Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.4% → 98.8%

Time: 12.2s

Alternatives: 10

Speedup: 3.0×

• could not determine a ground truth

  1. z = -2556498798696.163

Specification

\[z \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := \left(1 - z\right) - 0\\ t_2 := \frac{676.5203681218851}{t\_1}\\ t_3 := 0.9999999999996197 + \left(\frac{-457679.80848377093}{\left(0 - \left(1 - z\right)\right) \cdot t\_1} - 0.9999999999998099 \cdot t\_2\right)\\ t_4 := t\_0 + 7\\ t_5 := t\_4 + 0.5\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {t\_5}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{-t\_5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{0.9999999999994297}{t\_3} + \frac{{t\_2}^{3}}{t\_3}\right) + \frac{-1259.1392167224028}{t\_0 + 2}\right) + \frac{771.3234287776531}{t\_0 + 3}\right) + \frac{-176.6150291621406}{t\_0 + 4}\right) + \frac{12.507343278686905}{t\_0 + 5}\right) + \frac{-0.13857109526572012}{t\_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_4}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0))
        (t_1 (- (- 1.0 z) 0.0))
        (t_2 (/ 676.5203681218851 t_1))
        (t_3
         (+
          0.9999999999996197
          (-
           (/ -457679.80848377093 (* (- 0.0 (- 1.0 z)) t_1))
           (* 0.9999999999998099 t_2))))
        (t_4 (+ t_0 7.0))
        (t_5 (+ t_4 0.5)))
   (*
    (/ PI (sin (* PI z)))
    (*
     (* (* (sqrt (* PI 2.0)) (pow t_5 (+ t_0 0.5))) (exp (- t_5)))
     (+
      (+
       (+
        (+
         (+
          (+
           (+
            (+ (/ 0.9999999999994297 t_3) (/ (pow t_2 3.0) t_3))
            (/ -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_4))
      (/ 1.5056327351493116e-7 (+ t_0 8.0)))))))

C

double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = (1.0 - z) - 0.0;
	double t_2 = 676.5203681218851 / t_1;
	double t_3 = 0.9999999999996197 + ((-457679.80848377093 / ((0.0 - (1.0 - z)) * t_1)) - (0.9999999999998099 * t_2));
	double t_4 = t_0 + 7.0;
	double t_5 = t_4 + 0.5;
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt((((double) M_PI) * 2.0)) * pow(t_5, (t_0 + 0.5))) * exp(-t_5)) * (((((((((0.9999999999994297 / t_3) + (pow(t_2, 3.0) / t_3)) + (-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_4)) + (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) - 0.0;
	double t_2 = 676.5203681218851 / t_1;
	double t_3 = 0.9999999999996197 + ((-457679.80848377093 / ((0.0 - (1.0 - z)) * t_1)) - (0.9999999999998099 * t_2));
	double t_4 = t_0 + 7.0;
	double t_5 = t_4 + 0.5;
	return (Math.PI / Math.sin((Math.PI * z))) * (((Math.sqrt((Math.PI * 2.0)) * Math.pow(t_5, (t_0 + 0.5))) * Math.exp(-t_5)) * (((((((((0.9999999999994297 / t_3) + (Math.pow(t_2, 3.0) / t_3)) + (-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_4)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
}

Python

def code(z):
	t_0 = (1.0 - z) - 1.0
	t_1 = (1.0 - z) - 0.0
	t_2 = 676.5203681218851 / t_1
	t_3 = 0.9999999999996197 + ((-457679.80848377093 / ((0.0 - (1.0 - z)) * t_1)) - (0.9999999999998099 * t_2))
	t_4 = t_0 + 7.0
	t_5 = t_4 + 0.5
	return (math.pi / math.sin((math.pi * z))) * (((math.sqrt((math.pi * 2.0)) * math.pow(t_5, (t_0 + 0.5))) * math.exp(-t_5)) * (((((((((0.9999999999994297 / t_3) + (math.pow(t_2, 3.0) / t_3)) + (-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_4)) + (1.5056327351493116e-7 / (t_0 + 8.0))))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - 1.0)
	t_1 = Float64(Float64(1.0 - z) - 0.0)
	t_2 = Float64(676.5203681218851 / t_1)
	t_3 = Float64(0.9999999999996197 + Float64(Float64(-457679.80848377093 / Float64(Float64(0.0 - Float64(1.0 - z)) * t_1)) - Float64(0.9999999999998099 * t_2)))
	t_4 = Float64(t_0 + 7.0)
	t_5 = Float64(t_4 + 0.5)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (t_5 ^ Float64(t_0 + 0.5))) * exp(Float64(-t_5))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999994297 / t_3) + Float64((t_2 ^ 3.0) / t_3)) + 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_4)) + 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) - 0.0;
	t_2 = 676.5203681218851 / t_1;
	t_3 = 0.9999999999996197 + ((-457679.80848377093 / ((0.0 - (1.0 - z)) * t_1)) - (0.9999999999998099 * t_2));
	t_4 = t_0 + 7.0;
	t_5 = t_4 + 0.5;
	tmp = (pi / sin((pi * z))) * (((sqrt((pi * 2.0)) * (t_5 ^ (t_0 + 0.5))) * exp(-t_5)) * (((((((((0.9999999999994297 / t_3) + ((t_2 ^ 3.0) / t_3)) + (-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_4)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - z), $MachinePrecision] - 0.0), $MachinePrecision]}, Block[{t$95$2 = N[(676.5203681218851 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(0.9999999999996197 + N[(N[(-457679.80848377093 / N[(N[(0.0 - N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(0.9999999999998099 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 + 7.0), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 + 0.5), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$5, N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$5)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(N[(0.9999999999994297 / t$95$3), $MachinePrecision] + N[(N[Power[t$95$2, 3.0], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(t$95$0 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(t$95$0 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(t$95$0 + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(t$95$0 + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$4), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(t$95$0 + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Initial program 96.4%

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

   1. lift-+.f64 N/A

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

   2. flip3-+ N/A

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

   3. div-add 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(\left(\left(\color{blue}{\left(\frac{{\frac{9999999999998099}{10000000000000000}}^{3}}{\frac{9999999999998099}{10000000000000000} \cdot \frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} - \frac{9999999999998099}{10000000000000000} \cdot \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right)} + \frac{{\left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3}}{\frac{9999999999998099}{10000000000000000} \cdot \frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} - \frac{9999999999998099}{10000000000000000} \cdot \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right)}\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) \]

   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(\left(\left(\left(\left(\color{blue}{\left(\frac{{\frac{9999999999998099}{10000000000000000}}^{3}}{\frac{9999999999998099}{10000000000000000} \cdot \frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} - \frac{9999999999998099}{10000000000000000} \cdot \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right)} + \frac{{\left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3}}{\frac{9999999999998099}{10000000000000000} \cdot \frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} - \frac{9999999999998099}{10000000000000000} \cdot \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right)}\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 98.2%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\color{blue}{\left(\frac{0.9999999999994297}{0.9999999999996197 + \left(\frac{-457679.80848377093}{\left(0 - \left(1 - z\right)\right) \cdot \left(\left(1 - z\right) - 0\right)} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(1 - z\right) - 0}\right)} + \frac{{\left(\frac{676.5203681218851}{\left(1 - z\right) - 0}\right)}^{3}}{0.9999999999996197 + \left(\frac{-457679.80848377093}{\left(0 - \left(1 - z\right)\right) \cdot \left(\left(1 - z\right) - 0\right)} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(1 - z\right) - 0}\right)}\right)} + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
4. Add Preprocessing

Alternative 2: 98.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.4%

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

3. Applied rewrites 97.7%

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

5. Applied rewrites 98.5%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. count-2-rev N/A

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

   4. *-commutative N/A

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

   5. sqrt-prod N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-sqrt.f64 98.8

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

7. Applied rewrites 98.8%

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

8. Taylor expanded in z around inf

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

   1. lower-exp.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-log.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower--.f64 98.8

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

10. Applied rewrites 98.8%

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

Alternative 3: 98.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.4%

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

3. Applied rewrites 97.7%

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in z around inf

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

   1. lower-exp.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-log.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower--.f64 98.5

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

8. Applied rewrites 98.5%

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

Alternative 4: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - -6.5\\ \frac{\left(\pi \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{676.5203681218851}{1 - z} - -0.9999999999998099\right) - \frac{1259.1392167224028}{\left(1 - z\right) - -1}\right) - \frac{-771.3234287776531}{\left(1 - z\right) - -2}\right) - \frac{176.6150291621406}{\left(1 - z\right) - -3}\right) - \frac{-12.507343278686905}{\left(1 - z\right) - -4}\right) - \frac{0.13857109526572012}{\left(1 - z\right) - -5}\right) - \frac{-9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) - -1.8820409189366395 \cdot 10^{-8}\right)\right) \cdot \left(e^{\log t\_0 \cdot \left(\left(1 - z\right) - 0.5\right) - t\_0} \cdot \sqrt{\pi + \pi}\right)}{\sin \left(z \cdot \pi\right)} \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) -6.5)))
   (/
    (*
     (*
      PI
      (-
       (-
        (-
         (-
          (-
           (-
            (-
             (- (/ 676.5203681218851 (- 1.0 z)) -0.9999999999998099)
             (/ 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)))
       -1.8820409189366395e-8))
     (* (exp (- (* (log t_0) (- (- 1.0 z) 0.5)) t_0)) (sqrt (+ PI PI))))
    (sin (* z PI)))))

C

double code(double z) {
	double t_0 = (1.0 - z) - -6.5;
	return ((((double) M_PI) * (((((((((676.5203681218851 / (1.0 - z)) - -0.9999999999998099) - (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))) - -1.8820409189366395e-8)) * (exp(((log(t_0) * ((1.0 - z) - 0.5)) - t_0)) * sqrt((((double) M_PI) + ((double) M_PI))))) / sin((z * ((double) M_PI)));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - -6.5;
	return ((Math.PI * (((((((((676.5203681218851 / (1.0 - z)) - -0.9999999999998099) - (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))) - -1.8820409189366395e-8)) * (Math.exp(((Math.log(t_0) * ((1.0 - z) - 0.5)) - t_0)) * Math.sqrt((Math.PI + Math.PI)))) / Math.sin((z * Math.PI));
}

Python

def code(z):
	t_0 = (1.0 - z) - -6.5
	return ((math.pi * (((((((((676.5203681218851 / (1.0 - z)) - -0.9999999999998099) - (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))) - -1.8820409189366395e-8)) * (math.exp(((math.log(t_0) * ((1.0 - z) - 0.5)) - t_0)) * math.sqrt((math.pi + math.pi)))) / math.sin((z * math.pi))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - -6.5)
	return Float64(Float64(Float64(pi * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) - -0.9999999999998099) - Float64(1259.1392167224028 / Float64(Float64(1.0 - z) - -1.0))) - Float64(-771.3234287776531 / Float64(Float64(1.0 - z) - -2.0))) - Float64(176.6150291621406 / Float64(Float64(1.0 - z) - -3.0))) - Float64(-12.507343278686905 / Float64(Float64(1.0 - z) - -4.0))) - Float64(0.13857109526572012 / Float64(Float64(1.0 - z) - -5.0))) - Float64(-9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0))) - -1.8820409189366395e-8)) * Float64(exp(Float64(Float64(log(t_0) * Float64(Float64(1.0 - z) - 0.5)) - t_0)) * sqrt(Float64(pi + pi)))) / sin(Float64(z * pi)))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - -6.5;
	tmp = ((pi * (((((((((676.5203681218851 / (1.0 - z)) - -0.9999999999998099) - (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))) - -1.8820409189366395e-8)) * (exp(((log(t_0) * ((1.0 - z) - 0.5)) - t_0)) * sqrt((pi + pi)))) / sin((z * pi));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - -6.5), $MachinePrecision]}, N[(N[(N[(Pi * N[(N[(N[(N[(N[(N[(N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - -0.9999999999998099), $MachinePrecision] - N[(1259.1392167224028 / N[(N[(1.0 - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-771.3234287776531 / N[(N[(1.0 - z), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] - -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] - -5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.8820409189366395e-8), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(N[(N[Log[t$95$0], $MachinePrecision] * N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi + Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 96.4%

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

3. Applied rewrites 97.7%

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in z around 0

   \[\leadsto \frac{\left(\pi \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{-9999999999998099}{10000000000000000}\right) - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) - \frac{\frac{-7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) - \frac{\frac{-2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) - \frac{\frac{3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5}\right) - \frac{\frac{-2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) - \color{blue}{\frac{-3764081837873279}{200000000000000000000000}}\right)\right) \cdot \left(e^{\log \left(\left(1 - z\right) - \frac{-13}{2}\right) \cdot \left(\left(1 - z\right) - \frac{1}{2}\right) - \left(\left(1 - z\right) - \frac{-13}{2}\right)} \cdot \sqrt{\pi + \pi}\right)}{\sin \left(z \cdot \pi\right)} \]
7. Step-by-step derivation

8. Applied rewrites 98.2%

   \[\leadsto \frac{\left(\pi \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{676.5203681218851}{1 - z} - -0.9999999999998099\right) - \frac{1259.1392167224028}{\left(1 - z\right) - -1}\right) - \frac{-771.3234287776531}{\left(1 - z\right) - -2}\right) - \frac{176.6150291621406}{\left(1 - z\right) - -3}\right) - \frac{-12.507343278686905}{\left(1 - z\right) - -4}\right) - \frac{0.13857109526572012}{\left(1 - z\right) - -5}\right) - \frac{-9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) - \color{blue}{-1.8820409189366395 \cdot 10^{-8}}\right)\right) \cdot \left(e^{\log \left(\left(1 - z\right) - -6.5\right) \cdot \left(\left(1 - z\right) - 0.5\right) - \left(\left(1 - z\right) - -6.5\right)} \cdot \sqrt{\pi + \pi}\right)}{\sin \left(z \cdot \pi\right)} \]
9. Add Preprocessing

Alternative 5: 98.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - -6.5\\ \frac{\left(\pi \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + 545.0353078134797 \cdot z\right)\right) - \frac{-9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) - \frac{-1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(e^{\log t\_0 \cdot \left(\left(1 - z\right) - 0.5\right) - t\_0} \cdot \sqrt{\pi + \pi}\right)}{\sin \left(z \cdot \pi\right)} \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) -6.5)))
   (/
    (*
     (*
      PI
      (-
       (-
        (+
         263.3831855358925
         (* z (+ 436.8961723502244 (* 545.0353078134797 z))))
        (/ -9.984369578019572e-6 (- (- 1.0 z) -6.0)))
       (/ -1.5056327351493116e-7 (- (- 1.0 z) -7.0))))
     (* (exp (- (* (log t_0) (- (- 1.0 z) 0.5)) t_0)) (sqrt (+ PI PI))))
    (sin (* z PI)))))

C

double code(double z) {
	double t_0 = (1.0 - z) - -6.5;
	return ((((double) M_PI) * (((263.3831855358925 + (z * (436.8961723502244 + (545.0353078134797 * z)))) - (-9.984369578019572e-6 / ((1.0 - z) - -6.0))) - (-1.5056327351493116e-7 / ((1.0 - z) - -7.0)))) * (exp(((log(t_0) * ((1.0 - z) - 0.5)) - t_0)) * sqrt((((double) M_PI) + ((double) M_PI))))) / sin((z * ((double) M_PI)));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - -6.5;
	return ((Math.PI * (((263.3831855358925 + (z * (436.8961723502244 + (545.0353078134797 * z)))) - (-9.984369578019572e-6 / ((1.0 - z) - -6.0))) - (-1.5056327351493116e-7 / ((1.0 - z) - -7.0)))) * (Math.exp(((Math.log(t_0) * ((1.0 - z) - 0.5)) - t_0)) * Math.sqrt((Math.PI + Math.PI)))) / Math.sin((z * Math.PI));
}

Python

def code(z):
	t_0 = (1.0 - z) - -6.5
	return ((math.pi * (((263.3831855358925 + (z * (436.8961723502244 + (545.0353078134797 * z)))) - (-9.984369578019572e-6 / ((1.0 - z) - -6.0))) - (-1.5056327351493116e-7 / ((1.0 - z) - -7.0)))) * (math.exp(((math.log(t_0) * ((1.0 - z) - 0.5)) - t_0)) * math.sqrt((math.pi + math.pi)))) / math.sin((z * math.pi))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - -6.5)
	return Float64(Float64(Float64(pi * Float64(Float64(Float64(263.3831855358925 + Float64(z * Float64(436.8961723502244 + Float64(545.0353078134797 * z)))) - Float64(-9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0))) - Float64(-1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0)))) * Float64(exp(Float64(Float64(log(t_0) * Float64(Float64(1.0 - z) - 0.5)) - t_0)) * sqrt(Float64(pi + pi)))) / sin(Float64(z * pi)))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - -6.5;
	tmp = ((pi * (((263.3831855358925 + (z * (436.8961723502244 + (545.0353078134797 * z)))) - (-9.984369578019572e-6 / ((1.0 - z) - -6.0))) - (-1.5056327351493116e-7 / ((1.0 - z) - -7.0)))) * (exp(((log(t_0) * ((1.0 - z) - 0.5)) - t_0)) * sqrt((pi + pi)))) / sin((z * pi));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - -6.5), $MachinePrecision]}, N[(N[(N[(Pi * N[(N[(N[(263.3831855358925 + N[(z * N[(436.8961723502244 + N[(545.0353078134797 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(N[(N[Log[t$95$0], $MachinePrecision] * N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi + Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 96.4%

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

3. Applied rewrites 97.7%

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in z around 0

   \[\leadsto \frac{\left(\pi \cdot \left(\left(\color{blue}{\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + \frac{367898832774098786021}{675000000000000000} \cdot z\right)\right)} - \frac{\frac{-2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) - \frac{\frac{-3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(e^{\log \left(\left(1 - z\right) - \frac{-13}{2}\right) \cdot \left(\left(1 - z\right) - \frac{1}{2}\right) - \left(\left(1 - z\right) - \frac{-13}{2}\right)} \cdot \sqrt{\pi + \pi}\right)}{\sin \left(z \cdot \pi\right)} \]
7. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\left(\pi \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + \color{blue}{z \cdot \left(\frac{131068851705067315609}{300000000000000000} + \frac{367898832774098786021}{675000000000000000} \cdot z\right)}\right) - \frac{\frac{-2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) - \frac{\frac{-3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(e^{\log \left(\left(1 - z\right) - \frac{-13}{2}\right) \cdot \left(\left(1 - z\right) - \frac{1}{2}\right) - \left(\left(1 - z\right) - \frac{-13}{2}\right)} \cdot \sqrt{\pi + \pi}\right)}{\sin \left(z \cdot \pi\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\left(\pi \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \color{blue}{\left(\frac{131068851705067315609}{300000000000000000} + \frac{367898832774098786021}{675000000000000000} \cdot z\right)}\right) - \frac{\frac{-2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) - \frac{\frac{-3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(e^{\log \left(\left(1 - z\right) - \frac{-13}{2}\right) \cdot \left(\left(1 - z\right) - \frac{1}{2}\right) - \left(\left(1 - z\right) - \frac{-13}{2}\right)} \cdot \sqrt{\pi + \pi}\right)}{\sin \left(z \cdot \pi\right)} \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{\left(\pi \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + \color{blue}{\frac{367898832774098786021}{675000000000000000} \cdot z}\right)\right) - \frac{\frac{-2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) - \frac{\frac{-3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(e^{\log \left(\left(1 - z\right) - \frac{-13}{2}\right) \cdot \left(\left(1 - z\right) - \frac{1}{2}\right) - \left(\left(1 - z\right) - \frac{-13}{2}\right)} \cdot \sqrt{\pi + \pi}\right)}{\sin \left(z \cdot \pi\right)} \]

   4. lower-*.f64 98.6

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

8. Applied rewrites 98.6%

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

Alternative 6: 98.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - -6.5\\ \frac{\left(\pi \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right)\right) \cdot \left(e^{\log t\_0 \cdot \left(\left(1 - z\right) - 0.5\right) - t\_0} \cdot \sqrt{\pi + \pi}\right)}{\sin \left(z \cdot \pi\right)} \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) -6.5)))
   (/
    (*
     (*
      PI
      (+
       263.3831869810514
       (* z (+ 436.8961725563396 (* 545.0353078428827 z)))))
     (* (exp (- (* (log t_0) (- (- 1.0 z) 0.5)) t_0)) (sqrt (+ PI PI))))
    (sin (* z PI)))))

C

double code(double z) {
	double t_0 = (1.0 - z) - -6.5;
	return ((((double) M_PI) * (263.3831869810514 + (z * (436.8961725563396 + (545.0353078428827 * z))))) * (exp(((log(t_0) * ((1.0 - z) - 0.5)) - t_0)) * sqrt((((double) M_PI) + ((double) M_PI))))) / sin((z * ((double) M_PI)));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - -6.5;
	return ((Math.PI * (263.3831869810514 + (z * (436.8961725563396 + (545.0353078428827 * z))))) * (Math.exp(((Math.log(t_0) * ((1.0 - z) - 0.5)) - t_0)) * Math.sqrt((Math.PI + Math.PI)))) / Math.sin((z * Math.PI));
}

Python

def code(z):
	t_0 = (1.0 - z) - -6.5
	return ((math.pi * (263.3831869810514 + (z * (436.8961725563396 + (545.0353078428827 * z))))) * (math.exp(((math.log(t_0) * ((1.0 - z) - 0.5)) - t_0)) * math.sqrt((math.pi + math.pi)))) / math.sin((z * math.pi))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - -6.5)
	return Float64(Float64(Float64(pi * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(545.0353078428827 * z))))) * Float64(exp(Float64(Float64(log(t_0) * Float64(Float64(1.0 - z) - 0.5)) - t_0)) * sqrt(Float64(pi + pi)))) / sin(Float64(z * pi)))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - -6.5;
	tmp = ((pi * (263.3831869810514 + (z * (436.8961725563396 + (545.0353078428827 * z))))) * (exp(((log(t_0) * ((1.0 - z) - 0.5)) - t_0)) * sqrt((pi + pi)))) / sin((z * pi));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - -6.5), $MachinePrecision]}, N[(N[(N[(Pi * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(545.0353078428827 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(N[(N[Log[t$95$0], $MachinePrecision] * N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi + Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 96.4%

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

3. Applied rewrites 97.7%

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in z around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 98.6

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

8. Applied rewrites 98.6%

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

Alternative 7: 97.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.4%

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

   1. lift-+.f64 N/A

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

   2. flip3-+ N/A

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

   3. div-add 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(\left(\left(\color{blue}{\left(\frac{{\frac{9999999999998099}{10000000000000000}}^{3}}{\frac{9999999999998099}{10000000000000000} \cdot \frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} - \frac{9999999999998099}{10000000000000000} \cdot \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right)} + \frac{{\left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3}}{\frac{9999999999998099}{10000000000000000} \cdot \frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} - \frac{9999999999998099}{10000000000000000} \cdot \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right)}\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) \]

   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(\left(\left(\left(\left(\color{blue}{\left(\frac{{\frac{9999999999998099}{10000000000000000}}^{3}}{\frac{9999999999998099}{10000000000000000} \cdot \frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} - \frac{9999999999998099}{10000000000000000} \cdot \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right)} + \frac{{\left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3}}{\frac{9999999999998099}{10000000000000000} \cdot \frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} - \frac{9999999999998099}{10000000000000000} \cdot \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right)}\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 98.2%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\color{blue}{\left(\frac{0.9999999999994297}{0.9999999999996197 + \left(\frac{-457679.80848377093}{\left(0 - \left(1 - z\right)\right) \cdot \left(\left(1 - z\right) - 0\right)} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(1 - z\right) - 0}\right)} + \frac{{\left(\frac{676.5203681218851}{\left(1 - z\right) - 0}\right)}^{3}}{0.9999999999996197 + \left(\frac{-457679.80848377093}{\left(0 - \left(1 - z\right)\right) \cdot \left(\left(1 - z\right) - 0\right)} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(1 - z\right) - 0}\right)}\right)} + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Taylor expanded in z around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 96.7

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

6. Applied rewrites 96.7%

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

7. Taylor expanded in z around 0

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

   1. lower-/.f64 96.3

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

9. Applied rewrites 96.3%

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

11. Applied rewrites 97.7%

    \[\leadsto \color{blue}{\left(\frac{1}{z} \cdot \left(\sqrt{\pi + \pi} \cdot e^{\mathsf{fma}\left(\log \left(\left(\left(1 - z\right) - -6\right) - -0.5\right), \left(1 - z\right) - 0.5, -\left(\left(\left(1 - z\right) - -6\right) - -0.5\right)\right)}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.6766809167608, z, 545.0353078428827\right), z, 436.8961725563396\right), z, 263.3831869810514\right)} \]
12. Add Preprocessing

Alternative 8: 97.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.4%

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

   1. lift-+.f64 N/A

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

   2. flip3-+ N/A

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

   3. div-add 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(\left(\left(\color{blue}{\left(\frac{{\frac{9999999999998099}{10000000000000000}}^{3}}{\frac{9999999999998099}{10000000000000000} \cdot \frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} - \frac{9999999999998099}{10000000000000000} \cdot \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right)} + \frac{{\left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3}}{\frac{9999999999998099}{10000000000000000} \cdot \frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} - \frac{9999999999998099}{10000000000000000} \cdot \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right)}\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) \]

   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(\left(\left(\left(\left(\color{blue}{\left(\frac{{\frac{9999999999998099}{10000000000000000}}^{3}}{\frac{9999999999998099}{10000000000000000} \cdot \frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} - \frac{9999999999998099}{10000000000000000} \cdot \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right)} + \frac{{\left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right)}^{3}}{\frac{9999999999998099}{10000000000000000} \cdot \frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} \cdot \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} - \frac{9999999999998099}{10000000000000000} \cdot \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right)}\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 98.2%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\color{blue}{\left(\frac{0.9999999999994297}{0.9999999999996197 + \left(\frac{-457679.80848377093}{\left(0 - \left(1 - z\right)\right) \cdot \left(\left(1 - z\right) - 0\right)} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(1 - z\right) - 0}\right)} + \frac{{\left(\frac{676.5203681218851}{\left(1 - z\right) - 0}\right)}^{3}}{0.9999999999996197 + \left(\frac{-457679.80848377093}{\left(0 - \left(1 - z\right)\right) \cdot \left(\left(1 - z\right) - 0\right)} - 0.9999999999998099 \cdot \frac{676.5203681218851}{\left(1 - z\right) - 0}\right)}\right)} + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Taylor expanded in z around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 96.7

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

6. Applied rewrites 96.7%

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

7. Taylor expanded in z around 0

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

   1. lower-/.f64 96.3

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

9. Applied rewrites 96.3%

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

11. Applied rewrites 97.6%

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.6766809167608, z, 545.0353078428827\right), z, 436.8961725563396\right), z, 263.3831869810514\right) \cdot \left(\sqrt{\pi + \pi} \cdot e^{\mathsf{fma}\left(\log \left(\left(\left(1 - z\right) - -6\right) - -0.5\right), \left(1 - z\right) - 0.5, -\left(\left(\left(1 - z\right) - -6\right) - -0.5\right)\right)}\right)\right) \cdot \frac{1}{z}} \]
12. Add Preprocessing

Alternative 9: 96.4% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.4%

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

3. Applied rewrites 97.7%

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

4. Taylor expanded in z around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

6. Applied rewrites 96.4%

   \[\leadsto \color{blue}{263.3831869810514 \cdot \frac{e^{0.5 \cdot \log 7.5 - 7.5} \cdot \sqrt{2 \cdot \pi}}{z}} \]
7. Add Preprocessing

Alternative 10: 95.6% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.4%

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

2. Taylor expanded in z around 0

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

   1. 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 95.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 95.6

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

6. Applied rewrites 95.6%

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

7. Taylor expanded in z around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-exp.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 95.6

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

9. Applied rewrites 95.6%

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

Reproduce

herbie shell --seed 2025150 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z less than 0.5"
  :precision binary64
  :pre (<= z 0.5)
  (* (/ PI (sin (* PI z))) (* (* (* (sqrt (* PI 2.0)) (pow (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5) (+ (- (- 1.0 z) 1.0) 0.5))) (exp (- (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- (- 1.0 z) 1.0) 1.0))) (/ -1259.1392167224028 (+ (- (- 1.0 z) 1.0) 2.0))) (/ 771.3234287776531 (+ (- (- 1.0 z) 1.0) 3.0))) (/ -176.6150291621406 (+ (- (- 1.0 z) 1.0) 4.0))) (/ 12.507343278686905 (+ (- (- 1.0 z) 1.0) 5.0))) (/ -0.13857109526572012 (+ (- (- 1.0 z) 1.0) 6.0))) (/ 9.984369578019572e-6 (+ (- (- 1.0 z) 1.0) 7.0))) (/ 1.5056327351493116e-7 (+ (- (- 1.0 z) 1.0) 8.0))))))