Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.2% → 98.2%

Time: 10.9s

Alternatives: 9

Speedup: 1.9×

• could not determine a ground truth

  1. z = -1001912566.6269813

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: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

3. Applied rewrites 98.1%

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

5. Applied rewrites 98.2%

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

Alternative 2: 98.1% accurate, 1.1× speedup

Math

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

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 (- 1.0 z))
            (/ 1259.1392167224028 (- (- 1.0 z) -1.0)))
           (-
            (/ 771.3234287776531 (- (- 1.0 z) -2.0))
            (/ 176.6150291621406 (- (- 1.0 z) -3.0)))))
         (/ 12.507343278686905 (+ t_0 5.0)))
        (/ -0.13857109526572012 (+ t_0 6.0)))
       (/ 9.984369578019572e-6 t_1))
      (/ 1.5056327351493116e-7 (+ t_0 8.0)))))))

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 / (1.0 - z)) - (1259.1392167224028 / ((1.0 - z) - -1.0))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) - (176.6150291621406 / ((1.0 - z) - -3.0))))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = 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 / (1.0 - z)) - (1259.1392167224028 / ((1.0 - z) - -1.0))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) - (176.6150291621406 / ((1.0 - z) - -3.0))))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
}

Python

def code(z):
	t_0 = (1.0 - z) - 1.0
	t_1 = 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 / (1.0 - z)) - (1259.1392167224028 / ((1.0 - z) - -1.0))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) - (176.6150291621406 / ((1.0 - z) - -3.0))))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))))

Julia

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

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - 1.0;
	t_1 = 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 / (1.0 - z)) - (1259.1392167224028 / ((1.0 - z) - -1.0))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) - (176.6150291621406 / ((1.0 - z) - -3.0))))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - 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[(0.9999999999998099 + N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[(1259.1392167224028 / N[(N[(1.0 - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(N[(1.0 - z), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] - N[(176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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]]]]

Derivation

1. Initial program 96.2%

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

3. Applied rewrites 98.1%

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

Alternative 3: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

3. Applied rewrites 98.1%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. metadata-eval N/A

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

   4. lift--.f64 N/A

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

   5. sub-negate-rev N/A

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

   6. lift--.f64 N/A

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

   7. frac-2neg N/A

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

   8. lift-/.f64 N/A

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

   9. lower-*.f64 98.1

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

5. Applied rewrites 98.1%

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

6. Taylor expanded in z around 0

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

   1. lower-*.f64 98.1

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

8. Applied rewrites 98.1%

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

9. Taylor expanded in z around 0

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

    1. lower-*.f64 98.1

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

11. Applied rewrites 98.1%

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

12. Taylor expanded in z around 0

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

    1. lower-*.f64 98.1

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

14. Applied rewrites 98.1%

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

15. Taylor expanded in z around 0

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

    1. lower-*.f64 98.1

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

17. Applied rewrites 98.1%

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

18. Taylor expanded in z around 0

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

    1. lower-*.f64 98.1

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

20. Applied rewrites 98.1%

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

21. Taylor expanded in z around 0

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

    1. lower-*.f64 98.1

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

23. Applied rewrites 98.1%

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

24. Taylor expanded in z around 0

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

    1. lower-*.f64 98.1

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

26. Applied rewrites 98.1%

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

Alternative 4: 97.9% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (* (/ PI (sin (* PI z))) (exp (- -6.5 (- 1.0 z))))
   (* (pow (- (- 1.0 z) -6.5) (- (- 1.0 z) 0.5)) (sqrt (+ PI PI))))
  (-
   (-
    (-
     (-
      (-
       (-
        (/ 676.5203681218851 (- 1.0 z))
        (/ 1259.1392167224028 (- (- 1.0 z) -1.0)))
       (/ -771.3234287776531 (- (- 1.0 z) -2.0)))
      (- (/ 176.6150291621406 (- (- 1.0 z) -3.0)) 0.9999999999998099))
     (/ -12.507343278686905 (- (- 1.0 z) -4.0)))
    (/ 0.13857109526572012 (- (- 1.0 z) -5.0)))
   (-
    (/ -1.5056327351493116e-7 (- (- 1.0 z) -7.0))
    (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))))))

C

double code(double z) {
	return (((((double) M_PI) / sin((((double) M_PI) * z))) * exp((-6.5 - (1.0 - z)))) * (pow(((1.0 - z) - -6.5), ((1.0 - z) - 0.5)) * sqrt((((double) M_PI) + ((double) M_PI))))) * (((((((676.5203681218851 / (1.0 - z)) - (1259.1392167224028 / ((1.0 - z) - -1.0))) - (-771.3234287776531 / ((1.0 - z) - -2.0))) - ((176.6150291621406 / ((1.0 - z) - -3.0)) - 0.9999999999998099)) - (-12.507343278686905 / ((1.0 - z) - -4.0))) - (0.13857109526572012 / ((1.0 - z) - -5.0))) - ((-1.5056327351493116e-7 / ((1.0 - z) - -7.0)) - (9.984369578019572e-6 / ((1.0 - z) - -6.0))));
}

Java

public static double code(double z) {
	return (((Math.PI / Math.sin((Math.PI * z))) * Math.exp((-6.5 - (1.0 - z)))) * (Math.pow(((1.0 - z) - -6.5), ((1.0 - z) - 0.5)) * Math.sqrt((Math.PI + Math.PI)))) * (((((((676.5203681218851 / (1.0 - z)) - (1259.1392167224028 / ((1.0 - z) - -1.0))) - (-771.3234287776531 / ((1.0 - z) - -2.0))) - ((176.6150291621406 / ((1.0 - z) - -3.0)) - 0.9999999999998099)) - (-12.507343278686905 / ((1.0 - z) - -4.0))) - (0.13857109526572012 / ((1.0 - z) - -5.0))) - ((-1.5056327351493116e-7 / ((1.0 - z) - -7.0)) - (9.984369578019572e-6 / ((1.0 - z) - -6.0))));
}

Python

def code(z):
	return (((math.pi / math.sin((math.pi * z))) * math.exp((-6.5 - (1.0 - z)))) * (math.pow(((1.0 - z) - -6.5), ((1.0 - z) - 0.5)) * math.sqrt((math.pi + math.pi)))) * (((((((676.5203681218851 / (1.0 - z)) - (1259.1392167224028 / ((1.0 - z) - -1.0))) - (-771.3234287776531 / ((1.0 - z) - -2.0))) - ((176.6150291621406 / ((1.0 - z) - -3.0)) - 0.9999999999998099)) - (-12.507343278686905 / ((1.0 - z) - -4.0))) - (0.13857109526572012 / ((1.0 - z) - -5.0))) - ((-1.5056327351493116e-7 / ((1.0 - z) - -7.0)) - (9.984369578019572e-6 / ((1.0 - z) - -6.0))))

Julia

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

MATLAB

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

Wolfram

code[z_] := N[(N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(-6.5 - N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] - -6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi + Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[(1259.1392167224028 / N[(N[(1.0 - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-771.3234287776531 / N[(N[(1.0 - z), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision] - 0.9999999999998099), $MachinePrecision]), $MachinePrecision] - N[(-12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] - -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] - -5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision] - N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

3. Applied rewrites 98.1%

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

5. Applied rewrites 98.2%

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

7. Applied rewrites 97.9%

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

Alternative 5: 97.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(t\_0 + 7\right) + 0.5\right)}^{\left(t\_0 + 0.5\right)}\right) \cdot \left(e^{-7.5} + z \cdot e^{-7.5}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)))
   (*
    (/ PI (sin (* PI z)))
    (*
     (*
      (* (sqrt (* PI 2.0)) (pow (+ (+ t_0 7.0) 0.5) (+ t_0 0.5)))
      (+ (exp -7.5) (* z (exp -7.5))))
     (+
      263.3831869810514
      (*
       z
       (+
        436.8961725563396
        (* z (+ 545.0353078428827 (* 606.6766809167608 z))))))))))

C

double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt((((double) M_PI) * 2.0)) * pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5))) * (exp(-7.5) + (z * exp(-7.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z)))))));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	return (Math.PI / Math.sin((Math.PI * z))) * (((Math.sqrt((Math.PI * 2.0)) * Math.pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5))) * (Math.exp(-7.5) + (z * Math.exp(-7.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z)))))));
}

Python

def code(z):
	t_0 = (1.0 - z) - 1.0
	return (math.pi / math.sin((math.pi * z))) * (((math.sqrt((math.pi * 2.0)) * math.pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5))) * (math.exp(-7.5) + (z * math.exp(-7.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z)))))))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - 1.0)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (Float64(Float64(t_0 + 7.0) + 0.5) ^ Float64(t_0 + 0.5))) * Float64(exp(-7.5) + Float64(z * exp(-7.5)))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * Float64(545.0353078428827 + Float64(606.6766809167608 * z))))))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - 1.0;
	tmp = (pi / sin((pi * z))) * (((sqrt((pi * 2.0)) * (((t_0 + 7.0) + 0.5) ^ (t_0 + 0.5))) * (exp(-7.5) + (z * exp(-7.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z)))))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - 1.0), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[(t$95$0 + 7.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] + N[(z * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * N[(545.0353078428827 + N[(606.6766809167608 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 96.2%

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

2. Taylor expanded in z around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 96.4

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

4. Applied rewrites 96.4%

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

5. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \color{blue}{\left(e^{\frac{-15}{2}} + z \cdot e^{\frac{-15}{2}}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \color{blue}{z \cdot e^{\frac{-15}{2}}}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]

   2. lower-exp.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \color{blue}{z} \cdot e^{\frac{-15}{2}}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + z \cdot \color{blue}{e^{\frac{-15}{2}}}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]

   4. lower-exp.f64 97.3

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot \left(e^{-7.5} + z \cdot e^{-7.5}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)\right) \]

7. Applied rewrites 97.3%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot \color{blue}{\left(e^{-7.5} + z \cdot e^{-7.5}\right)}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)\right) \]
8. Add Preprocessing

Alternative 6: 96.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

2. Taylor expanded in z around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 96.4

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

4. Applied rewrites 96.4%

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

6. Applied rewrites 96.7%

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

Alternative 7: 96.1% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

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 95.4%

   \[\leadsto \color{blue}{263.3831869810514 \cdot \frac{e^{-7.5} \cdot \left(\sqrt{2 \cdot \pi} \cdot {7.5}^{0.5}\right)}{z}} \]
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 95.4

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

6. Applied rewrites 95.4%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. distribute-rgt-out N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval 95.4

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

8. Applied rewrites 95.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f64 N/A

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

   4. add-log-exp N/A

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

   5. log-pow-rev N/A

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

   6. lower-log.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lower-exp.f64 96.1

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

10. Applied rewrites 96.1%

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

Alternative 8: 95.5% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

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 95.4%

   \[\leadsto \color{blue}{263.3831869810514 \cdot \frac{e^{-7.5} \cdot \left(\sqrt{2 \cdot \pi} \cdot {7.5}^{0.5}\right)}{z}} \]
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 95.4

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

6. Applied rewrites 95.4%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. distribute-rgt-out N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval 95.4

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

8. Applied rewrites 95.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 95.5

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

10. Applied rewrites 95.5%

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

Alternative 9: 95.4% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

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 95.4%

   \[\leadsto \color{blue}{263.3831869810514 \cdot \frac{e^{-7.5} \cdot \left(\sqrt{2 \cdot \pi} \cdot {7.5}^{0.5}\right)}{z}} \]
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 95.4

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

6. Applied rewrites 95.4%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. distribute-rgt-out N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval 95.4

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

8. Applied rewrites 95.4%

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

Reproduce

herbie shell --seed 2025156 
(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))))))