Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.4% → 99.2%

Time: 12.1s

Alternatives: 12

Speedup: 2.7×

• could not determine a ground truth

  1. z = -16967974839.892467

Specification

\[z \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - -6\\ \frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{676.5203681218851}{1 - z} - -0.9999999999998099\right) - \frac{1259.1392167224028}{\left(1 - z\right) - -1}\right) - \frac{-771.3234287776531}{\left(1 - z\right) - -2}\right) - \frac{176.6150291621406}{\left(1 - z\right) - -3}\right) - \frac{-12.507343278686905}{\left(1 - z\right) - -4}\right) - \frac{-0.13857109526572012}{-6 + z}\right) - \frac{-9.984369578019572 \cdot 10^{-6}}{t\_0}\right) - \frac{1.5056327351493116 \cdot 10^{-7}}{-8 + z}\right) \cdot \left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot e^{\mathsf{fma}\left(\log \left(t\_0 - -0.5\right), \left(1 - z\right) - 0.5, z - 7.5\right)}\right)}{\frac{\sin \left(z \cdot \pi\right)}{\pi}} \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) -6.0)))
   (/
    (*
     (-
      (-
       (-
        (-
         (-
          (-
           (-
            (- (/ 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 (+ -6.0 z)))
       (/ -9.984369578019572e-6 t_0))
      (/ 1.5056327351493116e-7 (+ -8.0 z)))
     (*
      (* (sqrt PI) (sqrt 2.0))
      (exp (fma (log (- t_0 -0.5)) (- (- 1.0 z) 0.5) (- z 7.5)))))
    (/ (sin (* z PI)) PI))))

C

double code(double z) {
	double t_0 = (1.0 - z) - -6.0;
	return ((((((((((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 / (-6.0 + z))) - (-9.984369578019572e-6 / t_0)) - (1.5056327351493116e-7 / (-8.0 + z))) * ((sqrt(((double) M_PI)) * sqrt(2.0)) * exp(fma(log((t_0 - -0.5)), ((1.0 - z) - 0.5), (z - 7.5))))) / (sin((z * ((double) M_PI))) / ((double) M_PI));
}

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - -6.0)
	return Float64(Float64(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(-6.0 + z))) - Float64(-9.984369578019572e-6 / t_0)) - Float64(1.5056327351493116e-7 / Float64(-8.0 + z))) * Float64(Float64(sqrt(pi) * sqrt(2.0)) * exp(fma(log(Float64(t_0 - -0.5)), Float64(Float64(1.0 - z) - 0.5), Float64(z - 7.5))))) / Float64(sin(Float64(z * pi)) / pi))
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]}, N[(N[(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[(-6.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-9.984369578019572e-6 / t$95$0), $MachinePrecision]), $MachinePrecision] - N[(1.5056327351493116e-7 / N[(-8.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Log[N[(t$95$0 - -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision] + N[(z - 7.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision] / Pi), $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 97.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. sqrt-prod N/A

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

   5. lift-PI.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-sqrt.f64 99.2

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

5. Applied rewrites 99.2%

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

7. Applied rewrites 99.2%

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

8. Taylor expanded in z around 0

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

   1. lower--.f64 99.2

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

10. Applied rewrites 99.2%

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

Alternative 2: 99.2% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (/
  (*
   (-
    (-
     (-
      (-
       (-
        (-
         (-
          (- (/ 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 (+ -6.0 z)))
     (/ -9.984369578019572e-6 (- (- 1.0 z) -6.0)))
    (/ 1.5056327351493116e-7 (+ -8.0 z)))
   (*
    (* (sqrt PI) (sqrt 2.0))
    (exp (- (+ z (* (log (- 7.5 z)) (- 0.5 z))) 7.5))))
  (/ (sin (* z PI)) PI)))

C

double code(double z) {
	return ((((((((((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 / (-6.0 + z))) - (-9.984369578019572e-6 / ((1.0 - z) - -6.0))) - (1.5056327351493116e-7 / (-8.0 + z))) * ((sqrt(((double) M_PI)) * sqrt(2.0)) * exp(((z + (log((7.5 - z)) * (0.5 - z))) - 7.5)))) / (sin((z * ((double) M_PI))) / ((double) M_PI));
}

Java

public static double code(double z) {
	return ((((((((((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 / (-6.0 + z))) - (-9.984369578019572e-6 / ((1.0 - z) - -6.0))) - (1.5056327351493116e-7 / (-8.0 + z))) * ((Math.sqrt(Math.PI) * Math.sqrt(2.0)) * Math.exp(((z + (Math.log((7.5 - z)) * (0.5 - z))) - 7.5)))) / (Math.sin((z * Math.PI)) / Math.PI);
}

Python

def code(z):
	return ((((((((((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 / (-6.0 + z))) - (-9.984369578019572e-6 / ((1.0 - z) - -6.0))) - (1.5056327351493116e-7 / (-8.0 + z))) * ((math.sqrt(math.pi) * math.sqrt(2.0)) * math.exp(((z + (math.log((7.5 - z)) * (0.5 - z))) - 7.5)))) / (math.sin((z * math.pi)) / math.pi)

Julia

function code(z)
	return Float64(Float64(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(-6.0 + z))) - Float64(-9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0))) - Float64(1.5056327351493116e-7 / Float64(-8.0 + z))) * Float64(Float64(sqrt(pi) * sqrt(2.0)) * exp(Float64(Float64(z + Float64(log(Float64(7.5 - z)) * Float64(0.5 - z))) - 7.5)))) / Float64(sin(Float64(z * pi)) / pi))
end

MATLAB

function tmp = code(z)
	tmp = ((((((((((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 / (-6.0 + z))) - (-9.984369578019572e-6 / ((1.0 - z) - -6.0))) - (1.5056327351493116e-7 / (-8.0 + z))) * ((sqrt(pi) * sqrt(2.0)) * exp(((z + (log((7.5 - z)) * (0.5 - z))) - 7.5)))) / (sin((z * pi)) / pi);
end

Wolfram

code[z_] := N[(N[(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[(-6.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.5056327351493116e-7 / N[(-8.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision] / Pi), $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 97.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. sqrt-prod N/A

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

   5. lift-PI.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-sqrt.f64 99.2

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

5. Applied rewrites 99.2%

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

7. Applied rewrites 99.2%

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

8. Taylor expanded in z around inf

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

   1. lower-exp.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-log.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower--.f64 99.2

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

10. Applied rewrites 99.2%

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

Alternative 3: 98.4% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) -6.5)))
   (/
    (*
     (*
      (*
       (-
        (-
         (-
          (-
           (-
            (-
             (-
              (- (/ 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 (+ -6.0 z)))
         (/ -9.984369578019572e-6 (- (- 1.0 z) -6.0)))
        -1.8820409189366395e-8)
       (sqrt (+ PI PI)))
      (exp (- (* (log t_0) (- (- 1.0 z) 0.5)) t_0)))
     PI)
    (sin (* PI z)))))

C

double code(double z) {
	double t_0 = (1.0 - z) - -6.5;
	return ((((((((((((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 / (-6.0 + z))) - (-9.984369578019572e-6 / ((1.0 - z) - -6.0))) - -1.8820409189366395e-8) * sqrt((((double) M_PI) + ((double) M_PI)))) * exp(((log(t_0) * ((1.0 - z) - 0.5)) - t_0))) * ((double) M_PI)) / sin((((double) M_PI) * z));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - -6.5;
	return ((((((((((((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 / (-6.0 + z))) - (-9.984369578019572e-6 / ((1.0 - z) - -6.0))) - -1.8820409189366395e-8) * Math.sqrt((Math.PI + Math.PI))) * Math.exp(((Math.log(t_0) * ((1.0 - z) - 0.5)) - t_0))) * Math.PI) / Math.sin((Math.PI * z));
}

Python

def code(z):
	t_0 = (1.0 - z) - -6.5
	return ((((((((((((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 / (-6.0 + z))) - (-9.984369578019572e-6 / ((1.0 - z) - -6.0))) - -1.8820409189366395e-8) * math.sqrt((math.pi + math.pi))) * math.exp(((math.log(t_0) * ((1.0 - z) - 0.5)) - t_0))) * math.pi) / math.sin((math.pi * z))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - -6.5)
	return Float64(Float64(Float64(Float64(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(-6.0 + z))) - Float64(-9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0))) - -1.8820409189366395e-8) * sqrt(Float64(pi + pi))) * exp(Float64(Float64(log(t_0) * Float64(Float64(1.0 - z) - 0.5)) - t_0))) * pi) / sin(Float64(pi * z)))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - -6.5;
	tmp = ((((((((((((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 / (-6.0 + z))) - (-9.984369578019572e-6 / ((1.0 - z) - -6.0))) - -1.8820409189366395e-8) * sqrt((pi + pi))) * exp(((log(t_0) * ((1.0 - z) - 0.5)) - t_0))) * pi) / sin((pi * z));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - -6.5), $MachinePrecision]}, N[(N[(N[(N[(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[(-6.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.8820409189366395e-8), $MachinePrecision] * N[Sqrt[N[(Pi + Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision] * Pi), $MachinePrecision] / N[Sin[N[(Pi * z), $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 97.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. sqrt-prod N/A

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

   5. lift-PI.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-sqrt.f64 99.2

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

5. Applied rewrites 99.2%

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in z around 0

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

10. Applied rewrites 98.2%

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

Alternative 4: 98.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 96.6

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

4. Applied rewrites 96.6%

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

6. Applied rewrites 98.4%

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

Alternative 5: 97.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot z\right)\\ \mathbf{if}\;z \leq -3.8:\\ \;\;\;\;\frac{\left(\sqrt{\pi + \pi} \cdot e^{\mathsf{fma}\left(0.5 - z, \log \left(7.5 - z\right), z\right) - 7.5}\right) \cdot 0.9999999999998099}{t\_0} \cdot \pi\\ \mathbf{else}:\\ \;\;\;\;\frac{263.3831869810514 \cdot \left(\pi \cdot \left(e^{0.5 \cdot \log 7.5 - 7.5} \cdot \sqrt{2 \cdot \pi}\right)\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sin (* PI z))))
   (if (<= z -3.8)
     (*
      (/
       (*
        (* (sqrt (+ PI PI)) (exp (- (fma (- 0.5 z) (log (- 7.5 z)) z) 7.5)))
        0.9999999999998099)
       t_0)
      PI)
     (/
      (*
       263.3831869810514
       (* PI (* (exp (- (* 0.5 (log 7.5)) 7.5)) (sqrt (* 2.0 PI)))))
      t_0))))

C

double code(double z) {
	double t_0 = sin((((double) M_PI) * z));
	double tmp;
	if (z <= -3.8) {
		tmp = (((sqrt((((double) M_PI) + ((double) M_PI))) * exp((fma((0.5 - z), log((7.5 - z)), z) - 7.5))) * 0.9999999999998099) / t_0) * ((double) M_PI);
	} else {
		tmp = (263.3831869810514 * (((double) M_PI) * (exp(((0.5 * log(7.5)) - 7.5)) * sqrt((2.0 * ((double) M_PI)))))) / t_0;
	}
	return tmp;
}

Julia

function code(z)
	t_0 = sin(Float64(pi * z))
	tmp = 0.0
	if (z <= -3.8)
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(pi + pi)) * exp(Float64(fma(Float64(0.5 - z), log(Float64(7.5 - z)), z) - 7.5))) * 0.9999999999998099) / t_0) * pi);
	else
		tmp = Float64(Float64(263.3831869810514 * Float64(pi * Float64(exp(Float64(Float64(0.5 * log(7.5)) - 7.5)) * sqrt(Float64(2.0 * pi))))) / t_0);
	end
	return tmp
end

Wolfram

code[z_] := Block[{t$95$0 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -3.8], N[(N[(N[(N[(N[Sqrt[N[(Pi + Pi), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[(0.5 - z), $MachinePrecision] * N[Log[N[(7.5 - z), $MachinePrecision]], $MachinePrecision] + z), $MachinePrecision] - 7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.9999999999998099), $MachinePrecision] / t$95$0), $MachinePrecision] * Pi), $MachinePrecision], N[(N[(263.3831869810514 * N[(Pi * 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]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.7999999999999998

   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 97.8%

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

   4. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{\frac{9999999999998099}{10000000000000000} \cdot \left(e^{\left(z + \log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)\right) - \frac{15}{2}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)}}{\frac{\sin \left(z \cdot \pi\right)}{\pi}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\frac{9999999999998099}{10000000000000000} \cdot \color{blue}{\left(e^{\left(z + \log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)\right) - \frac{15}{2}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)}}{\frac{\sin \left(z \cdot \pi\right)}{\pi}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{9999999999998099}{10000000000000000} \cdot \left(e^{\left(z + \log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)\right) - \frac{15}{2}} \cdot \color{blue}{\sqrt{2 \cdot \mathsf{PI}\left(\right)}}\right)}{\frac{\sin \left(z \cdot \pi\right)}{\pi}} \]

   6. Applied rewrites 15.0%

      \[\leadsto \frac{\color{blue}{0.9999999999998099 \cdot \left(e^{\left(z + \log \left(7.5 - z\right) \cdot \left(0.5 - z\right)\right) - 7.5} \cdot \sqrt{2 \cdot \pi}\right)}}{\frac{\sin \left(z \cdot \pi\right)}{\pi}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{9999999999998099}{10000000000000000} \cdot \left(e^{\left(z + \log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)\right) - \frac{15}{2}} \cdot \sqrt{2 \cdot \pi}\right)}{\frac{\sin \left(z \cdot \pi\right)}{\pi}}} \]

   8. Applied rewrites 15.0%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\pi + \pi} \cdot e^{\mathsf{fma}\left(0.5 - z, \log \left(7.5 - z\right), z\right) - 7.5}\right) \cdot 0.9999999999998099}{\sin \left(\pi \cdot z\right)} \cdot \pi} \]

   if -3.7999999999999998 < z

   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 97.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. sqrt-prod N/A

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

      5. lift-PI.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 99.2

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 98.4%

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

   8. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

   10. Applied rewrites 96.3%

       \[\leadsto \frac{\color{blue}{263.3831869810514 \cdot \left(\pi \cdot \left(e^{0.5 \cdot \log 7.5 - 7.5} \cdot \sqrt{2 \cdot \pi}\right)\right)}}{\sin \left(\pi \cdot z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 97.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot z\right)\\ \mathbf{if}\;z \leq -7:\\ \;\;\;\;\left(\left(e^{\left(\left(-\log \left(-z\right)\right) \cdot z + z\right) - 7.5} \cdot \sqrt{\pi + \pi}\right) \cdot 0.9999999999998099\right) \cdot \frac{\pi}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{263.3831869810514 \cdot \left(\pi \cdot \left(e^{0.5 \cdot \log 7.5 - 7.5} \cdot \sqrt{2 \cdot \pi}\right)\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sin (* PI z))))
   (if (<= z -7.0)
     (*
      (*
       (* (exp (- (+ (* (- (log (- z))) z) z) 7.5)) (sqrt (+ PI PI)))
       0.9999999999998099)
      (/ PI t_0))
     (/
      (*
       263.3831869810514
       (* PI (* (exp (- (* 0.5 (log 7.5)) 7.5)) (sqrt (* 2.0 PI)))))
      t_0))))

C

double code(double z) {
	double t_0 = sin((((double) M_PI) * z));
	double tmp;
	if (z <= -7.0) {
		tmp = ((exp((((-log(-z) * z) + z) - 7.5)) * sqrt((((double) M_PI) + ((double) M_PI)))) * 0.9999999999998099) * (((double) M_PI) / t_0);
	} else {
		tmp = (263.3831869810514 * (((double) M_PI) * (exp(((0.5 * log(7.5)) - 7.5)) * sqrt((2.0 * ((double) M_PI)))))) / t_0;
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.sin((Math.PI * z));
	double tmp;
	if (z <= -7.0) {
		tmp = ((Math.exp((((-Math.log(-z) * z) + z) - 7.5)) * Math.sqrt((Math.PI + Math.PI))) * 0.9999999999998099) * (Math.PI / t_0);
	} else {
		tmp = (263.3831869810514 * (Math.PI * (Math.exp(((0.5 * Math.log(7.5)) - 7.5)) * Math.sqrt((2.0 * Math.PI))))) / t_0;
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.sin((math.pi * z))
	tmp = 0
	if z <= -7.0:
		tmp = ((math.exp((((-math.log(-z) * z) + z) - 7.5)) * math.sqrt((math.pi + math.pi))) * 0.9999999999998099) * (math.pi / t_0)
	else:
		tmp = (263.3831869810514 * (math.pi * (math.exp(((0.5 * math.log(7.5)) - 7.5)) * math.sqrt((2.0 * math.pi))))) / t_0
	return tmp

Julia

function code(z)
	t_0 = sin(Float64(pi * z))
	tmp = 0.0
	if (z <= -7.0)
		tmp = Float64(Float64(Float64(exp(Float64(Float64(Float64(Float64(-log(Float64(-z))) * z) + z) - 7.5)) * sqrt(Float64(pi + pi))) * 0.9999999999998099) * Float64(pi / t_0));
	else
		tmp = Float64(Float64(263.3831869810514 * Float64(pi * Float64(exp(Float64(Float64(0.5 * log(7.5)) - 7.5)) * sqrt(Float64(2.0 * pi))))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = sin((pi * z));
	tmp = 0.0;
	if (z <= -7.0)
		tmp = ((exp((((-log(-z) * z) + z) - 7.5)) * sqrt((pi + pi))) * 0.9999999999998099) * (pi / t_0);
	else
		tmp = (263.3831869810514 * (pi * (exp(((0.5 * log(7.5)) - 7.5)) * sqrt((2.0 * pi))))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -7.0], N[(N[(N[(N[Exp[N[(N[(N[((-N[Log[(-z)], $MachinePrecision]) * z), $MachinePrecision] + z), $MachinePrecision] - 7.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi + Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.9999999999998099), $MachinePrecision] * N[(Pi / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(263.3831869810514 * N[(Pi * 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]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -7

   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 97.8%

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

   4. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{\frac{9999999999998099}{10000000000000000} \cdot \left(e^{\left(z + \log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)\right) - \frac{15}{2}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)}}{\frac{\sin \left(z \cdot \pi\right)}{\pi}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\frac{9999999999998099}{10000000000000000} \cdot \color{blue}{\left(e^{\left(z + \log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)\right) - \frac{15}{2}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)}}{\frac{\sin \left(z \cdot \pi\right)}{\pi}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{9999999999998099}{10000000000000000} \cdot \left(e^{\left(z + \log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)\right) - \frac{15}{2}} \cdot \color{blue}{\sqrt{2 \cdot \mathsf{PI}\left(\right)}}\right)}{\frac{\sin \left(z \cdot \pi\right)}{\pi}} \]

   6. Applied rewrites 15.0%

      \[\leadsto \frac{\color{blue}{0.9999999999998099 \cdot \left(e^{\left(z + \log \left(7.5 - z\right) \cdot \left(0.5 - z\right)\right) - 7.5} \cdot \sqrt{2 \cdot \pi}\right)}}{\frac{\sin \left(z \cdot \pi\right)}{\pi}} \]

   7. Taylor expanded in z around -inf

      \[\leadsto \frac{\frac{9999999999998099}{10000000000000000} \cdot \left(e^{\left(z + z \cdot \log \left(\frac{-1}{z}\right)\right) - \frac{15}{2}} \cdot \sqrt{2 \cdot \pi}\right)}{\frac{\sin \left(z \cdot \pi\right)}{\pi}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\frac{9999999999998099}{10000000000000000} \cdot \left(e^{\left(z + z \cdot \log \left(\frac{-1}{z}\right)\right) - \frac{15}{2}} \cdot \sqrt{2 \cdot \pi}\right)}{\frac{\sin \left(z \cdot \pi\right)}{\pi}} \]

      2. lower-log.f64 N/A

         \[\leadsto \frac{\frac{9999999999998099}{10000000000000000} \cdot \left(e^{\left(z + z \cdot \log \left(\frac{-1}{z}\right)\right) - \frac{15}{2}} \cdot \sqrt{2 \cdot \pi}\right)}{\frac{\sin \left(z \cdot \pi\right)}{\pi}} \]

      3. lower-/.f64 7.8

         \[\leadsto \frac{0.9999999999998099 \cdot \left(e^{\left(z + z \cdot \log \left(\frac{-1}{z}\right)\right) - 7.5} \cdot \sqrt{2 \cdot \pi}\right)}{\frac{\sin \left(z \cdot \pi\right)}{\pi}} \]

   9. Applied rewrites 7.8%

      \[\leadsto \frac{0.9999999999998099 \cdot \left(e^{\left(z + z \cdot \log \left(\frac{-1}{z}\right)\right) - 7.5} \cdot \sqrt{2 \cdot \pi}\right)}{\frac{\sin \left(z \cdot \pi\right)}{\pi}} \]

   11. Applied rewrites 7.8%

       \[\leadsto \color{blue}{\left(\left(e^{\left(\left(-\log \left(-z\right)\right) \cdot z + z\right) - 7.5} \cdot \sqrt{\pi + \pi}\right) \cdot 0.9999999999998099\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}} \]

   if -7 < z

   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 97.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. sqrt-prod N/A

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

      5. lift-PI.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 99.2

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 98.4%

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

   8. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

   10. Applied rewrites 96.3%

       \[\leadsto \frac{\color{blue}{263.3831869810514 \cdot \left(\pi \cdot \left(e^{0.5 \cdot \log 7.5 - 7.5} \cdot \sqrt{2 \cdot \pi}\right)\right)}}{\sin \left(\pi \cdot z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 96.5% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - -6.5), $MachinePrecision]}, N[(N[(N[(N[(263.3831869810514 * N[Sqrt[N[(Pi + Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision] * Pi), $MachinePrecision] / N[Sin[N[(Pi * z), $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 97.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. sqrt-prod N/A

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

   5. lift-PI.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-sqrt.f64 99.2

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

5. Applied rewrites 99.2%

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in z around 0

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

10. Applied rewrites 96.5%

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

Alternative 8: 96.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]}, N[(N[(N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * N[(545.0353078428827 + N[(606.6766809167608 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Log[N[(t$95$0 - -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision] + N[(-0.5 - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(z * Pi), $MachinePrecision] / Pi), $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 97.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. sqrt-prod N/A

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

   5. lift-PI.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-sqrt.f64 99.2

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

5. Applied rewrites 99.2%

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

6. Taylor expanded in z around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 96.0

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

8. Applied rewrites 96.0%

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

9. Taylor expanded in z around 0

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

    1. lower-*.f64 N/A

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

    2. lower-PI.f64 96.0

       \[\leadsto \frac{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right) \cdot \left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot e^{\mathsf{fma}\left(\log \left(\left(\left(1 - z\right) - -6\right) - -0.5\right), \left(1 - z\right) - 0.5, -0.5 - \left(\left(1 - z\right) - -6\right)\right)}\right)}{\frac{z \cdot \pi}{\pi}} \]

11. Applied rewrites 96.0%

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

Alternative 9: 96.3% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[z_] := N[(N[(263.3831869810514 * N[(Pi * 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]), $MachinePrecision]), $MachinePrecision] / N[Sin[N[(Pi * z), $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 97.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. sqrt-prod N/A

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

   5. lift-PI.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-sqrt.f64 99.2

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

5. Applied rewrites 99.2%

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in z around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-PI.f64 N/A

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

   4. lower-*.f64 N/A

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

10. Applied rewrites 96.3%

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

Alternative 10: 96.0% 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 97.8%

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

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 96.3%

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

Alternative 11: 95.6% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{z}{\sqrt{15 \cdot \pi} \cdot e^{-7.5}}} \cdot 263.3831869810514 \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (* (/ 1.0 (/ z (* (sqrt (* 15.0 PI)) (exp -7.5)))) 263.3831869810514))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[z_] := N[(N[(1.0 / N[(z / N[(N[Sqrt[N[(15.0 * Pi), $MachinePrecision]], $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. Step-by-step derivation

    1. lift-/.f64 N/A

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

    2. div-flip N/A

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

    3. lower-/.f64 N/A

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

    4. lower-/.f64 95.6

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

    5. lift-*.f64 N/A

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

    6. *-commutative N/A

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

    7. lower-*.f64 95.6

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

11. Applied rewrites 95.6%

    \[\leadsto \frac{1}{\frac{z}{\sqrt{15 \cdot \pi} \cdot e^{-7.5}}} \cdot 263.3831869810514 \]
12. Add Preprocessing

Alternative 12: 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 2025148 
(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))))))