \pi + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \mathsf{PI}\left(\right) + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \mathsf{PI}\left(\right) + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \left(z \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(\left(e^{\left(0 + z\right) + \frac{-15}{2}} \cdot \sqrt{\pi + \pi}\right) \cdot {\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right)\right) \cdot \frac{1}{\sin \left(z \cdot \pi\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \pi + z \cdot \color{blue}{\left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \mathsf{PI}\left(\right) + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \mathsf{PI}\left(\right) + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \left(z \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \cdot \left(\left(e^{\left(0 + z\right) + \frac{-15}{2}} \cdot \sqrt{\pi + \pi}\right) \cdot {\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right)\right) \cdot \frac{1}{\sin \left(z \cdot \pi\right)} \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \pi + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \mathsf{PI}\left(\right) + \color{blue}{z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \mathsf{PI}\left(\right) + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \left(z \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \cdot \left(\left(e^{\left(0 + z\right) + \frac{-15}{2}} \cdot \sqrt{\pi + \pi}\right) \cdot {\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right)\right) \cdot \frac{1}{\sin \left(z \cdot \pi\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \pi + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \mathsf{PI}\left(\right) + \color{blue}{z} \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \mathsf{PI}\left(\right) + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \left(z \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(\left(e^{\left(0 + z\right) + \frac{-15}{2}} \cdot \sqrt{\pi + \pi}\right) \cdot {\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right)\right) \cdot \frac{1}{\sin \left(z \cdot \pi\right)} \]

   7. lower-PI.f64 N/A

      \[\leadsto \left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \pi + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \pi + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \mathsf{PI}\left(\right) + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \left(z \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(\left(e^{\left(0 + z\right) + \frac{-15}{2}} \cdot \sqrt{\pi + \pi}\right) \cdot {\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right)\right) \cdot \frac{1}{\sin \left(z \cdot \pi\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \pi + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \pi + z \cdot \color{blue}{\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \mathsf{PI}\left(\right) + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \left(z \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \cdot \left(\left(e^{\left(0 + z\right) + \frac{-15}{2}} \cdot \sqrt{\pi + \pi}\right) \cdot {\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right)\right) \cdot \frac{1}{\sin \left(z \cdot \pi\right)} \]

6. Applied rewrites 96.6%

   \[\leadsto \left(\color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \pi + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \pi + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \pi + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \left(z \cdot \pi\right)\right)\right)\right)} \cdot \left(\left(e^{\left(0 + z\right) + \frac{-15}{2}} \cdot \sqrt{\pi + \pi}\right) \cdot {\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right)\right) \cdot \frac{1}{\sin \left(z \cdot \pi\right)} \]

7. Taylor expanded in z around 0

   \[\leadsto \left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \pi + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \pi + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \pi + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \left(z \cdot \pi\right)\right)\right)\right) \cdot \left(\left(e^{\left(0 + z\right) + \frac{-15}{2}} \cdot \sqrt{\pi + \pi}\right) \cdot {\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right)\right) \cdot \color{blue}{\frac{1}{z \cdot \pi}} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \pi + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \pi + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \pi + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \left(z \cdot \pi\right)\right)\right)\right) \cdot \left(\left(e^{\left(0 + z\right) + \frac{-15}{2}} \cdot \sqrt{\pi + \pi}\right) \cdot {\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right)\right) \cdot \frac{1}{\color{blue}{z \cdot \mathsf{PI}\left(\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \pi + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \pi + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \pi + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \left(z \cdot \pi\right)\right)\right)\right) \cdot \left(\left(e^{\left(0 + z\right) + \frac{-15}{2}} \cdot \sqrt{\pi + \pi}\right) \cdot {\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right)\right) \cdot \frac{1}{z \cdot \color{blue}{\mathsf{PI}\left(\right)}} \]

   3. lower-PI.f64 96.2%

      \[\leadsto \left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \pi + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \pi + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \pi + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \left(z \cdot \pi\right)\right)\right)\right) \cdot \left(\left(e^{\left(0 + z\right) + \frac{-15}{2}} \cdot \sqrt{\pi + \pi}\right) \cdot {\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right)\right) \cdot \frac{1}{z \cdot \pi} \]

9. Applied rewrites 96.2%

   \[\leadsto \left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \pi + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \pi + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \pi + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \left(z \cdot \pi\right)\right)\right)\right) \cdot \left(\left(e^{\left(0 + z\right) + \frac{-15}{2}} \cdot \sqrt{\pi + \pi}\right) \cdot {\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right)\right) \cdot \color{blue}{\frac{1}{z \cdot \pi}} \]
10. Add Preprocessing

Alternative 10: 96.2% accurate, 2.3× speedup

Math

\[\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)}{z} \]

FPCore

(FPCore (z)
  :precision binary64
  (*
 1106209385320415913103082059/4200000000000000000000000
 (/ (* (exp -15/2) (* (sqrt 2) (* (sqrt PI) (pow 15/2 1/2)))) z)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[z_] := N[(1106209385320415913103082059/4200000000000000000000000 * N[(N[(N[Exp[-15/2], $MachinePrecision] * N[(N[Sqrt[2], $MachinePrecision] * N[(N[Sqrt[Pi], $MachinePrecision] * N[Power[15/2, 1/2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.4%

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

2. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\color{blue}{\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + \color{blue}{z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \color{blue}{\left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + \color{blue}{z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \color{blue}{\left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \color{blue}{\frac{65521081538557082921549}{108000000000000000000} \cdot z}\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower-*.f64 96.6%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot \color{blue}{z}\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 96.6%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\color{blue}{\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\sqrt{\pi \cdot 2}} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\pi \cdot 2}} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. sqrt-prod N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lift-PI.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-unsound-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lift-PI.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\color{blue}{\pi}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lower-unsound-sqrt.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\color{blue}{\sqrt{\pi}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lower-unsound-sqrt.f64 97.4%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \color{blue}{\sqrt{2}}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Applied rewrites 97.4%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

7. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)}{z}} \]
8. 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

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

9. Applied rewrites 96.4%

   \[\leadsto \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)}{z}} \]
10. Add Preprocessing

Alternative 11: 95.7% accurate, 3.9× speedup

Math

\[\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{1}{\frac{z}{\sqrt{15 \cdot \pi} \cdot e^{\frac{-15}{2}}}} \]

FPCore

(FPCore (z)
  :precision binary64
  (*
 1106209385320415913103082059/4200000000000000000000000
 (/ 1 (/ z (* (sqrt (* 15 PI)) (exp -15/2))))))

C

double code(double z) {
	return 263.3831869810514 * (1.0 / (z / (sqrt((15.0 * ((double) M_PI))) * exp(-7.5))));
}

Java

public static double code(double z) {
	return 263.3831869810514 * (1.0 / (z / (Math.sqrt((15.0 * Math.PI)) * Math.exp(-7.5))));
}

Python

def code(z):
	return 263.3831869810514 * (1.0 / (z / (math.sqrt((15.0 * math.pi)) * math.exp(-7.5))))

Julia

function code(z)
	return Float64(263.3831869810514 * Float64(1.0 / Float64(z / Float64(sqrt(Float64(15.0 * pi)) * exp(-7.5)))))
end

MATLAB

function tmp = code(z)
	tmp = 263.3831869810514 * (1.0 / (z / (sqrt((15.0 * pi)) * exp(-7.5))));
end

Wolfram

code[z_] := N[(1106209385320415913103082059/4200000000000000000000000 * N[(1 / N[(z / N[(N[Sqrt[N[(15 * Pi), $MachinePrecision]], $MachinePrecision] * N[Exp[-15/2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.4%

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

2. Taylor expanded in z around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

4. Applied rewrites 95.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 95.4%

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow1/2 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. sqrt-unprod N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-*.f64 95.4%

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

   14. lift-*.f64 N/A

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

   15. count-2-rev N/A

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

   16. lift-+.f64 95.4%

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

6. Applied rewrites 95.4%

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

7. Taylor expanded in z around 0

   \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \pi}}{\color{blue}{z}} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

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

   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-exp.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 95.7%

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \pi}}{z} \]

9. Applied rewrites 95.7%

   \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \pi}}{\color{blue}{z}} \]
10. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \pi}}{z} \]

    2. div-flip N/A

       \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{1}{\frac{z}{\color{blue}{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \pi}}}} \]

    3. lower-unsound-/.f64 N/A

       \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{1}{\frac{z}{\color{blue}{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \pi}}}} \]

    4. lower-unsound-/.f64 95.7%

       \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{1}{\frac{z}{e^{\frac{-15}{2}} \cdot \color{blue}{\sqrt{15 \cdot \pi}}}} \]

    5. lift-*.f64 N/A

       \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{1}{\frac{z}{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \pi}}} \]

    6. *-commutative N/A

       \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{1}{\frac{z}{\sqrt{15 \cdot \pi} \cdot e^{\frac{-15}{2}}}} \]

    7. lower-*.f64 95.7%

       \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{1}{\frac{z}{\sqrt{15 \cdot \pi} \cdot e^{\frac{-15}{2}}}} \]

11. Applied rewrites 95.7%

    \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{1}{\frac{z}{\color{blue}{\sqrt{15 \cdot \pi} \cdot e^{\frac{-15}{2}}}}} \]
12. Add Preprocessing

Alternative 12: 95.7% accurate, 4.2× speedup

Math

\[\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \pi}}{z} \]

FPCore

(FPCore (z)
  :precision binary64
  (*
 1106209385320415913103082059/4200000000000000000000000
 (/ (* (exp -15/2) (sqrt (* 15 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[(1106209385320415913103082059/4200000000000000000000000 * N[(N[(N[Exp[-15/2], $MachinePrecision] * N[Sqrt[N[(15 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.4%

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

2. Taylor expanded in z around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

4. Applied rewrites 95.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 95.4%

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow1/2 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. sqrt-unprod N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-*.f64 95.4%

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

   14. lift-*.f64 N/A

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

   15. count-2-rev N/A

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

   16. lift-+.f64 95.4%

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

6. Applied rewrites 95.4%

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

7. Taylor expanded in z around 0

   \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \pi}}{\color{blue}{z}} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

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

   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-exp.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 95.7%

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \pi}}{z} \]

9. Applied rewrites 95.7%

   \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \pi}}{\color{blue}{z}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (z)
  :name "Jmat.Real.gamma, branch z less than 0.5"
  :precision binary64
  :pre (<= z 1/2)
  (* (/ PI (sin (* PI z))) (* (* (* (sqrt (* PI 2)) (pow (+ (+ (- (- 1 z) 1) 7) 1/2) (+ (- (- 1 z) 1) 1/2))) (exp (- (+ (+ (- (- 1 z) 1) 7) 1/2)))) (+ (+ (+ (+ (+ (+ (+ (+ 9999999999998099/10000000000000000 (/ 6765203681218851/10000000000000 (+ (- (- 1 z) 1) 1))) (/ -3147848041806007/2500000000000 (+ (- (- 1 z) 1) 2))) (/ 7713234287776531/10000000000000 (+ (- (- 1 z) 1) 3))) (/ -883075145810703/5000000000000 (+ (- (- 1 z) 1) 4))) (/ 2501468655737381/200000000000000 (+ (- (- 1 z) 1) 5))) (/ -3464277381643003/25000000000000000 (+ (- (- 1 z) 1) 6))) (/ 2496092394504893/250000000000000000000 (+ (- (- 1 z) 1) 7))) (/ 3764081837873279/25000000000000000000000 (+ (- (- 1 z) 1) 8))))))