Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.4% → 98.8%

Time: 1.0min

Alternatives: 11

Speedup: 1.5×

• could not determine a ground truth

  1. z = -1.5908572976966669e+199

Specification

\[z \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-176.6150291621406}{4 - z}\\ t_1 := \left(1 - z\right) + -1\\ t_2 := t\_1 + 7\\ t_3 := \frac{-0.13857109526572012}{6 - z}\\ t_4 := \frac{\pi}{\sin \left(\pi \cdot z\right)}\\ t_5 := \frac{12.507343278686905}{5 - z}\\ t_6 := \sqrt{\pi \cdot 2}\\ t_7 := \left(z + -1\right) - -1\\ t_8 := \frac{-1259.1392167224028}{z - 2}\\ \mathbf{if}\;t\_4 \cdot \left(\left(\left(t\_6 \cdot {\left(t\_2 + 0.5\right)}^{\left(t\_1 + 0.5\right)}\right) \cdot e^{\left(t\_7 - 7\right) - 0.5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{-1 + t\_7}\right) + \frac{-1259.1392167224028}{2 - t\_7}\right) + \frac{771.3234287776531}{t\_1 + 3}\right) + \frac{-176.6150291621406}{t\_1 + 4}\right) + \frac{12.507343278686905}{5 - t\_7}\right) + \frac{-0.13857109526572012}{6 - t\_7}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_2}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_1 + 8}\right)\right) \leq 5 \cdot 10^{+307}:\\ \;\;\;\;t\_4 \cdot \left(\left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(0.9999999999998099 - \left(t\_8 - \frac{771.3234287776531}{3 - z}\right)\right) + t\_0\right) + \left(t\_3 + \left(t\_5 + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \cdot \left(t\_6 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_6 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(z, e^{-7.5} \cdot \left(\left(1 + \mathsf{fma}\left(-1, \log 7.5, -0.06666666666666667\right)\right) \cdot \sqrt{7.5}\right), e^{-7.5} \cdot \sqrt{7.5}\right)\right)\right)\right) \cdot \left(t\_4 \cdot \left(\left(t\_3 + \left(t\_0 + t\_5\right)\right) - \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z - 7} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - 8}\right) + \left(\left(\frac{676.5203681218851}{z + -1} + t\_8\right) + \left(\frac{771.3234287776531}{z - 3} - 0.9999999999998099\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ -176.6150291621406 (- 4.0 z)))
        (t_1 (+ (- 1.0 z) -1.0))
        (t_2 (+ t_1 7.0))
        (t_3 (/ -0.13857109526572012 (- 6.0 z)))
        (t_4 (/ PI (sin (* PI z))))
        (t_5 (/ 12.507343278686905 (- 5.0 z)))
        (t_6 (sqrt (* PI 2.0)))
        (t_7 (- (+ z -1.0) -1.0))
        (t_8 (/ -1259.1392167224028 (- z 2.0))))
   (if (<=
        (*
         t_4
         (*
          (* (* t_6 (pow (+ t_2 0.5) (+ t_1 0.5))) (exp (- (- t_7 7.0) 0.5)))
          (+
           (+
            (+
             (+
              (+
               (+
                (+
                 (- 0.9999999999998099 (/ 676.5203681218851 (+ -1.0 t_7)))
                 (/ -1259.1392167224028 (- 2.0 t_7)))
                (/ 771.3234287776531 (+ t_1 3.0)))
               (/ -176.6150291621406 (+ t_1 4.0)))
              (/ 12.507343278686905 (- 5.0 t_7)))
             (/ -0.13857109526572012 (- 6.0 t_7)))
            (/ 9.984369578019572e-6 t_2))
           (/ 1.5056327351493116e-7 (+ t_1 8.0)))))
        5e+307)
     (*
      t_4
      (*
       (+
        (/ 676.5203681218851 (- 1.0 z))
        (+
         (+ (- 0.9999999999998099 (- t_8 (/ 771.3234287776531 (- 3.0 z)))) t_0)
         (+
          t_3
          (+
           t_5
           (+
            (/ 9.984369578019572e-6 (- 7.0 z))
            (/ 1.5056327351493116e-7 (- 8.0 z)))))))
       (* t_6 (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5))))))
     (*
      (*
       t_6
       (log1p
        (expm1
         (fma
          z
          (*
           (exp -7.5)
           (* (+ 1.0 (fma -1.0 (log 7.5) -0.06666666666666667)) (sqrt 7.5)))
          (* (exp -7.5) (sqrt 7.5))))))
      (*
       t_4
       (-
        (+ t_3 (+ t_0 t_5))
        (+
         (+
          (/ 9.984369578019572e-6 (- z 7.0))
          (/ 1.5056327351493116e-7 (- z 8.0)))
         (+
          (+ (/ 676.5203681218851 (+ z -1.0)) t_8)
          (- (/ 771.3234287776531 (- z 3.0)) 0.9999999999998099)))))))))

C

double code(double z) {
	double t_0 = -176.6150291621406 / (4.0 - z);
	double t_1 = (1.0 - z) + -1.0;
	double t_2 = t_1 + 7.0;
	double t_3 = -0.13857109526572012 / (6.0 - z);
	double t_4 = ((double) M_PI) / sin((((double) M_PI) * z));
	double t_5 = 12.507343278686905 / (5.0 - z);
	double t_6 = sqrt((((double) M_PI) * 2.0));
	double t_7 = (z + -1.0) - -1.0;
	double t_8 = -1259.1392167224028 / (z - 2.0);
	double tmp;
	if ((t_4 * (((t_6 * pow((t_2 + 0.5), (t_1 + 0.5))) * exp(((t_7 - 7.0) - 0.5))) * ((((((((0.9999999999998099 - (676.5203681218851 / (-1.0 + t_7))) + (-1259.1392167224028 / (2.0 - t_7))) + (771.3234287776531 / (t_1 + 3.0))) + (-176.6150291621406 / (t_1 + 4.0))) + (12.507343278686905 / (5.0 - t_7))) + (-0.13857109526572012 / (6.0 - t_7))) + (9.984369578019572e-6 / t_2)) + (1.5056327351493116e-7 / (t_1 + 8.0))))) <= 5e+307) {
		tmp = t_4 * (((676.5203681218851 / (1.0 - z)) + (((0.9999999999998099 - (t_8 - (771.3234287776531 / (3.0 - z)))) + t_0) + (t_3 + (t_5 + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))))) * (t_6 * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5)))));
	} else {
		tmp = (t_6 * log1p(expm1(fma(z, (exp(-7.5) * ((1.0 + fma(-1.0, log(7.5), -0.06666666666666667)) * sqrt(7.5))), (exp(-7.5) * sqrt(7.5)))))) * (t_4 * ((t_3 + (t_0 + t_5)) - (((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0))) + (((676.5203681218851 / (z + -1.0)) + t_8) + ((771.3234287776531 / (z - 3.0)) - 0.9999999999998099)))));
	}
	return tmp;
}

Julia

function code(z)
	t_0 = Float64(-176.6150291621406 / Float64(4.0 - z))
	t_1 = Float64(Float64(1.0 - z) + -1.0)
	t_2 = Float64(t_1 + 7.0)
	t_3 = Float64(-0.13857109526572012 / Float64(6.0 - z))
	t_4 = Float64(pi / sin(Float64(pi * z)))
	t_5 = Float64(12.507343278686905 / Float64(5.0 - z))
	t_6 = sqrt(Float64(pi * 2.0))
	t_7 = Float64(Float64(z + -1.0) - -1.0)
	t_8 = Float64(-1259.1392167224028 / Float64(z - 2.0))
	tmp = 0.0
	if (Float64(t_4 * Float64(Float64(Float64(t_6 * (Float64(t_2 + 0.5) ^ Float64(t_1 + 0.5))) * exp(Float64(Float64(t_7 - 7.0) - 0.5))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 - Float64(676.5203681218851 / Float64(-1.0 + t_7))) + Float64(-1259.1392167224028 / Float64(2.0 - t_7))) + Float64(771.3234287776531 / Float64(t_1 + 3.0))) + Float64(-176.6150291621406 / Float64(t_1 + 4.0))) + Float64(12.507343278686905 / Float64(5.0 - t_7))) + Float64(-0.13857109526572012 / Float64(6.0 - t_7))) + Float64(9.984369578019572e-6 / t_2)) + Float64(1.5056327351493116e-7 / Float64(t_1 + 8.0))))) <= 5e+307)
		tmp = Float64(t_4 * Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(Float64(0.9999999999998099 - Float64(t_8 - Float64(771.3234287776531 / Float64(3.0 - z)))) + t_0) + Float64(t_3 + Float64(t_5 + Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))))))) * Float64(t_6 * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5))))));
	else
		tmp = Float64(Float64(t_6 * log1p(expm1(fma(z, Float64(exp(-7.5) * Float64(Float64(1.0 + fma(-1.0, log(7.5), -0.06666666666666667)) * sqrt(7.5))), Float64(exp(-7.5) * sqrt(7.5)))))) * Float64(t_4 * Float64(Float64(t_3 + Float64(t_0 + t_5)) - Float64(Float64(Float64(9.984369578019572e-6 / Float64(z - 7.0)) + Float64(1.5056327351493116e-7 / Float64(z - 8.0))) + Float64(Float64(Float64(676.5203681218851 / Float64(z + -1.0)) + t_8) + Float64(Float64(771.3234287776531 / Float64(z - 3.0)) - 0.9999999999998099))))));
	end
	return tmp
end

Wolfram

code[z_] := Block[{t$95$0 = N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - z), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + 7.0), $MachinePrecision]}, Block[{t$95$3 = N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$7 = N[(N[(z + -1.0), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$8 = N[(-1259.1392167224028 / N[(z - 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$4 * N[(N[(N[(t$95$6 * N[Power[N[(t$95$2 + 0.5), $MachinePrecision], N[(t$95$1 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t$95$7 - 7.0), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 - N[(676.5203681218851 / N[(-1.0 + t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(t$95$1 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(t$95$1 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(5.0 - t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(6.0 - t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$2), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(t$95$1 + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+307], N[(t$95$4 * N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.9999999999998099 - N[(t$95$8 - N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + N[(t$95$3 + N[(t$95$5 + N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$6 * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$6 * N[Log[1 + N[(Exp[N[(z * N[(N[Exp[-7.5], $MachinePrecision] * N[(N[(1.0 + N[(-1.0 * N[Log[7.5], $MachinePrecision] + -0.06666666666666667), $MachinePrecision]), $MachinePrecision] * N[Sqrt[7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Exp[-7.5], $MachinePrecision] * N[Sqrt[7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$4 * N[(N[(t$95$3 + N[(t$95$0 + t$95$5), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(9.984369578019572e-6 / N[(z - 7.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(z - 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(676.5203681218851 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision] + t$95$8), $MachinePrecision] + N[(N[(771.3234287776531 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision] - 0.9999999999998099), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (PI.f64) (sin.f64 (*.f64 (PI.f64) z))) (*.f64 (*.f64 (*.f64 (sqrt.f64 (*.f64 (PI.f64) #s(literal 2 binary64))) (pow.f64 (+.f64 (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)) #s(literal 1/2 binary64)) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 1/2 binary64)))) (exp.f64 (neg.f64 (+.f64 (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)) #s(literal 1/2 binary64))))) (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 #s(literal 9999999999998099/10000000000000000 binary64) (/.f64 #s(literal 6765203681218851/10000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 1 binary64)))) (/.f64 #s(literal -3147848041806007/2500000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 2 binary64)))) (/.f64 #s(literal 7713234287776531/10000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 3 binary64)))) (/.f64 #s(literal -883075145810703/5000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 4 binary64)))) (/.f64 #s(literal 2501468655737381/200000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 5 binary64)))) (/.f64 #s(literal -3464277381643003/25000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 6 binary64)))) (/.f64 #s(literal 2496092394504893/250000000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)))) (/.f64 #s(literal 3764081837873279/25000000000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 8 binary64)))))) < 5e307

   1. Initial program 97.4%

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

   4. Applied egg-rr 98.5%

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

   6. Simplified 99.2%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)\right)} \]

   if 5e307 < (*.f64 (/.f64 (PI.f64) (sin.f64 (*.f64 (PI.f64) z))) (*.f64 (*.f64 (*.f64 (sqrt.f64 (*.f64 (PI.f64) #s(literal 2 binary64))) (pow.f64 (+.f64 (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)) #s(literal 1/2 binary64)) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 1/2 binary64)))) (exp.f64 (neg.f64 (+.f64 (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)) #s(literal 1/2 binary64))))) (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 #s(literal 9999999999998099/10000000000000000 binary64) (/.f64 #s(literal 6765203681218851/10000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 1 binary64)))) (/.f64 #s(literal -3147848041806007/2500000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 2 binary64)))) (/.f64 #s(literal 7713234287776531/10000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 3 binary64)))) (/.f64 #s(literal -883075145810703/5000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 4 binary64)))) (/.f64 #s(literal 2501468655737381/200000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 5 binary64)))) (/.f64 #s(literal -3464277381643003/25000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 6 binary64)))) (/.f64 #s(literal 2496092394504893/250000000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)))) (/.f64 #s(literal 3764081837873279/25000000000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 8 binary64))))))

   1. Initial program 0.0%

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

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 3.2%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \color{blue}{\left(z \cdot \left(e^{-7.5} \cdot \sqrt{7.5} + e^{-7.5} \cdot \left(\sqrt{7.5} \cdot \left(-1 \cdot \log 7.5 - 0.06666666666666667\right)\right)\right) + e^{-7.5} \cdot \sqrt{7.5}\right)}\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. fma-define 3.2%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \color{blue}{\mathsf{fma}\left(z, e^{-7.5} \cdot \sqrt{7.5} + e^{-7.5} \cdot \left(\sqrt{7.5} \cdot \left(-1 \cdot \log 7.5 - 0.06666666666666667\right)\right), e^{-7.5} \cdot \sqrt{7.5}\right)}\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]

      2. distribute-lft-out 3.2%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \mathsf{fma}\left(z, \color{blue}{e^{-7.5} \cdot \left(\sqrt{7.5} + \sqrt{7.5} \cdot \left(-1 \cdot \log 7.5 - 0.06666666666666667\right)\right)}, e^{-7.5} \cdot \sqrt{7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]

      3. *-commutative 3.2%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \mathsf{fma}\left(z, e^{-7.5} \cdot \left(\sqrt{7.5} + \color{blue}{\left(-1 \cdot \log 7.5 - 0.06666666666666667\right) \cdot \sqrt{7.5}}\right), e^{-7.5} \cdot \sqrt{7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]

      4. distribute-rgt1-in 3.2%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \mathsf{fma}\left(z, e^{-7.5} \cdot \color{blue}{\left(\left(\left(-1 \cdot \log 7.5 - 0.06666666666666667\right) + 1\right) \cdot \sqrt{7.5}\right)}, e^{-7.5} \cdot \sqrt{7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]

      5. fma-neg 3.2%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \mathsf{fma}\left(z, e^{-7.5} \cdot \left(\left(\color{blue}{\mathsf{fma}\left(-1, \log 7.5, -0.06666666666666667\right)} + 1\right) \cdot \sqrt{7.5}\right), e^{-7.5} \cdot \sqrt{7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]

      6. metadata-eval 3.2%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \mathsf{fma}\left(z, e^{-7.5} \cdot \left(\left(\mathsf{fma}\left(-1, \log 7.5, \color{blue}{-0.06666666666666667}\right) + 1\right) \cdot \sqrt{7.5}\right), e^{-7.5} \cdot \sqrt{7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]

   7. Simplified 3.2%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \color{blue}{\mathsf{fma}\left(z, e^{-7.5} \cdot \left(\left(\mathsf{fma}\left(-1, \log 7.5, -0.06666666666666667\right) + 1\right) \cdot \sqrt{7.5}\right), e^{-7.5} \cdot \sqrt{7.5}\right)}\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. log1p-expm1-u 83.9%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(z, e^{-7.5} \cdot \left(\left(\mathsf{fma}\left(-1, \log 7.5, -0.06666666666666667\right) + 1\right) \cdot \sqrt{7.5}\right), e^{-7.5} \cdot \sqrt{7.5}\right)\right)\right)}\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]

      2. +-commutative 83.9%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(z, e^{-7.5} \cdot \left(\color{blue}{\left(1 + \mathsf{fma}\left(-1, \log 7.5, -0.06666666666666667\right)\right)} \cdot \sqrt{7.5}\right), e^{-7.5} \cdot \sqrt{7.5}\right)\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]

   9. Applied egg-rr 83.9%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(z, e^{-7.5} \cdot \left(\left(1 + \mathsf{fma}\left(-1, \log 7.5, -0.06666666666666667\right)\right) \cdot \sqrt{7.5}\right), e^{-7.5} \cdot \sqrt{7.5}\right)\right)\right)}\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\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(z + -1\right) - -1\right) - 7\right) - 0.5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{-1 + \left(\left(z + -1\right) - -1\right)}\right) + \frac{-1259.1392167224028}{2 - \left(\left(z + -1\right) - -1\right)}\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}{5 - \left(\left(z + -1\right) - -1\right)}\right) + \frac{-0.13857109526572012}{6 - \left(\left(z + -1\right) - -1\right)}\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) \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(0.9999999999998099 - \left(\frac{-1259.1392167224028}{z - 2} - \frac{771.3234287776531}{3 - z}\right)\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(z, e^{-7.5} \cdot \left(\left(1 + \mathsf{fma}\left(-1, \log 7.5, -0.06666666666666667\right)\right) \cdot \sqrt{7.5}\right), e^{-7.5} \cdot \sqrt{7.5}\right)\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right) - \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z - 7} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - 8}\right) + \left(\left(\frac{676.5203681218851}{z + -1} + \frac{-1259.1392167224028}{z - 2}\right) + \left(\frac{771.3234287776531}{z - 3} - 0.9999999999998099\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) + -1\\ t_1 := t\_0 + 7\\ t_2 := \frac{\pi}{\sin \left(\pi \cdot z\right)}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \left(z + -1\right) - -1\\ \mathbf{if}\;t\_2 \cdot \left(\left(\left(t\_3 \cdot {\left(t\_1 + 0.5\right)}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{\left(t\_4 - 7\right) - 0.5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{-1 + t\_4}\right) + \frac{-1259.1392167224028}{2 - t\_4}\right) + \frac{771.3234287776531}{t\_0 + 3}\right) + \frac{-176.6150291621406}{t\_0 + 4}\right) + \frac{12.507343278686905}{5 - t\_4}\right) + \frac{-0.13857109526572012}{6 - t\_4}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_1}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\right)\right) \leq 5 \cdot 10^{+307}:\\ \;\;\;\;t\_2 \cdot \left(\left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(0.9999999999998099 - \left(\frac{-1259.1392167224028}{z - 2} - \frac{771.3234287776531}{3 - z}\right)\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \cdot \left(t\_3 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_3 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(e^{-7.5} \cdot \left(z + 1\right)\right)\right)\right) \cdot \left(t\_2 \cdot \left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot 544.9358906000987\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ (- 1.0 z) -1.0))
        (t_1 (+ t_0 7.0))
        (t_2 (/ PI (sin (* PI z))))
        (t_3 (sqrt (* PI 2.0)))
        (t_4 (- (+ z -1.0) -1.0)))
   (if (<=
        (*
         t_2
         (*
          (* (* t_3 (pow (+ t_1 0.5) (+ t_0 0.5))) (exp (- (- t_4 7.0) 0.5)))
          (+
           (+
            (+
             (+
              (+
               (+
                (+
                 (- 0.9999999999998099 (/ 676.5203681218851 (+ -1.0 t_4)))
                 (/ -1259.1392167224028 (- 2.0 t_4)))
                (/ 771.3234287776531 (+ t_0 3.0)))
               (/ -176.6150291621406 (+ t_0 4.0)))
              (/ 12.507343278686905 (- 5.0 t_4)))
             (/ -0.13857109526572012 (- 6.0 t_4)))
            (/ 9.984369578019572e-6 t_1))
           (/ 1.5056327351493116e-7 (+ t_0 8.0)))))
        5e+307)
     (*
      t_2
      (*
       (+
        (/ 676.5203681218851 (- 1.0 z))
        (+
         (+
          (-
           0.9999999999998099
           (-
            (/ -1259.1392167224028 (- z 2.0))
            (/ 771.3234287776531 (- 3.0 z))))
          (/ -176.6150291621406 (- 4.0 z)))
         (+
          (/ -0.13857109526572012 (- 6.0 z))
          (+
           (/ 12.507343278686905 (- 5.0 z))
           (+
            (/ 9.984369578019572e-6 (- 7.0 z))
            (/ 1.5056327351493116e-7 (- 8.0 z)))))))
       (* t_3 (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5))))))
     (*
      (*
       t_3
       (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (* (exp -7.5) (+ z 1.0))))
      (*
       t_2
       (+
        (+
         260.9048120626994
         (* z (+ 436.3997278161676 (* z 544.9358906000987))))
        (+
         2.4783749183520145
         (*
          z
          (+
           0.49644474017195733
           (* z (+ 0.09941724278406093 (* z 0.01990483129967024))))))))))))

C

double code(double z) {
	double t_0 = (1.0 - z) + -1.0;
	double t_1 = t_0 + 7.0;
	double t_2 = ((double) M_PI) / sin((((double) M_PI) * z));
	double t_3 = sqrt((((double) M_PI) * 2.0));
	double t_4 = (z + -1.0) - -1.0;
	double tmp;
	if ((t_2 * (((t_3 * pow((t_1 + 0.5), (t_0 + 0.5))) * exp(((t_4 - 7.0) - 0.5))) * ((((((((0.9999999999998099 - (676.5203681218851 / (-1.0 + t_4))) + (-1259.1392167224028 / (2.0 - t_4))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (5.0 - t_4))) + (-0.13857109526572012 / (6.0 - t_4))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))))) <= 5e+307) {
		tmp = t_2 * (((676.5203681218851 / (1.0 - z)) + (((0.9999999999998099 - ((-1259.1392167224028 / (z - 2.0)) - (771.3234287776531 / (3.0 - z)))) + (-176.6150291621406 / (4.0 - z))) + ((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))))) * (t_3 * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5)))));
	} else {
		tmp = (t_3 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * (exp(-7.5) * (z + 1.0)))) * (t_2 * ((260.9048120626994 + (z * (436.3997278161676 + (z * 544.9358906000987)))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * (0.09941724278406093 + (z * 0.01990483129967024))))))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) + -1.0;
	double t_1 = t_0 + 7.0;
	double t_2 = Math.PI / Math.sin((Math.PI * z));
	double t_3 = Math.sqrt((Math.PI * 2.0));
	double t_4 = (z + -1.0) - -1.0;
	double tmp;
	if ((t_2 * (((t_3 * Math.pow((t_1 + 0.5), (t_0 + 0.5))) * Math.exp(((t_4 - 7.0) - 0.5))) * ((((((((0.9999999999998099 - (676.5203681218851 / (-1.0 + t_4))) + (-1259.1392167224028 / (2.0 - t_4))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (5.0 - t_4))) + (-0.13857109526572012 / (6.0 - t_4))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))))) <= 5e+307) {
		tmp = t_2 * (((676.5203681218851 / (1.0 - z)) + (((0.9999999999998099 - ((-1259.1392167224028 / (z - 2.0)) - (771.3234287776531 / (3.0 - z)))) + (-176.6150291621406 / (4.0 - z))) + ((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))))) * (t_3 * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5)))));
	} else {
		tmp = (t_3 * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * (Math.exp(-7.5) * (z + 1.0)))) * (t_2 * ((260.9048120626994 + (z * (436.3997278161676 + (z * 544.9358906000987)))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * (0.09941724278406093 + (z * 0.01990483129967024))))))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = (1.0 - z) + -1.0
	t_1 = t_0 + 7.0
	t_2 = math.pi / math.sin((math.pi * z))
	t_3 = math.sqrt((math.pi * 2.0))
	t_4 = (z + -1.0) - -1.0
	tmp = 0
	if (t_2 * (((t_3 * math.pow((t_1 + 0.5), (t_0 + 0.5))) * math.exp(((t_4 - 7.0) - 0.5))) * ((((((((0.9999999999998099 - (676.5203681218851 / (-1.0 + t_4))) + (-1259.1392167224028 / (2.0 - t_4))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (5.0 - t_4))) + (-0.13857109526572012 / (6.0 - t_4))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))))) <= 5e+307:
		tmp = t_2 * (((676.5203681218851 / (1.0 - z)) + (((0.9999999999998099 - ((-1259.1392167224028 / (z - 2.0)) - (771.3234287776531 / (3.0 - z)))) + (-176.6150291621406 / (4.0 - z))) + ((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))))) * (t_3 * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5)))))
	else:
		tmp = (t_3 * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * (math.exp(-7.5) * (z + 1.0)))) * (t_2 * ((260.9048120626994 + (z * (436.3997278161676 + (z * 544.9358906000987)))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * (0.09941724278406093 + (z * 0.01990483129967024))))))))
	return tmp

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) + -1.0)
	t_1 = Float64(t_0 + 7.0)
	t_2 = Float64(pi / sin(Float64(pi * z)))
	t_3 = sqrt(Float64(pi * 2.0))
	t_4 = Float64(Float64(z + -1.0) - -1.0)
	tmp = 0.0
	if (Float64(t_2 * Float64(Float64(Float64(t_3 * (Float64(t_1 + 0.5) ^ Float64(t_0 + 0.5))) * exp(Float64(Float64(t_4 - 7.0) - 0.5))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 - Float64(676.5203681218851 / Float64(-1.0 + t_4))) + Float64(-1259.1392167224028 / Float64(2.0 - t_4))) + Float64(771.3234287776531 / Float64(t_0 + 3.0))) + Float64(-176.6150291621406 / Float64(t_0 + 4.0))) + Float64(12.507343278686905 / Float64(5.0 - t_4))) + Float64(-0.13857109526572012 / Float64(6.0 - t_4))) + Float64(9.984369578019572e-6 / t_1)) + Float64(1.5056327351493116e-7 / Float64(t_0 + 8.0))))) <= 5e+307)
		tmp = Float64(t_2 * Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(Float64(0.9999999999998099 - Float64(Float64(-1259.1392167224028 / Float64(z - 2.0)) - Float64(771.3234287776531 / Float64(3.0 - z)))) + Float64(-176.6150291621406 / Float64(4.0 - z))) + Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))))))) * Float64(t_3 * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5))))));
	else
		tmp = Float64(Float64(t_3 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * Float64(exp(-7.5) * Float64(z + 1.0)))) * Float64(t_2 * Float64(Float64(260.9048120626994 + Float64(z * Float64(436.3997278161676 + Float64(z * 544.9358906000987)))) + Float64(2.4783749183520145 + Float64(z * Float64(0.49644474017195733 + Float64(z * Float64(0.09941724278406093 + Float64(z * 0.01990483129967024)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = (1.0 - z) + -1.0;
	t_1 = t_0 + 7.0;
	t_2 = pi / sin((pi * z));
	t_3 = sqrt((pi * 2.0));
	t_4 = (z + -1.0) - -1.0;
	tmp = 0.0;
	if ((t_2 * (((t_3 * ((t_1 + 0.5) ^ (t_0 + 0.5))) * exp(((t_4 - 7.0) - 0.5))) * ((((((((0.9999999999998099 - (676.5203681218851 / (-1.0 + t_4))) + (-1259.1392167224028 / (2.0 - t_4))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (5.0 - t_4))) + (-0.13857109526572012 / (6.0 - t_4))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))))) <= 5e+307)
		tmp = t_2 * (((676.5203681218851 / (1.0 - z)) + (((0.9999999999998099 - ((-1259.1392167224028 / (z - 2.0)) - (771.3234287776531 / (3.0 - z)))) + (-176.6150291621406 / (4.0 - z))) + ((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))))) * (t_3 * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5)))));
	else
		tmp = (t_3 * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * (exp(-7.5) * (z + 1.0)))) * (t_2 * ((260.9048120626994 + (z * (436.3997278161676 + (z * 544.9358906000987)))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * (0.09941724278406093 + (z * 0.01990483129967024))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 7.0), $MachinePrecision]}, Block[{t$95$2 = N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(z + -1.0), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[N[(t$95$2 * N[(N[(N[(t$95$3 * N[Power[N[(t$95$1 + 0.5), $MachinePrecision], N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t$95$4 - 7.0), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 - N[(676.5203681218851 / N[(-1.0 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - t$95$4), $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[(5.0 - t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(6.0 - t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(t$95$0 + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+307], N[(t$95$2 * N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.9999999999998099 - N[(N[(-1259.1392167224028 / N[(z - 2.0), $MachinePrecision]), $MachinePrecision] - N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$3 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(N[(260.9048120626994 + N[(z * N[(436.3997278161676 + N[(z * 544.9358906000987), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.4783749183520145 + N[(z * N[(0.49644474017195733 + N[(z * N[(0.09941724278406093 + N[(z * 0.01990483129967024), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (PI.f64) (sin.f64 (*.f64 (PI.f64) z))) (*.f64 (*.f64 (*.f64 (sqrt.f64 (*.f64 (PI.f64) #s(literal 2 binary64))) (pow.f64 (+.f64 (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)) #s(literal 1/2 binary64)) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 1/2 binary64)))) (exp.f64 (neg.f64 (+.f64 (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)) #s(literal 1/2 binary64))))) (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 #s(literal 9999999999998099/10000000000000000 binary64) (/.f64 #s(literal 6765203681218851/10000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 1 binary64)))) (/.f64 #s(literal -3147848041806007/2500000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 2 binary64)))) (/.f64 #s(literal 7713234287776531/10000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 3 binary64)))) (/.f64 #s(literal -883075145810703/5000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 4 binary64)))) (/.f64 #s(literal 2501468655737381/200000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 5 binary64)))) (/.f64 #s(literal -3464277381643003/25000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 6 binary64)))) (/.f64 #s(literal 2496092394504893/250000000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)))) (/.f64 #s(literal 3764081837873279/25000000000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 8 binary64)))))) < 5e307

   1. Initial program 97.4%

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

   4. Applied egg-rr 98.5%

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

   6. Simplified 99.2%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)\right)} \]

   if 5e307 < (*.f64 (/.f64 (PI.f64) (sin.f64 (*.f64 (PI.f64) z))) (*.f64 (*.f64 (*.f64 (sqrt.f64 (*.f64 (PI.f64) #s(literal 2 binary64))) (pow.f64 (+.f64 (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)) #s(literal 1/2 binary64)) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 1/2 binary64)))) (exp.f64 (neg.f64 (+.f64 (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)) #s(literal 1/2 binary64))))) (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 #s(literal 9999999999998099/10000000000000000 binary64) (/.f64 #s(literal 6765203681218851/10000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 1 binary64)))) (/.f64 #s(literal -3147848041806007/2500000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 2 binary64)))) (/.f64 #s(literal 7713234287776531/10000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 3 binary64)))) (/.f64 #s(literal -883075145810703/5000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 4 binary64)))) (/.f64 #s(literal 2501468655737381/200000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 5 binary64)))) (/.f64 #s(literal -3464277381643003/25000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 6 binary64)))) (/.f64 #s(literal 2496092394504893/250000000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)))) (/.f64 #s(literal 3764081837873279/25000000000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 8 binary64))))))

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in z around 0 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \color{blue}{\left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + 0.01990483129967024 \cdot z\right)\right)\right)}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + \color{blue}{z \cdot 0.01990483129967024}\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   7. Simplified 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \color{blue}{\left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. Taylor expanded in z around 0 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(260.9048120626994 + z \cdot \left(436.3997278161676 + 544.9358906000987 \cdot z\right)\right)} + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + \color{blue}{z \cdot 544.9358906000987}\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   10. Simplified 0.0%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot 544.9358906000987\right)\right)} + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   11. Taylor expanded in z around 0 83.3%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(e^{-7.5} + z \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot 544.9358906000987\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   12. Step-by-step derivation

       1. distribute-rgt1-in 83.3%

          \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(\left(z + 1\right) \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot 544.9358906000987\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   13. Simplified 83.3%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(\left(z + 1\right) \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot 544.9358906000987\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\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(z + -1\right) - -1\right) - 7\right) - 0.5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{-1 + \left(\left(z + -1\right) - -1\right)}\right) + \frac{-1259.1392167224028}{2 - \left(\left(z + -1\right) - -1\right)}\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}{5 - \left(\left(z + -1\right) - -1\right)}\right) + \frac{-0.13857109526572012}{6 - \left(\left(z + -1\right) - -1\right)}\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) \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(0.9999999999998099 - \left(\frac{-1259.1392167224028}{z - 2} - \frac{771.3234287776531}{3 - z}\right)\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(e^{-7.5} \cdot \left(z + 1\right)\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot 544.9358906000987\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ t_1 := \sin \left(\pi \cdot z\right)\\ \mathbf{if}\;z \leq -740:\\ \;\;\;\;\left(t\_0 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(e^{-7.5} \cdot \left(z + 1\right)\right)\right)\right) \cdot \left(\frac{\pi}{t\_1} \cdot \left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot 544.9358906000987\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(0.9999999999998099 - \left(\frac{-1259.1392167224028}{z - 2} - \frac{771.3234287776531}{3 - z}\right)\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \cdot \left(t\_0 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 2.0))) (t_1 (sin (* PI z))))
   (if (<= z -740.0)
     (*
      (*
       t_0
       (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (* (exp -7.5) (+ z 1.0))))
      (*
       (/ PI t_1)
       (+
        (+
         260.9048120626994
         (* z (+ 436.3997278161676 (* z 544.9358906000987))))
        (+
         2.4783749183520145
         (*
          z
          (+
           0.49644474017195733
           (* z (+ 0.09941724278406093 (* z 0.01990483129967024)))))))))
     (*
      PI
      (/
       (*
        (+
         (/ 676.5203681218851 (- 1.0 z))
         (+
          (+
           (-
            0.9999999999998099
            (-
             (/ -1259.1392167224028 (- z 2.0))
             (/ 771.3234287776531 (- 3.0 z))))
           (/ -176.6150291621406 (- 4.0 z)))
          (+
           (/ -0.13857109526572012 (- 6.0 z))
           (+
            (/ 12.507343278686905 (- 5.0 z))
            (+
             (/ 9.984369578019572e-6 (- 7.0 z))
             (/ 1.5056327351493116e-7 (- 8.0 z)))))))
        (* t_0 (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5)))))
       t_1)))))

C

double code(double z) {
	double t_0 = sqrt((((double) M_PI) * 2.0));
	double t_1 = sin((((double) M_PI) * z));
	double tmp;
	if (z <= -740.0) {
		tmp = (t_0 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * (exp(-7.5) * (z + 1.0)))) * ((((double) M_PI) / t_1) * ((260.9048120626994 + (z * (436.3997278161676 + (z * 544.9358906000987)))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * (0.09941724278406093 + (z * 0.01990483129967024))))))));
	} else {
		tmp = ((double) M_PI) * ((((676.5203681218851 / (1.0 - z)) + (((0.9999999999998099 - ((-1259.1392167224028 / (z - 2.0)) - (771.3234287776531 / (3.0 - z)))) + (-176.6150291621406 / (4.0 - z))) + ((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))))) * (t_0 * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5))))) / t_1);
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.sqrt((Math.PI * 2.0));
	double t_1 = Math.sin((Math.PI * z));
	double tmp;
	if (z <= -740.0) {
		tmp = (t_0 * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * (Math.exp(-7.5) * (z + 1.0)))) * ((Math.PI / t_1) * ((260.9048120626994 + (z * (436.3997278161676 + (z * 544.9358906000987)))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * (0.09941724278406093 + (z * 0.01990483129967024))))))));
	} else {
		tmp = Math.PI * ((((676.5203681218851 / (1.0 - z)) + (((0.9999999999998099 - ((-1259.1392167224028 / (z - 2.0)) - (771.3234287776531 / (3.0 - z)))) + (-176.6150291621406 / (4.0 - z))) + ((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))))) * (t_0 * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5))))) / t_1);
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.sqrt((math.pi * 2.0))
	t_1 = math.sin((math.pi * z))
	tmp = 0
	if z <= -740.0:
		tmp = (t_0 * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * (math.exp(-7.5) * (z + 1.0)))) * ((math.pi / t_1) * ((260.9048120626994 + (z * (436.3997278161676 + (z * 544.9358906000987)))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * (0.09941724278406093 + (z * 0.01990483129967024))))))))
	else:
		tmp = math.pi * ((((676.5203681218851 / (1.0 - z)) + (((0.9999999999998099 - ((-1259.1392167224028 / (z - 2.0)) - (771.3234287776531 / (3.0 - z)))) + (-176.6150291621406 / (4.0 - z))) + ((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))))) * (t_0 * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5))))) / t_1)
	return tmp

Julia

function code(z)
	t_0 = sqrt(Float64(pi * 2.0))
	t_1 = sin(Float64(pi * z))
	tmp = 0.0
	if (z <= -740.0)
		tmp = Float64(Float64(t_0 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * Float64(exp(-7.5) * Float64(z + 1.0)))) * Float64(Float64(pi / t_1) * Float64(Float64(260.9048120626994 + Float64(z * Float64(436.3997278161676 + Float64(z * 544.9358906000987)))) + Float64(2.4783749183520145 + Float64(z * Float64(0.49644474017195733 + Float64(z * Float64(0.09941724278406093 + Float64(z * 0.01990483129967024)))))))));
	else
		tmp = Float64(pi * Float64(Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(Float64(0.9999999999998099 - Float64(Float64(-1259.1392167224028 / Float64(z - 2.0)) - Float64(771.3234287776531 / Float64(3.0 - z)))) + Float64(-176.6150291621406 / Float64(4.0 - z))) + Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))))))) * Float64(t_0 * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5))))) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = sqrt((pi * 2.0));
	t_1 = sin((pi * z));
	tmp = 0.0;
	if (z <= -740.0)
		tmp = (t_0 * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * (exp(-7.5) * (z + 1.0)))) * ((pi / t_1) * ((260.9048120626994 + (z * (436.3997278161676 + (z * 544.9358906000987)))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * (0.09941724278406093 + (z * 0.01990483129967024))))))));
	else
		tmp = pi * ((((676.5203681218851 / (1.0 - z)) + (((0.9999999999998099 - ((-1259.1392167224028 / (z - 2.0)) - (771.3234287776531 / (3.0 - z)))) + (-176.6150291621406 / (4.0 - z))) + ((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))))) * (t_0 * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5))))) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -740.0], N[(N[(t$95$0 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / t$95$1), $MachinePrecision] * N[(N[(260.9048120626994 + N[(z * N[(436.3997278161676 + N[(z * 544.9358906000987), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.4783749183520145 + N[(z * N[(0.49644474017195733 + N[(z * N[(0.09941724278406093 + N[(z * 0.01990483129967024), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.9999999999998099 - N[(N[(-1259.1392167224028 / N[(z - 2.0), $MachinePrecision]), $MachinePrecision] - N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < -740

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in z around 0 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \color{blue}{\left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + 0.01990483129967024 \cdot z\right)\right)\right)}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + \color{blue}{z \cdot 0.01990483129967024}\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   7. Simplified 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \color{blue}{\left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. Taylor expanded in z around 0 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(260.9048120626994 + z \cdot \left(436.3997278161676 + 544.9358906000987 \cdot z\right)\right)} + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + \color{blue}{z \cdot 544.9358906000987}\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   10. Simplified 0.0%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot 544.9358906000987\right)\right)} + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   11. Taylor expanded in z around 0 83.3%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(e^{-7.5} + z \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot 544.9358906000987\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   12. Step-by-step derivation

       1. distribute-rgt1-in 83.3%

          \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(\left(z + 1\right) \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot 544.9358906000987\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   13. Simplified 83.3%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(\left(z + 1\right) \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot 544.9358906000987\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   if -740 < z

   1. Initial program 97.4%

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

   3. Simplified 96.3%

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

   6. Applied egg-rr 98.7%

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

   8. Simplified 99.2%

      \[\leadsto \color{blue}{\pi \cdot \frac{\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)}{\sin \left(z \cdot \pi\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -740:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(e^{-7.5} \cdot \left(z + 1\right)\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot 544.9358906000987\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(0.9999999999998099 - \left(\frac{-1259.1392167224028}{z - 2} - \frac{771.3234287776531}{3 - z}\right)\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right)}{\sin \left(\pi \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\left(t\_0 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(e^{-7.5} \cdot \left(z + 1\right)\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot 544.9358906000987\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(0.9999999999998099 - \left(\frac{-1259.1392167224028}{z - 2} - \frac{771.3234287776531}{3 - z}\right)\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \cdot \left(t\_0 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right)\right) \cdot \frac{1}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 2.0))))
   (if (<= z -2e-16)
     (*
      (*
       t_0
       (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (* (exp -7.5) (+ z 1.0))))
      (*
       (/ PI (sin (* PI z)))
       (+
        (+
         260.9048120626994
         (* z (+ 436.3997278161676 (* z 544.9358906000987))))
        (+
         2.4783749183520145
         (*
          z
          (+
           0.49644474017195733
           (* z (+ 0.09941724278406093 (* z 0.01990483129967024)))))))))
     (*
      (*
       (+
        (/ 676.5203681218851 (- 1.0 z))
        (+
         (+
          (-
           0.9999999999998099
           (-
            (/ -1259.1392167224028 (- z 2.0))
            (/ 771.3234287776531 (- 3.0 z))))
          (/ -176.6150291621406 (- 4.0 z)))
         (+
          (/ -0.13857109526572012 (- 6.0 z))
          (+
           (/ 12.507343278686905 (- 5.0 z))
           (+
            (/ 9.984369578019572e-6 (- 7.0 z))
            (/ 1.5056327351493116e-7 (- 8.0 z)))))))
       (* t_0 (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5)))))
      (/ 1.0 z)))))

C

double code(double z) {
	double t_0 = sqrt((((double) M_PI) * 2.0));
	double tmp;
	if (z <= -2e-16) {
		tmp = (t_0 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * (exp(-7.5) * (z + 1.0)))) * ((((double) M_PI) / sin((((double) M_PI) * z))) * ((260.9048120626994 + (z * (436.3997278161676 + (z * 544.9358906000987)))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * (0.09941724278406093 + (z * 0.01990483129967024))))))));
	} else {
		tmp = (((676.5203681218851 / (1.0 - z)) + (((0.9999999999998099 - ((-1259.1392167224028 / (z - 2.0)) - (771.3234287776531 / (3.0 - z)))) + (-176.6150291621406 / (4.0 - z))) + ((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))))) * (t_0 * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5))))) * (1.0 / z);
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.sqrt((Math.PI * 2.0));
	double tmp;
	if (z <= -2e-16) {
		tmp = (t_0 * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * (Math.exp(-7.5) * (z + 1.0)))) * ((Math.PI / Math.sin((Math.PI * z))) * ((260.9048120626994 + (z * (436.3997278161676 + (z * 544.9358906000987)))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * (0.09941724278406093 + (z * 0.01990483129967024))))))));
	} else {
		tmp = (((676.5203681218851 / (1.0 - z)) + (((0.9999999999998099 - ((-1259.1392167224028 / (z - 2.0)) - (771.3234287776531 / (3.0 - z)))) + (-176.6150291621406 / (4.0 - z))) + ((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))))) * (t_0 * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5))))) * (1.0 / z);
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.sqrt((math.pi * 2.0))
	tmp = 0
	if z <= -2e-16:
		tmp = (t_0 * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * (math.exp(-7.5) * (z + 1.0)))) * ((math.pi / math.sin((math.pi * z))) * ((260.9048120626994 + (z * (436.3997278161676 + (z * 544.9358906000987)))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * (0.09941724278406093 + (z * 0.01990483129967024))))))))
	else:
		tmp = (((676.5203681218851 / (1.0 - z)) + (((0.9999999999998099 - ((-1259.1392167224028 / (z - 2.0)) - (771.3234287776531 / (3.0 - z)))) + (-176.6150291621406 / (4.0 - z))) + ((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))))) * (t_0 * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5))))) * (1.0 / z)
	return tmp

Julia

function code(z)
	t_0 = sqrt(Float64(pi * 2.0))
	tmp = 0.0
	if (z <= -2e-16)
		tmp = Float64(Float64(t_0 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * Float64(exp(-7.5) * Float64(z + 1.0)))) * Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(260.9048120626994 + Float64(z * Float64(436.3997278161676 + Float64(z * 544.9358906000987)))) + Float64(2.4783749183520145 + Float64(z * Float64(0.49644474017195733 + Float64(z * Float64(0.09941724278406093 + Float64(z * 0.01990483129967024)))))))));
	else
		tmp = Float64(Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(Float64(0.9999999999998099 - Float64(Float64(-1259.1392167224028 / Float64(z - 2.0)) - Float64(771.3234287776531 / Float64(3.0 - z)))) + Float64(-176.6150291621406 / Float64(4.0 - z))) + Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))))))) * Float64(t_0 * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5))))) * Float64(1.0 / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = sqrt((pi * 2.0));
	tmp = 0.0;
	if (z <= -2e-16)
		tmp = (t_0 * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * (exp(-7.5) * (z + 1.0)))) * ((pi / sin((pi * z))) * ((260.9048120626994 + (z * (436.3997278161676 + (z * 544.9358906000987)))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * (0.09941724278406093 + (z * 0.01990483129967024))))))));
	else
		tmp = (((676.5203681218851 / (1.0 - z)) + (((0.9999999999998099 - ((-1259.1392167224028 / (z - 2.0)) - (771.3234287776531 / (3.0 - z)))) + (-176.6150291621406 / (4.0 - z))) + ((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))))) * (t_0 * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5))))) * (1.0 / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -2e-16], N[(N[(t$95$0 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(260.9048120626994 + N[(z * N[(436.3997278161676 + N[(z * 544.9358906000987), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.4783749183520145 + N[(z * N[(0.49644474017195733 + N[(z * N[(0.09941724278406093 + N[(z * 0.01990483129967024), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.9999999999998099 - N[(N[(-1259.1392167224028 / N[(z - 2.0), $MachinePrecision]), $MachinePrecision] - N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2e-16

   1. Initial program 54.8%

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

   3. Simplified 54.7%

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

   5. Taylor expanded in z around 0 49.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \color{blue}{\left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + 0.01990483129967024 \cdot z\right)\right)\right)}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative 49.0%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + \color{blue}{z \cdot 0.01990483129967024}\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   7. Simplified 49.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \color{blue}{\left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. Taylor expanded in z around 0 48.2%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(260.9048120626994 + z \cdot \left(436.3997278161676 + 544.9358906000987 \cdot z\right)\right)} + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative 48.2%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + \color{blue}{z \cdot 544.9358906000987}\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   10. Simplified 48.2%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot 544.9358906000987\right)\right)} + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   11. Taylor expanded in z around 0 81.8%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(e^{-7.5} + z \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot 544.9358906000987\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   12. Step-by-step derivation

       1. distribute-rgt1-in 81.8%

          \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(\left(z + 1\right) \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot 544.9358906000987\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   13. Simplified 81.8%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(\left(z + 1\right) \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot 544.9358906000987\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   if -2e-16 < z

   1. Initial program 97.4%

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

   4. Applied egg-rr 98.5%

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

   6. Simplified 99.3%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)\right)} \]

   7. Taylor expanded in z around 0 99.2%

      \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(e^{-7.5} \cdot \left(z + 1\right)\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot 544.9358906000987\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(0.9999999999998099 - \left(\frac{-1259.1392167224028}{z - 2} - \frac{771.3234287776531}{3 - z}\right)\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right)\right) \cdot \frac{1}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(0.9999999999998099 - \left(\frac{-1259.1392167224028}{z - 2} - \frac{771.3234287776531}{3 - z}\right)\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right)\right) \cdot \frac{1}{z} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (+
    (/ 676.5203681218851 (- 1.0 z))
    (+
     (+
      (-
       0.9999999999998099
       (- (/ -1259.1392167224028 (- z 2.0)) (/ 771.3234287776531 (- 3.0 z))))
      (/ -176.6150291621406 (- 4.0 z)))
     (+
      (/ -0.13857109526572012 (- 6.0 z))
      (+
       (/ 12.507343278686905 (- 5.0 z))
       (+
        (/ 9.984369578019572e-6 (- 7.0 z))
        (/ 1.5056327351493116e-7 (- 8.0 z)))))))
   (* (sqrt (* PI 2.0)) (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5)))))
  (/ 1.0 z)))

C

double code(double z) {
	return (((676.5203681218851 / (1.0 - z)) + (((0.9999999999998099 - ((-1259.1392167224028 / (z - 2.0)) - (771.3234287776531 / (3.0 - z)))) + (-176.6150291621406 / (4.0 - z))) + ((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))))) * (sqrt((((double) M_PI) * 2.0)) * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5))))) * (1.0 / z);
}

Java

public static double code(double z) {
	return (((676.5203681218851 / (1.0 - z)) + (((0.9999999999998099 - ((-1259.1392167224028 / (z - 2.0)) - (771.3234287776531 / (3.0 - z)))) + (-176.6150291621406 / (4.0 - z))) + ((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))))) * (Math.sqrt((Math.PI * 2.0)) * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5))))) * (1.0 / z);
}

Python

def code(z):
	return (((676.5203681218851 / (1.0 - z)) + (((0.9999999999998099 - ((-1259.1392167224028 / (z - 2.0)) - (771.3234287776531 / (3.0 - z)))) + (-176.6150291621406 / (4.0 - z))) + ((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))))) * (math.sqrt((math.pi * 2.0)) * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5))))) * (1.0 / z)

Julia

function code(z)
	return Float64(Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(Float64(0.9999999999998099 - Float64(Float64(-1259.1392167224028 / Float64(z - 2.0)) - Float64(771.3234287776531 / Float64(3.0 - z)))) + Float64(-176.6150291621406 / Float64(4.0 - z))) + Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))))))) * Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5))))) * Float64(1.0 / z))
end

MATLAB

function tmp = code(z)
	tmp = (((676.5203681218851 / (1.0 - z)) + (((0.9999999999998099 - ((-1259.1392167224028 / (z - 2.0)) - (771.3234287776531 / (3.0 - z)))) + (-176.6150291621406 / (4.0 - z))) + ((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))))) * (sqrt((pi * 2.0)) * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5))))) * (1.0 / z);
end

Wolfram

code[z_] := N[(N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.9999999999998099 - N[(N[(-1259.1392167224028 / N[(z - 2.0), $MachinePrecision]), $MachinePrecision] - N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.1%

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

4. Applied egg-rr 96.2%

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

6. Simplified 96.9%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)\right)} \]

7. Taylor expanded in z around 0 96.2%

   \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)\right) \]

8. Final simplification 96.2%

   \[\leadsto \left(\left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(0.9999999999998099 - \left(\frac{-1259.1392167224028}{z - 2} - \frac{771.3234287776531}{3 - z}\right)\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right)\right) \cdot \frac{1}{z} \]
9. Add Preprocessing

Alternative 6: 96.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{z} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - \left(\left(z + -1\right) - -1\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{-0.5 + \left(-6 + \left(z + -1\right)\right)}\right)\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (/ 1.0 z)
   (*
    (sqrt (* PI 2.0))
    (*
     (pow (- 7.5 (- (+ z -1.0) -1.0)) (- (- 1.0 z) 0.5))
     (exp (+ -0.5 (+ -6.0 (+ z -1.0)))))))
  (+
   (+ 263.3831855358925 (* z (+ 436.8961723502244 (* z 545.0353078134797))))
   (+
    (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))
    (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0))))))

C

double code(double z) {
	return ((1.0 / z) * (sqrt((((double) M_PI) * 2.0)) * (pow((7.5 - ((z + -1.0) - -1.0)), ((1.0 - z) - 0.5)) * exp((-0.5 + (-6.0 + (z + -1.0))))))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * 545.0353078134797)))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))));
}

Java

public static double code(double z) {
	return ((1.0 / z) * (Math.sqrt((Math.PI * 2.0)) * (Math.pow((7.5 - ((z + -1.0) - -1.0)), ((1.0 - z) - 0.5)) * Math.exp((-0.5 + (-6.0 + (z + -1.0))))))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * 545.0353078134797)))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))));
}

Python

def code(z):
	return ((1.0 / z) * (math.sqrt((math.pi * 2.0)) * (math.pow((7.5 - ((z + -1.0) - -1.0)), ((1.0 - z) - 0.5)) * math.exp((-0.5 + (-6.0 + (z + -1.0))))))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * 545.0353078134797)))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))))

Julia

function code(z)
	return Float64(Float64(Float64(1.0 / z) * Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(7.5 - Float64(Float64(z + -1.0) - -1.0)) ^ Float64(Float64(1.0 - z) - 0.5)) * exp(Float64(-0.5 + Float64(-6.0 + Float64(z + -1.0))))))) * Float64(Float64(263.3831855358925 + Float64(z * Float64(436.8961723502244 + Float64(z * 545.0353078134797)))) + Float64(Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0)) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0)))))
end

MATLAB

function tmp = code(z)
	tmp = ((1.0 / z) * (sqrt((pi * 2.0)) * (((7.5 - ((z + -1.0) - -1.0)) ^ ((1.0 - z) - 0.5)) * exp((-0.5 + (-6.0 + (z + -1.0))))))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * 545.0353078134797)))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))));
end

Wolfram

code[z_] := N[(N[(N[(1.0 / z), $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(7.5 - N[(N[(z + -1.0), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-0.5 + N[(-6.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(263.3831855358925 + N[(z * N[(436.8961723502244 + N[(z * 545.0353078134797), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.1%

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

3. Simplified 96.6%

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

5. Taylor expanded in z around 0 96.0%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + 545.0353078134797 \cdot z\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
6. Step-by-step derivation

   1. *-commutative 96.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + \color{blue}{z \cdot 545.0353078134797}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

7. Simplified 96.0%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

8. Taylor expanded in z around 0 95.9%

   \[\leadsto \left(\color{blue}{\frac{1}{z}} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

9. Final simplification 95.9%

   \[\leadsto \left(\frac{1}{z} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - \left(\left(z + -1\right) - -1\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{-0.5 + \left(-6 + \left(z + -1\right)\right)}\right)\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
10. Add Preprocessing

Alternative 7: 96.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(\frac{\pi}{\pi \cdot z} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (+
   (+ 263.3831855358925 (* z (+ 436.8961723502244 (* z 545.0353078134797))))
   (+
    (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))
    (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0))))
  (*
   (/ PI (* PI z))
   (* (sqrt (* PI 2.0)) (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5)))))))

C

double code(double z) {
	return ((263.3831855358925 + (z * (436.8961723502244 + (z * 545.0353078134797)))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)))) * ((((double) M_PI) / (((double) M_PI) * z)) * (sqrt((((double) M_PI) * 2.0)) * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5)))));
}

Java

public static double code(double z) {
	return ((263.3831855358925 + (z * (436.8961723502244 + (z * 545.0353078134797)))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)))) * ((Math.PI / (Math.PI * z)) * (Math.sqrt((Math.PI * 2.0)) * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5)))));
}

Python

def code(z):
	return ((263.3831855358925 + (z * (436.8961723502244 + (z * 545.0353078134797)))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)))) * ((math.pi / (math.pi * z)) * (math.sqrt((math.pi * 2.0)) * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5)))))

Julia

function code(z)
	return Float64(Float64(Float64(263.3831855358925 + Float64(z * Float64(436.8961723502244 + Float64(z * 545.0353078134797)))) + Float64(Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0)) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0)))) * Float64(Float64(pi / Float64(pi * z)) * Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5))))))
end

MATLAB

function tmp = code(z)
	tmp = ((263.3831855358925 + (z * (436.8961723502244 + (z * 545.0353078134797)))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)))) * ((pi / (pi * z)) * (sqrt((pi * 2.0)) * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5)))));
end

Wolfram

code[z_] := N[(N[(N[(263.3831855358925 + N[(z * N[(436.8961723502244 + N[(z * 545.0353078134797), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / N[(Pi * z), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.1%

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

3. Simplified 96.6%

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

5. Taylor expanded in z around 0 96.0%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + 545.0353078134797 \cdot z\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
6. Step-by-step derivation

   1. *-commutative 96.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + \color{blue}{z \cdot 545.0353078134797}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

7. Simplified 96.0%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

8. Taylor expanded in z around 0 95.8%

   \[\leadsto \left(\frac{\pi}{\color{blue}{z \cdot \pi}} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

9. Taylor expanded in z around inf 95.8%

   \[\leadsto \left(\frac{\pi}{z \cdot \pi} \cdot \left(\sqrt{\pi \cdot 2} \cdot \color{blue}{\left(e^{\log \left(7.5 - z\right) \cdot \left(0.5 - z\right)} \cdot e^{z - 7.5}\right)}\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
10. Step-by-step derivation

    1. exp-to-pow 95.8%

       \[\leadsto \left(\frac{\pi}{z \cdot \pi} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\color{blue}{{\left(7.5 - z\right)}^{\left(0.5 - z\right)}} \cdot e^{z - 7.5}\right)\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    2. sub-neg 95.8%

       \[\leadsto \left(\frac{\pi}{z \cdot \pi} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\color{blue}{z + \left(-7.5\right)}}\right)\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    3. metadata-eval 95.8%

       \[\leadsto \left(\frac{\pi}{z \cdot \pi} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + \color{blue}{-7.5}}\right)\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    4. +-commutative 95.8%

       \[\leadsto \left(\frac{\pi}{z \cdot \pi} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\color{blue}{-7.5 + z}}\right)\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

11. Simplified 95.8%

    \[\leadsto \left(\frac{\pi}{z \cdot \pi} \cdot \left(\sqrt{\pi \cdot 2} \cdot \color{blue}{\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right)}\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

12. Final simplification 95.8%

    \[\leadsto \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(\frac{\pi}{\pi \cdot z} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right)\right) \]
13. Add Preprocessing

Alternative 8: 96.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - \left(\left(z + -1\right) - -1\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{-0.5 + \left(-6 + \left(z + -1\right)\right)}\right)\right) \cdot \frac{\pi}{\pi \cdot z}\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right) + 1.4451589203350195 \cdot 10^{-6}\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (*
    (sqrt (* PI 2.0))
    (*
     (pow (- 7.5 (- (+ z -1.0) -1.0)) (- (- 1.0 z) 0.5))
     (exp (+ -0.5 (+ -6.0 (+ z -1.0))))))
   (/ PI (* PI z)))
  (+
   (+ 263.3831855358925 (* z (+ 436.8961723502244 (* z 545.0353078134797))))
   1.4451589203350195e-6)))

C

double code(double z) {
	return ((sqrt((((double) M_PI) * 2.0)) * (pow((7.5 - ((z + -1.0) - -1.0)), ((1.0 - z) - 0.5)) * exp((-0.5 + (-6.0 + (z + -1.0)))))) * (((double) M_PI) / (((double) M_PI) * z))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * 545.0353078134797)))) + 1.4451589203350195e-6);
}

Java

public static double code(double z) {
	return ((Math.sqrt((Math.PI * 2.0)) * (Math.pow((7.5 - ((z + -1.0) - -1.0)), ((1.0 - z) - 0.5)) * Math.exp((-0.5 + (-6.0 + (z + -1.0)))))) * (Math.PI / (Math.PI * z))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * 545.0353078134797)))) + 1.4451589203350195e-6);
}

Python

def code(z):
	return ((math.sqrt((math.pi * 2.0)) * (math.pow((7.5 - ((z + -1.0) - -1.0)), ((1.0 - z) - 0.5)) * math.exp((-0.5 + (-6.0 + (z + -1.0)))))) * (math.pi / (math.pi * z))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * 545.0353078134797)))) + 1.4451589203350195e-6)

Julia

function code(z)
	return Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(7.5 - Float64(Float64(z + -1.0) - -1.0)) ^ Float64(Float64(1.0 - z) - 0.5)) * exp(Float64(-0.5 + Float64(-6.0 + Float64(z + -1.0)))))) * Float64(pi / Float64(pi * z))) * Float64(Float64(263.3831855358925 + Float64(z * Float64(436.8961723502244 + Float64(z * 545.0353078134797)))) + 1.4451589203350195e-6))
end

MATLAB

function tmp = code(z)
	tmp = ((sqrt((pi * 2.0)) * (((7.5 - ((z + -1.0) - -1.0)) ^ ((1.0 - z) - 0.5)) * exp((-0.5 + (-6.0 + (z + -1.0)))))) * (pi / (pi * z))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * 545.0353078134797)))) + 1.4451589203350195e-6);
end

Wolfram

code[z_] := N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(7.5 - N[(N[(z + -1.0), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-0.5 + N[(-6.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi / N[(Pi * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(263.3831855358925 + N[(z * N[(436.8961723502244 + N[(z * 545.0353078134797), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.4451589203350195e-6), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.1%

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

3. Simplified 96.6%

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

5. Taylor expanded in z around 0 96.0%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + 545.0353078134797 \cdot z\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
6. Step-by-step derivation

   1. *-commutative 96.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + \color{blue}{z \cdot 545.0353078134797}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

7. Simplified 96.0%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

8. Taylor expanded in z around 0 95.8%

   \[\leadsto \left(\frac{\pi}{\color{blue}{z \cdot \pi}} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

9. Taylor expanded in z around 0 95.8%

   \[\leadsto \left(\frac{\pi}{z \cdot \pi} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right) + \color{blue}{1.4451589203350195 \cdot 10^{-6}}\right) \]

10. Final simplification 95.8%

    \[\leadsto \left(\left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - \left(\left(z + -1\right) - -1\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{-0.5 + \left(-6 + \left(z + -1\right)\right)}\right)\right) \cdot \frac{\pi}{\pi \cdot z}\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right) + 1.4451589203350195 \cdot 10^{-6}\right) \]
11. Add Preprocessing

Alternative 9: 96.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - \left(\left(z + -1\right) - -1\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{-0.5 + \left(-6 + \left(z + -1\right)\right)}\right)\right) \cdot \frac{\pi}{\pi \cdot z}\right) \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (*
    (sqrt (* PI 2.0))
    (*
     (pow (- 7.5 (- (+ z -1.0) -1.0)) (- (- 1.0 z) 0.5))
     (exp (+ -0.5 (+ -6.0 (+ z -1.0))))))
   (/ PI (* PI z)))
  (+ 263.3831869810514 (* z 436.8961725563396))))

C

double code(double z) {
	return ((sqrt((((double) M_PI) * 2.0)) * (pow((7.5 - ((z + -1.0) - -1.0)), ((1.0 - z) - 0.5)) * exp((-0.5 + (-6.0 + (z + -1.0)))))) * (((double) M_PI) / (((double) M_PI) * z))) * (263.3831869810514 + (z * 436.8961725563396));
}

Java

public static double code(double z) {
	return ((Math.sqrt((Math.PI * 2.0)) * (Math.pow((7.5 - ((z + -1.0) - -1.0)), ((1.0 - z) - 0.5)) * Math.exp((-0.5 + (-6.0 + (z + -1.0)))))) * (Math.PI / (Math.PI * z))) * (263.3831869810514 + (z * 436.8961725563396));
}

Python

def code(z):
	return ((math.sqrt((math.pi * 2.0)) * (math.pow((7.5 - ((z + -1.0) - -1.0)), ((1.0 - z) - 0.5)) * math.exp((-0.5 + (-6.0 + (z + -1.0)))))) * (math.pi / (math.pi * z))) * (263.3831869810514 + (z * 436.8961725563396))

Julia

function code(z)
	return Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(7.5 - Float64(Float64(z + -1.0) - -1.0)) ^ Float64(Float64(1.0 - z) - 0.5)) * exp(Float64(-0.5 + Float64(-6.0 + Float64(z + -1.0)))))) * Float64(pi / Float64(pi * z))) * Float64(263.3831869810514 + Float64(z * 436.8961725563396)))
end

MATLAB

function tmp = code(z)
	tmp = ((sqrt((pi * 2.0)) * (((7.5 - ((z + -1.0) - -1.0)) ^ ((1.0 - z) - 0.5)) * exp((-0.5 + (-6.0 + (z + -1.0)))))) * (pi / (pi * z))) * (263.3831869810514 + (z * 436.8961725563396));
end

Wolfram

code[z_] := N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(7.5 - N[(N[(z + -1.0), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-0.5 + N[(-6.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi / N[(Pi * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * 436.8961725563396), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.1%

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

3. Simplified 96.6%

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

5. Taylor expanded in z around 0 96.0%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + 545.0353078134797 \cdot z\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
6. Step-by-step derivation

   1. *-commutative 96.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + \color{blue}{z \cdot 545.0353078134797}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

7. Simplified 96.0%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

8. Taylor expanded in z around 0 95.8%

   \[\leadsto \left(\frac{\pi}{\color{blue}{z \cdot \pi}} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

9. Taylor expanded in z around 0 95.7%

   \[\leadsto \left(\frac{\pi}{z \cdot \pi} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \color{blue}{\left(263.3831869810514 + 436.8961725563396 \cdot z\right)} \]
10. Step-by-step derivation

    1. *-commutative 95.7%

       \[\leadsto \left(\frac{\pi}{z \cdot \pi} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(263.3831869810514 + \color{blue}{z \cdot 436.8961725563396}\right) \]

11. Simplified 95.7%

    \[\leadsto \left(\frac{\pi}{z \cdot \pi} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot 436.8961725563396\right)} \]

12. Final simplification 95.7%

    \[\leadsto \left(\left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - \left(\left(z + -1\right) - -1\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{-0.5 + \left(-6 + \left(z + -1\right)\right)}\right)\right) \cdot \frac{\pi}{\pi \cdot z}\right) \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right) \]
13. Add Preprocessing

Alternative 10: 96.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.1%

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

3. Simplified 94.0%

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

5. Taylor expanded in z around 0 93.1%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\frac{263.3831869810514}{z}} \]

6. Taylor expanded in z around 0 93.3%

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

   1. associate-*r* 93.8%

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

8. Simplified 93.8%

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

   1. pow1 93.8%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{\pi} \cdot e^{-7.5}\right) \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)\right)}^{1}} \cdot \frac{263.3831869810514}{z} \]

   2. associate-*l* 93.3%

      \[\leadsto {\color{blue}{\left(\sqrt{\pi} \cdot \left(e^{-7.5} \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)\right)\right)}}^{1} \cdot \frac{263.3831869810514}{z} \]

   3. sqrt-unprod 93.3%

      \[\leadsto {\left(\sqrt{\pi} \cdot \left(e^{-7.5} \cdot \color{blue}{\sqrt{2 \cdot 7.5}}\right)\right)}^{1} \cdot \frac{263.3831869810514}{z} \]

   4. metadata-eval 93.3%

      \[\leadsto {\left(\sqrt{\pi} \cdot \left(e^{-7.5} \cdot \sqrt{\color{blue}{15}}\right)\right)}^{1} \cdot \frac{263.3831869810514}{z} \]

10. Applied egg-rr 93.3%

    \[\leadsto \color{blue}{{\left(\sqrt{\pi} \cdot \left(e^{-7.5} \cdot \sqrt{15}\right)\right)}^{1}} \cdot \frac{263.3831869810514}{z} \]
11. Step-by-step derivation

    1. unpow1 93.3%

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

12. Simplified 93.3%

    \[\leadsto \color{blue}{\left(\sqrt{\pi} \cdot \left(e^{-7.5} \cdot \sqrt{15}\right)\right)} \cdot \frac{263.3831869810514}{z} \]
13. Step-by-step derivation

    1. associate-*r/ 94.2%

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

    2. associate-*r* 95.0%

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

14. Applied egg-rr 95.0%

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

15. Final simplification 95.0%

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

Alternative 11: 95.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.1%

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

3. Simplified 94.0%

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

5. Taylor expanded in z around 0 93.1%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\frac{263.3831869810514}{z}} \]

6. Taylor expanded in z around 0 93.3%

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

   1. associate-*r* 93.8%

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

8. Simplified 93.8%

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

   1. pow1 93.8%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{\pi} \cdot e^{-7.5}\right) \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)\right)}^{1}} \cdot \frac{263.3831869810514}{z} \]

   2. associate-*l* 93.3%

      \[\leadsto {\color{blue}{\left(\sqrt{\pi} \cdot \left(e^{-7.5} \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)\right)\right)}}^{1} \cdot \frac{263.3831869810514}{z} \]

   3. sqrt-unprod 93.3%

      \[\leadsto {\left(\sqrt{\pi} \cdot \left(e^{-7.5} \cdot \color{blue}{\sqrt{2 \cdot 7.5}}\right)\right)}^{1} \cdot \frac{263.3831869810514}{z} \]

   4. metadata-eval 93.3%

      \[\leadsto {\left(\sqrt{\pi} \cdot \left(e^{-7.5} \cdot \sqrt{\color{blue}{15}}\right)\right)}^{1} \cdot \frac{263.3831869810514}{z} \]

10. Applied egg-rr 93.3%

    \[\leadsto \color{blue}{{\left(\sqrt{\pi} \cdot \left(e^{-7.5} \cdot \sqrt{15}\right)\right)}^{1}} \cdot \frac{263.3831869810514}{z} \]
11. Step-by-step derivation

    1. unpow1 93.3%

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

12. Simplified 93.3%

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

13. Taylor expanded in z around 0 94.6%

    \[\leadsto \color{blue}{263.3831869810514 \cdot \left(\frac{e^{-7.5} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}\right)} \]
14. Step-by-step derivation

    1. *-commutative 94.6%

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

    2. associate-/l* 94.5%

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

15. Simplified 94.5%

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

Reproduce

herbie shell --seed 2024135 
(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))))))