Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.4% → 99.0%

Time: 57.2s

Alternatives: 12

Speedup: 1.1×

• could not determine a ground truth

  1. z = -2.464706267941683e+281

Specification

\[z \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{\sin \left(\pi \cdot z\right)}\\ t_1 := \sqrt{\pi \cdot 2}\\ t_2 := \left(z + -1\right) - -1\\ t_3 := \left(1 - z\right) + -1\\ t_4 := t\_3 + 7\\ t_5 := {\left(t\_4 + 0.5\right)}^{\left(t\_3 + 0.5\right)} \cdot t\_1\\ t_6 := \frac{1.5056327351493116 \cdot 10^{-7}}{t\_3 + 8}\\ \mathbf{if}\;t\_0 \cdot \left(\left(t\_5 \cdot e^{\left(t\_2 - 7\right) - 0.5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{-1 + t\_2}\right) + \frac{-1259.1392167224028}{2 - t\_2}\right) + \frac{771.3234287776531}{t\_3 + 3}\right) + \frac{-176.6150291621406}{t\_3 + 4}\right) + \frac{12.507343278686905}{5 - t\_2}\right) + \frac{-0.13857109526572012}{6 - t\_2}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_4}\right) + t\_6\right)\right) \leq 5 \cdot 10^{+302}:\\ \;\;\;\;e^{\log \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot t\_1\right)\right)} \cdot \left(t\_0 \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) - \frac{93.9015195213674 + z \cdot 582.6188486005177}{\left(2 - z\right) \cdot \left(z + -1\right)}\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(t\_5 \cdot e^{-7.5}\right) \cdot \left(t\_6 + \left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ PI (sin (* PI z))))
        (t_1 (sqrt (* PI 2.0)))
        (t_2 (- (+ z -1.0) -1.0))
        (t_3 (+ (- 1.0 z) -1.0))
        (t_4 (+ t_3 7.0))
        (t_5 (* (pow (+ t_4 0.5) (+ t_3 0.5)) t_1))
        (t_6 (/ 1.5056327351493116e-7 (+ t_3 8.0))))
   (if (<=
        (*
         t_0
         (*
          (* t_5 (exp (- (- t_2 7.0) 0.5)))
          (+
           (+
            (+
             (+
              (+
               (+
                (+
                 (- 0.9999999999998099 (/ 676.5203681218851 (+ -1.0 t_2)))
                 (/ -1259.1392167224028 (- 2.0 t_2)))
                (/ 771.3234287776531 (+ t_3 3.0)))
               (/ -176.6150291621406 (+ t_3 4.0)))
              (/ 12.507343278686905 (- 5.0 t_2)))
             (/ -0.13857109526572012 (- 6.0 t_2)))
            (/ 9.984369578019572e-6 t_4))
           t_6)))
        5e+302)
     (*
      (exp (log (* (pow (- 7.5 z) (- 0.5 z)) (* (exp (+ z -7.5)) t_1))))
      (*
       t_0
       (+
        (+
         (+
          (/ 9.984369578019572e-6 (- 7.0 z))
          (/ 1.5056327351493116e-7 (- 8.0 z)))
         (-
          (+ 0.9999999999998099 (/ 771.3234287776531 (- 3.0 z)))
          (/
           (+ 93.9015195213674 (* z 582.6188486005177))
           (* (- 2.0 z) (+ z -1.0)))))
        (+
         (/ -0.13857109526572012 (- 6.0 z))
         (+
          (/ -176.6150291621406 (- 4.0 z))
          (/ 12.507343278686905 (- 5.0 z)))))))
     (*
      t_0
      (*
       (* t_5 (exp -7.5))
       (+
        t_6
        (+
         263.383186962231
         (* z (+ 436.896172553987 (* z 545.0353078425886))))))))))

C

double code(double z) {
	double t_0 = ((double) M_PI) / sin((((double) M_PI) * z));
	double t_1 = sqrt((((double) M_PI) * 2.0));
	double t_2 = (z + -1.0) - -1.0;
	double t_3 = (1.0 - z) + -1.0;
	double t_4 = t_3 + 7.0;
	double t_5 = pow((t_4 + 0.5), (t_3 + 0.5)) * t_1;
	double t_6 = 1.5056327351493116e-7 / (t_3 + 8.0);
	double tmp;
	if ((t_0 * ((t_5 * exp(((t_2 - 7.0) - 0.5))) * ((((((((0.9999999999998099 - (676.5203681218851 / (-1.0 + t_2))) + (-1259.1392167224028 / (2.0 - t_2))) + (771.3234287776531 / (t_3 + 3.0))) + (-176.6150291621406 / (t_3 + 4.0))) + (12.507343278686905 / (5.0 - t_2))) + (-0.13857109526572012 / (6.0 - t_2))) + (9.984369578019572e-6 / t_4)) + t_6))) <= 5e+302) {
		tmp = exp(log((pow((7.5 - z), (0.5 - z)) * (exp((z + -7.5)) * t_1)))) * (t_0 * ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) - ((93.9015195213674 + (z * 582.6188486005177)) / ((2.0 - z) * (z + -1.0))))) + ((-0.13857109526572012 / (6.0 - z)) + ((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))))));
	} else {
		tmp = t_0 * ((t_5 * exp(-7.5)) * (t_6 + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.PI / Math.sin((Math.PI * z));
	double t_1 = Math.sqrt((Math.PI * 2.0));
	double t_2 = (z + -1.0) - -1.0;
	double t_3 = (1.0 - z) + -1.0;
	double t_4 = t_3 + 7.0;
	double t_5 = Math.pow((t_4 + 0.5), (t_3 + 0.5)) * t_1;
	double t_6 = 1.5056327351493116e-7 / (t_3 + 8.0);
	double tmp;
	if ((t_0 * ((t_5 * Math.exp(((t_2 - 7.0) - 0.5))) * ((((((((0.9999999999998099 - (676.5203681218851 / (-1.0 + t_2))) + (-1259.1392167224028 / (2.0 - t_2))) + (771.3234287776531 / (t_3 + 3.0))) + (-176.6150291621406 / (t_3 + 4.0))) + (12.507343278686905 / (5.0 - t_2))) + (-0.13857109526572012 / (6.0 - t_2))) + (9.984369578019572e-6 / t_4)) + t_6))) <= 5e+302) {
		tmp = Math.exp(Math.log((Math.pow((7.5 - z), (0.5 - z)) * (Math.exp((z + -7.5)) * t_1)))) * (t_0 * ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) - ((93.9015195213674 + (z * 582.6188486005177)) / ((2.0 - z) * (z + -1.0))))) + ((-0.13857109526572012 / (6.0 - z)) + ((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))))));
	} else {
		tmp = t_0 * ((t_5 * Math.exp(-7.5)) * (t_6 + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.pi / math.sin((math.pi * z))
	t_1 = math.sqrt((math.pi * 2.0))
	t_2 = (z + -1.0) - -1.0
	t_3 = (1.0 - z) + -1.0
	t_4 = t_3 + 7.0
	t_5 = math.pow((t_4 + 0.5), (t_3 + 0.5)) * t_1
	t_6 = 1.5056327351493116e-7 / (t_3 + 8.0)
	tmp = 0
	if (t_0 * ((t_5 * math.exp(((t_2 - 7.0) - 0.5))) * ((((((((0.9999999999998099 - (676.5203681218851 / (-1.0 + t_2))) + (-1259.1392167224028 / (2.0 - t_2))) + (771.3234287776531 / (t_3 + 3.0))) + (-176.6150291621406 / (t_3 + 4.0))) + (12.507343278686905 / (5.0 - t_2))) + (-0.13857109526572012 / (6.0 - t_2))) + (9.984369578019572e-6 / t_4)) + t_6))) <= 5e+302:
		tmp = math.exp(math.log((math.pow((7.5 - z), (0.5 - z)) * (math.exp((z + -7.5)) * t_1)))) * (t_0 * ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) - ((93.9015195213674 + (z * 582.6188486005177)) / ((2.0 - z) * (z + -1.0))))) + ((-0.13857109526572012 / (6.0 - z)) + ((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))))))
	else:
		tmp = t_0 * ((t_5 * math.exp(-7.5)) * (t_6 + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))))
	return tmp

Julia

function code(z)
	t_0 = Float64(pi / sin(Float64(pi * z)))
	t_1 = sqrt(Float64(pi * 2.0))
	t_2 = Float64(Float64(z + -1.0) - -1.0)
	t_3 = Float64(Float64(1.0 - z) + -1.0)
	t_4 = Float64(t_3 + 7.0)
	t_5 = Float64((Float64(t_4 + 0.5) ^ Float64(t_3 + 0.5)) * t_1)
	t_6 = Float64(1.5056327351493116e-7 / Float64(t_3 + 8.0))
	tmp = 0.0
	if (Float64(t_0 * Float64(Float64(t_5 * exp(Float64(Float64(t_2 - 7.0) - 0.5))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 - Float64(676.5203681218851 / Float64(-1.0 + t_2))) + Float64(-1259.1392167224028 / Float64(2.0 - t_2))) + Float64(771.3234287776531 / Float64(t_3 + 3.0))) + Float64(-176.6150291621406 / Float64(t_3 + 4.0))) + Float64(12.507343278686905 / Float64(5.0 - t_2))) + Float64(-0.13857109526572012 / Float64(6.0 - t_2))) + Float64(9.984369578019572e-6 / t_4)) + t_6))) <= 5e+302)
		tmp = Float64(exp(log(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * Float64(exp(Float64(z + -7.5)) * t_1)))) * Float64(t_0 * Float64(Float64(Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))) + Float64(Float64(0.9999999999998099 + Float64(771.3234287776531 / Float64(3.0 - z))) - Float64(Float64(93.9015195213674 + Float64(z * 582.6188486005177)) / Float64(Float64(2.0 - z) * Float64(z + -1.0))))) + Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(12.507343278686905 / Float64(5.0 - z)))))));
	else
		tmp = Float64(t_0 * Float64(Float64(t_5 * exp(-7.5)) * Float64(t_6 + Float64(263.383186962231 + Float64(z * Float64(436.896172553987 + Float64(z * 545.0353078425886)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = pi / sin((pi * z));
	t_1 = sqrt((pi * 2.0));
	t_2 = (z + -1.0) - -1.0;
	t_3 = (1.0 - z) + -1.0;
	t_4 = t_3 + 7.0;
	t_5 = ((t_4 + 0.5) ^ (t_3 + 0.5)) * t_1;
	t_6 = 1.5056327351493116e-7 / (t_3 + 8.0);
	tmp = 0.0;
	if ((t_0 * ((t_5 * exp(((t_2 - 7.0) - 0.5))) * ((((((((0.9999999999998099 - (676.5203681218851 / (-1.0 + t_2))) + (-1259.1392167224028 / (2.0 - t_2))) + (771.3234287776531 / (t_3 + 3.0))) + (-176.6150291621406 / (t_3 + 4.0))) + (12.507343278686905 / (5.0 - t_2))) + (-0.13857109526572012 / (6.0 - t_2))) + (9.984369578019572e-6 / t_4)) + t_6))) <= 5e+302)
		tmp = exp(log((((7.5 - z) ^ (0.5 - z)) * (exp((z + -7.5)) * t_1)))) * (t_0 * ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) - ((93.9015195213674 + (z * 582.6188486005177)) / ((2.0 - z) * (z + -1.0))))) + ((-0.13857109526572012 / (6.0 - z)) + ((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))))));
	else
		tmp = t_0 * ((t_5 * exp(-7.5)) * (t_6 + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(z + -1.0), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(1.0 - z), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + 7.0), $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[N[(t$95$4 + 0.5), $MachinePrecision], N[(t$95$3 + 0.5), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$6 = N[(1.5056327351493116e-7 / N[(t$95$3 + 8.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(N[(t$95$5 * N[Exp[N[(N[(t$95$2 - 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$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(t$95$3 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(t$95$3 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(5.0 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(6.0 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$4), $MachinePrecision]), $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+302], N[(N[Exp[N[Log[N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(N[(N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.9999999999998099 + N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(93.9015195213674 + N[(z * 582.6188486005177), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 - z), $MachinePrecision] * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(t$95$5 * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision] * N[(t$95$6 + N[(263.383186962231 + N[(z * N[(436.896172553987 + N[(z * 545.0353078425886), $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)))))) < 5e302

   1. Initial program 97.3%

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

   3. Simplified 97.6%

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

      1. add-exp-log 99.3%

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

      2. *-commutative 99.3%

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

      3. *-commutative 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. frac-add 99.3%

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

   8. Applied egg-rr 99.3%

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

   9. Taylor expanded in z around 0 99.3%

      \[\leadsto e^{\log \left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \sqrt{2 \cdot \pi}\right)} \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{\color{blue}{93.9015195213674 + 582.6188486005177 \cdot z}}{\left(1 - z\right) \cdot \left(2 - z\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. rem-exp-log 99.3%

          \[\leadsto e^{\log \color{blue}{\left(e^{\log \left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \sqrt{2 \cdot \pi}\right)}\right)}} \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{93.9015195213674 + 582.6188486005177 \cdot z}{\left(1 - z\right) \cdot \left(2 - z\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]

       2. *-un-lft-identity 99.3%

          \[\leadsto e^{\color{blue}{1 \cdot \log \left(e^{\log \left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \sqrt{2 \cdot \pi}\right)}\right)}} \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{93.9015195213674 + 582.6188486005177 \cdot z}{\left(1 - z\right) \cdot \left(2 - z\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]

       3. rem-exp-log 99.3%

          \[\leadsto e^{1 \cdot \log \color{blue}{\left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \sqrt{2 \cdot \pi}\right)}} \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{93.9015195213674 + 582.6188486005177 \cdot z}{\left(1 - z\right) \cdot \left(2 - z\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]

       4. associate-*l* 99.3%

          \[\leadsto e^{1 \cdot \log \color{blue}{\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \sqrt{2 \cdot \pi}\right)\right)}} \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{93.9015195213674 + 582.6188486005177 \cdot z}{\left(1 - z\right) \cdot \left(2 - z\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]

   11. Applied egg-rr 99.3%

       \[\leadsto e^{\color{blue}{1 \cdot \log \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \sqrt{2 \cdot \pi}\right)\right)}} \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{93.9015195213674 + 582.6188486005177 \cdot z}{\left(1 - z\right) \cdot \left(2 - z\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]
   12. Step-by-step derivation

       1. *-lft-identity 99.3%

          \[\leadsto e^{\color{blue}{\log \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \sqrt{2 \cdot \pi}\right)\right)}} \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{93.9015195213674 + 582.6188486005177 \cdot z}{\left(1 - z\right) \cdot \left(2 - z\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]

       2. +-commutative 99.3%

          \[\leadsto e^{\log \left({\color{blue}{\left(7.5 + \left(-z\right)\right)}}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \sqrt{2 \cdot \pi}\right)\right)} \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{93.9015195213674 + 582.6188486005177 \cdot z}{\left(1 - z\right) \cdot \left(2 - z\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]

       3. sub-neg 99.3%

          \[\leadsto e^{\log \left({\color{blue}{\left(7.5 - z\right)}}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \sqrt{2 \cdot \pi}\right)\right)} \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{93.9015195213674 + 582.6188486005177 \cdot z}{\left(1 - z\right) \cdot \left(2 - z\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]

       4. +-commutative 99.3%

          \[\leadsto e^{\log \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(e^{\color{blue}{-7.5 + z}} \cdot \sqrt{2 \cdot \pi}\right)\right)} \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{93.9015195213674 + 582.6188486005177 \cdot z}{\left(1 - z\right) \cdot \left(2 - z\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]

       5. *-commutative 99.3%

          \[\leadsto e^{\log \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(e^{-7.5 + z} \cdot \sqrt{\color{blue}{\pi \cdot 2}}\right)\right)} \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{93.9015195213674 + 582.6188486005177 \cdot z}{\left(1 - z\right) \cdot \left(2 - z\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]

   13. Simplified 99.3%

       \[\leadsto e^{\color{blue}{\log \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(e^{-7.5 + z} \cdot \sqrt{\pi \cdot 2}\right)\right)}} \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{93.9015195213674 + 582.6188486005177 \cdot z}{\left(1 - z\right) \cdot \left(2 - z\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]

   if 5e302 < (*.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 1.2%

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

   3. Taylor expanded in z around 0 1.2%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(263.383186962231 + z \cdot \left(436.896172553987 + 545.0353078425886 \cdot z\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative 1.2%

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + \color{blue}{z \cdot 545.0353078425886}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. Simplified 1.2%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. Taylor expanded in z around 0 84.5%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot \color{blue}{e^{-7.5}}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\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)} \cdot \sqrt{\pi \cdot 2}\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^{+302}:\\ \;\;\;\;e^{\log \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \sqrt{\pi \cdot 2}\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(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) - \frac{93.9015195213674 + z \cdot 582.6188486005177}{\left(2 - z\right) \cdot \left(z + -1\right)}\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\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)} \cdot \sqrt{\pi \cdot 2}\right) \cdot e^{-7.5}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) + -1\right) + 8} + \left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{\sin \left(\pi \cdot z\right)}\\ t_1 := \sqrt{\pi \cdot 2}\\ t_2 := \left(z + -1\right) - -1\\ t_3 := \left(1 - z\right) + -1\\ t_4 := t\_3 + 7\\ t_5 := {\left(t\_4 + 0.5\right)}^{\left(t\_3 + 0.5\right)} \cdot t\_1\\ t_6 := \frac{1.5056327351493116 \cdot 10^{-7}}{t\_3 + 8}\\ \mathbf{if}\;t\_0 \cdot \left(\left(t\_5 \cdot e^{\left(t\_2 - 7\right) - 0.5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{-1 + t\_2}\right) + \frac{-1259.1392167224028}{2 - t\_2}\right) + \frac{771.3234287776531}{t\_3 + 3}\right) + \frac{-176.6150291621406}{t\_3 + 4}\right) + \frac{12.507343278686905}{5 - t\_2}\right) + \frac{-0.13857109526572012}{6 - t\_2}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_4}\right) + t\_6\right)\right) \leq 5 \cdot 10^{+302}:\\ \;\;\;\;e^{\log \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot t\_1\right)} \cdot \left(t\_0 \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(t\_5 \cdot e^{-7.5}\right) \cdot \left(t\_6 + \left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ PI (sin (* PI z))))
        (t_1 (sqrt (* PI 2.0)))
        (t_2 (- (+ z -1.0) -1.0))
        (t_3 (+ (- 1.0 z) -1.0))
        (t_4 (+ t_3 7.0))
        (t_5 (* (pow (+ t_4 0.5) (+ t_3 0.5)) t_1))
        (t_6 (/ 1.5056327351493116e-7 (+ t_3 8.0))))
   (if (<=
        (*
         t_0
         (*
          (* t_5 (exp (- (- t_2 7.0) 0.5)))
          (+
           (+
            (+
             (+
              (+
               (+
                (+
                 (- 0.9999999999998099 (/ 676.5203681218851 (+ -1.0 t_2)))
                 (/ -1259.1392167224028 (- 2.0 t_2)))
                (/ 771.3234287776531 (+ t_3 3.0)))
               (/ -176.6150291621406 (+ t_3 4.0)))
              (/ 12.507343278686905 (- 5.0 t_2)))
             (/ -0.13857109526572012 (- 6.0 t_2)))
            (/ 9.984369578019572e-6 t_4))
           t_6)))
        5e+302)
     (*
      (exp (log (* (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5))) t_1)))
      (*
       t_0
       (+
        (+
         (/ -0.13857109526572012 (- 6.0 z))
         (+ (/ -176.6150291621406 (- 4.0 z)) (/ 12.507343278686905 (- 5.0 z))))
        (+
         (+
          (/ 9.984369578019572e-6 (- 7.0 z))
          (/ 1.5056327351493116e-7 (- 8.0 z)))
         (+
          (+ 0.9999999999998099 (/ 771.3234287776531 (- 3.0 z)))
          (+
           (/ 676.5203681218851 (- 1.0 z))
           (/ -1259.1392167224028 (- 2.0 z))))))))
     (*
      t_0
      (*
       (* t_5 (exp -7.5))
       (+
        t_6
        (+
         263.383186962231
         (* z (+ 436.896172553987 (* z 545.0353078425886))))))))))

C

double code(double z) {
	double t_0 = ((double) M_PI) / sin((((double) M_PI) * z));
	double t_1 = sqrt((((double) M_PI) * 2.0));
	double t_2 = (z + -1.0) - -1.0;
	double t_3 = (1.0 - z) + -1.0;
	double t_4 = t_3 + 7.0;
	double t_5 = pow((t_4 + 0.5), (t_3 + 0.5)) * t_1;
	double t_6 = 1.5056327351493116e-7 / (t_3 + 8.0);
	double tmp;
	if ((t_0 * ((t_5 * exp(((t_2 - 7.0) - 0.5))) * ((((((((0.9999999999998099 - (676.5203681218851 / (-1.0 + t_2))) + (-1259.1392167224028 / (2.0 - t_2))) + (771.3234287776531 / (t_3 + 3.0))) + (-176.6150291621406 / (t_3 + 4.0))) + (12.507343278686905 / (5.0 - t_2))) + (-0.13857109526572012 / (6.0 - t_2))) + (9.984369578019572e-6 / t_4)) + t_6))) <= 5e+302) {
		tmp = exp(log(((pow((7.5 - z), (0.5 - z)) * exp((z + -7.5))) * t_1))) * (t_0 * (((-0.13857109526572012 / (6.0 - z)) + ((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z)))) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))))));
	} else {
		tmp = t_0 * ((t_5 * exp(-7.5)) * (t_6 + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.PI / Math.sin((Math.PI * z));
	double t_1 = Math.sqrt((Math.PI * 2.0));
	double t_2 = (z + -1.0) - -1.0;
	double t_3 = (1.0 - z) + -1.0;
	double t_4 = t_3 + 7.0;
	double t_5 = Math.pow((t_4 + 0.5), (t_3 + 0.5)) * t_1;
	double t_6 = 1.5056327351493116e-7 / (t_3 + 8.0);
	double tmp;
	if ((t_0 * ((t_5 * Math.exp(((t_2 - 7.0) - 0.5))) * ((((((((0.9999999999998099 - (676.5203681218851 / (-1.0 + t_2))) + (-1259.1392167224028 / (2.0 - t_2))) + (771.3234287776531 / (t_3 + 3.0))) + (-176.6150291621406 / (t_3 + 4.0))) + (12.507343278686905 / (5.0 - t_2))) + (-0.13857109526572012 / (6.0 - t_2))) + (9.984369578019572e-6 / t_4)) + t_6))) <= 5e+302) {
		tmp = Math.exp(Math.log(((Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5))) * t_1))) * (t_0 * (((-0.13857109526572012 / (6.0 - z)) + ((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z)))) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))))));
	} else {
		tmp = t_0 * ((t_5 * Math.exp(-7.5)) * (t_6 + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.pi / math.sin((math.pi * z))
	t_1 = math.sqrt((math.pi * 2.0))
	t_2 = (z + -1.0) - -1.0
	t_3 = (1.0 - z) + -1.0
	t_4 = t_3 + 7.0
	t_5 = math.pow((t_4 + 0.5), (t_3 + 0.5)) * t_1
	t_6 = 1.5056327351493116e-7 / (t_3 + 8.0)
	tmp = 0
	if (t_0 * ((t_5 * math.exp(((t_2 - 7.0) - 0.5))) * ((((((((0.9999999999998099 - (676.5203681218851 / (-1.0 + t_2))) + (-1259.1392167224028 / (2.0 - t_2))) + (771.3234287776531 / (t_3 + 3.0))) + (-176.6150291621406 / (t_3 + 4.0))) + (12.507343278686905 / (5.0 - t_2))) + (-0.13857109526572012 / (6.0 - t_2))) + (9.984369578019572e-6 / t_4)) + t_6))) <= 5e+302:
		tmp = math.exp(math.log(((math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5))) * t_1))) * (t_0 * (((-0.13857109526572012 / (6.0 - z)) + ((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z)))) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))))))
	else:
		tmp = t_0 * ((t_5 * math.exp(-7.5)) * (t_6 + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))))
	return tmp

Julia

function code(z)
	t_0 = Float64(pi / sin(Float64(pi * z)))
	t_1 = sqrt(Float64(pi * 2.0))
	t_2 = Float64(Float64(z + -1.0) - -1.0)
	t_3 = Float64(Float64(1.0 - z) + -1.0)
	t_4 = Float64(t_3 + 7.0)
	t_5 = Float64((Float64(t_4 + 0.5) ^ Float64(t_3 + 0.5)) * t_1)
	t_6 = Float64(1.5056327351493116e-7 / Float64(t_3 + 8.0))
	tmp = 0.0
	if (Float64(t_0 * Float64(Float64(t_5 * exp(Float64(Float64(t_2 - 7.0) - 0.5))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 - Float64(676.5203681218851 / Float64(-1.0 + t_2))) + Float64(-1259.1392167224028 / Float64(2.0 - t_2))) + Float64(771.3234287776531 / Float64(t_3 + 3.0))) + Float64(-176.6150291621406 / Float64(t_3 + 4.0))) + Float64(12.507343278686905 / Float64(5.0 - t_2))) + Float64(-0.13857109526572012 / Float64(6.0 - t_2))) + Float64(9.984369578019572e-6 / t_4)) + t_6))) <= 5e+302)
		tmp = Float64(exp(log(Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5))) * t_1))) * Float64(t_0 * Float64(Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(12.507343278686905 / Float64(5.0 - z)))) + Float64(Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))) + Float64(Float64(0.9999999999998099 + Float64(771.3234287776531 / Float64(3.0 - z))) + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(-1259.1392167224028 / Float64(2.0 - z))))))));
	else
		tmp = Float64(t_0 * Float64(Float64(t_5 * exp(-7.5)) * Float64(t_6 + Float64(263.383186962231 + Float64(z * Float64(436.896172553987 + Float64(z * 545.0353078425886)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = pi / sin((pi * z));
	t_1 = sqrt((pi * 2.0));
	t_2 = (z + -1.0) - -1.0;
	t_3 = (1.0 - z) + -1.0;
	t_4 = t_3 + 7.0;
	t_5 = ((t_4 + 0.5) ^ (t_3 + 0.5)) * t_1;
	t_6 = 1.5056327351493116e-7 / (t_3 + 8.0);
	tmp = 0.0;
	if ((t_0 * ((t_5 * exp(((t_2 - 7.0) - 0.5))) * ((((((((0.9999999999998099 - (676.5203681218851 / (-1.0 + t_2))) + (-1259.1392167224028 / (2.0 - t_2))) + (771.3234287776531 / (t_3 + 3.0))) + (-176.6150291621406 / (t_3 + 4.0))) + (12.507343278686905 / (5.0 - t_2))) + (-0.13857109526572012 / (6.0 - t_2))) + (9.984369578019572e-6 / t_4)) + t_6))) <= 5e+302)
		tmp = exp(log(((((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5))) * t_1))) * (t_0 * (((-0.13857109526572012 / (6.0 - z)) + ((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z)))) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))))));
	else
		tmp = t_0 * ((t_5 * exp(-7.5)) * (t_6 + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(z + -1.0), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(1.0 - z), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + 7.0), $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[N[(t$95$4 + 0.5), $MachinePrecision], N[(t$95$3 + 0.5), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$6 = N[(1.5056327351493116e-7 / N[(t$95$3 + 8.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(N[(t$95$5 * N[Exp[N[(N[(t$95$2 - 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$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(t$95$3 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(t$95$3 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(5.0 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(6.0 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$4), $MachinePrecision]), $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+302], N[(N[Exp[N[Log[N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.9999999999998099 + N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(t$95$5 * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision] * N[(t$95$6 + N[(263.383186962231 + N[(z * N[(436.896172553987 + N[(z * 545.0353078425886), $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)))))) < 5e302

   1. Initial program 97.3%

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

   3. Simplified 97.6%

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

      1. add-exp-log 99.3%

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

      2. *-commutative 99.3%

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

      3. *-commutative 99.3%

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

   6. Applied egg-rr 99.3%

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

   if 5e302 < (*.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 1.2%

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

   3. Taylor expanded in z around 0 1.2%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(263.383186962231 + z \cdot \left(436.896172553987 + 545.0353078425886 \cdot z\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative 1.2%

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + \color{blue}{z \cdot 545.0353078425886}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. Simplified 1.2%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. Taylor expanded in z around 0 84.5%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot \color{blue}{e^{-7.5}}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\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)} \cdot \sqrt{\pi \cdot 2}\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^{+302}:\\ \;\;\;\;e^{\log \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \sqrt{\pi \cdot 2}\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}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\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)} \cdot \sqrt{\pi \cdot 2}\right) \cdot e^{-7.5}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) + -1\right) + 8} + \left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z + -1\right) - -1\\ t_1 := \sin \left(\pi \cdot z\right)\\ t_2 := \frac{\pi}{t\_1}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \left(1 - z\right) + -1\\ t_5 := t\_4 + 7\\ t_6 := {\left(t\_5 + 0.5\right)}^{\left(t\_4 + 0.5\right)} \cdot t\_3\\ t_7 := \frac{1.5056327351493116 \cdot 10^{-7}}{t\_4 + 8}\\ \mathbf{if}\;t\_2 \cdot \left(\left(t\_6 \cdot e^{\left(t\_0 - 7\right) - 0.5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{-1 + t\_0}\right) + \frac{-1259.1392167224028}{2 - t\_0}\right) + \frac{771.3234287776531}{t\_4 + 3}\right) + \frac{-176.6150291621406}{t\_4 + 4}\right) + \frac{12.507343278686905}{5 - t\_0}\right) + \frac{-0.13857109526572012}{6 - t\_0}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_5}\right) + t\_7\right)\right) \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{\pi \cdot \left(t\_3 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(z + -1\right) + -6.5}\right)\right)}{t\_1} \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(\left(t\_6 \cdot e^{-7.5}\right) \cdot \left(t\_7 + \left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (+ z -1.0) -1.0))
        (t_1 (sin (* PI z)))
        (t_2 (/ PI t_1))
        (t_3 (sqrt (* PI 2.0)))
        (t_4 (+ (- 1.0 z) -1.0))
        (t_5 (+ t_4 7.0))
        (t_6 (* (pow (+ t_5 0.5) (+ t_4 0.5)) t_3))
        (t_7 (/ 1.5056327351493116e-7 (+ t_4 8.0))))
   (if (<=
        (*
         t_2
         (*
          (* t_6 (exp (- (- t_0 7.0) 0.5)))
          (+
           (+
            (+
             (+
              (+
               (+
                (+
                 (- 0.9999999999998099 (/ 676.5203681218851 (+ -1.0 t_0)))
                 (/ -1259.1392167224028 (- 2.0 t_0)))
                (/ 771.3234287776531 (+ t_4 3.0)))
               (/ -176.6150291621406 (+ t_4 4.0)))
              (/ 12.507343278686905 (- 5.0 t_0)))
             (/ -0.13857109526572012 (- 6.0 t_0)))
            (/ 9.984369578019572e-6 t_5))
           t_7)))
        5e+302)
     (*
      (/
       (*
        PI
        (*
         t_3
         (*
          (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5))
          (exp (+ (+ z -1.0) -6.5)))))
       t_1)
      (+
       (+
        (+
         (+
          (/ -1259.1392167224028 (- (- 1.0 z) -1.0))
          (+ 0.9999999999998099 (/ 676.5203681218851 (- 1.0 z))))
         (+
          (/ 771.3234287776531 (- (- 1.0 z) -2.0))
          (/ -176.6150291621406 (- (- 1.0 z) -3.0))))
        (+
         (/ 12.507343278686905 (- (- 1.0 z) -4.0))
         (/ -0.13857109526572012 (- (- 1.0 z) -5.0))))
       (+
        (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))
        (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0)))))
     (*
      t_2
      (*
       (* t_6 (exp -7.5))
       (+
        t_7
        (+
         263.383186962231
         (* z (+ 436.896172553987 (* z 545.0353078425886))))))))))

C

double code(double z) {
	double t_0 = (z + -1.0) - -1.0;
	double t_1 = sin((((double) M_PI) * z));
	double t_2 = ((double) M_PI) / t_1;
	double t_3 = sqrt((((double) M_PI) * 2.0));
	double t_4 = (1.0 - z) + -1.0;
	double t_5 = t_4 + 7.0;
	double t_6 = pow((t_5 + 0.5), (t_4 + 0.5)) * t_3;
	double t_7 = 1.5056327351493116e-7 / (t_4 + 8.0);
	double tmp;
	if ((t_2 * ((t_6 * exp(((t_0 - 7.0) - 0.5))) * ((((((((0.9999999999998099 - (676.5203681218851 / (-1.0 + t_0))) + (-1259.1392167224028 / (2.0 - t_0))) + (771.3234287776531 / (t_4 + 3.0))) + (-176.6150291621406 / (t_4 + 4.0))) + (12.507343278686905 / (5.0 - t_0))) + (-0.13857109526572012 / (6.0 - t_0))) + (9.984369578019572e-6 / t_5)) + t_7))) <= 5e+302) {
		tmp = ((((double) M_PI) * (t_3 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(((z + -1.0) + -6.5))))) / t_1) * (((((-1259.1392167224028 / ((1.0 - z) - -1.0)) + (0.9999999999998099 + (676.5203681218851 / (1.0 - z)))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) + (-176.6150291621406 / ((1.0 - z) - -3.0)))) + ((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0)))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))));
	} else {
		tmp = t_2 * ((t_6 * exp(-7.5)) * (t_7 + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = (z + -1.0) - -1.0;
	double t_1 = Math.sin((Math.PI * z));
	double t_2 = Math.PI / t_1;
	double t_3 = Math.sqrt((Math.PI * 2.0));
	double t_4 = (1.0 - z) + -1.0;
	double t_5 = t_4 + 7.0;
	double t_6 = Math.pow((t_5 + 0.5), (t_4 + 0.5)) * t_3;
	double t_7 = 1.5056327351493116e-7 / (t_4 + 8.0);
	double tmp;
	if ((t_2 * ((t_6 * Math.exp(((t_0 - 7.0) - 0.5))) * ((((((((0.9999999999998099 - (676.5203681218851 / (-1.0 + t_0))) + (-1259.1392167224028 / (2.0 - t_0))) + (771.3234287776531 / (t_4 + 3.0))) + (-176.6150291621406 / (t_4 + 4.0))) + (12.507343278686905 / (5.0 - t_0))) + (-0.13857109526572012 / (6.0 - t_0))) + (9.984369578019572e-6 / t_5)) + t_7))) <= 5e+302) {
		tmp = ((Math.PI * (t_3 * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(((z + -1.0) + -6.5))))) / t_1) * (((((-1259.1392167224028 / ((1.0 - z) - -1.0)) + (0.9999999999998099 + (676.5203681218851 / (1.0 - z)))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) + (-176.6150291621406 / ((1.0 - z) - -3.0)))) + ((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0)))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))));
	} else {
		tmp = t_2 * ((t_6 * Math.exp(-7.5)) * (t_7 + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = (z + -1.0) - -1.0
	t_1 = math.sin((math.pi * z))
	t_2 = math.pi / t_1
	t_3 = math.sqrt((math.pi * 2.0))
	t_4 = (1.0 - z) + -1.0
	t_5 = t_4 + 7.0
	t_6 = math.pow((t_5 + 0.5), (t_4 + 0.5)) * t_3
	t_7 = 1.5056327351493116e-7 / (t_4 + 8.0)
	tmp = 0
	if (t_2 * ((t_6 * math.exp(((t_0 - 7.0) - 0.5))) * ((((((((0.9999999999998099 - (676.5203681218851 / (-1.0 + t_0))) + (-1259.1392167224028 / (2.0 - t_0))) + (771.3234287776531 / (t_4 + 3.0))) + (-176.6150291621406 / (t_4 + 4.0))) + (12.507343278686905 / (5.0 - t_0))) + (-0.13857109526572012 / (6.0 - t_0))) + (9.984369578019572e-6 / t_5)) + t_7))) <= 5e+302:
		tmp = ((math.pi * (t_3 * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(((z + -1.0) + -6.5))))) / t_1) * (((((-1259.1392167224028 / ((1.0 - z) - -1.0)) + (0.9999999999998099 + (676.5203681218851 / (1.0 - z)))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) + (-176.6150291621406 / ((1.0 - z) - -3.0)))) + ((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0)))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))))
	else:
		tmp = t_2 * ((t_6 * math.exp(-7.5)) * (t_7 + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))))
	return tmp

Julia

function code(z)
	t_0 = Float64(Float64(z + -1.0) - -1.0)
	t_1 = sin(Float64(pi * z))
	t_2 = Float64(pi / t_1)
	t_3 = sqrt(Float64(pi * 2.0))
	t_4 = Float64(Float64(1.0 - z) + -1.0)
	t_5 = Float64(t_4 + 7.0)
	t_6 = Float64((Float64(t_5 + 0.5) ^ Float64(t_4 + 0.5)) * t_3)
	t_7 = Float64(1.5056327351493116e-7 / Float64(t_4 + 8.0))
	tmp = 0.0
	if (Float64(t_2 * Float64(Float64(t_6 * exp(Float64(Float64(t_0 - 7.0) - 0.5))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 - Float64(676.5203681218851 / Float64(-1.0 + t_0))) + Float64(-1259.1392167224028 / Float64(2.0 - t_0))) + Float64(771.3234287776531 / Float64(t_4 + 3.0))) + Float64(-176.6150291621406 / Float64(t_4 + 4.0))) + Float64(12.507343278686905 / Float64(5.0 - t_0))) + Float64(-0.13857109526572012 / Float64(6.0 - t_0))) + Float64(9.984369578019572e-6 / t_5)) + t_7))) <= 5e+302)
		tmp = Float64(Float64(Float64(pi * Float64(t_3 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(Float64(Float64(z + -1.0) + -6.5))))) / t_1) * Float64(Float64(Float64(Float64(Float64(-1259.1392167224028 / Float64(Float64(1.0 - z) - -1.0)) + Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 - z)))) + Float64(Float64(771.3234287776531 / Float64(Float64(1.0 - z) - -2.0)) + Float64(-176.6150291621406 / Float64(Float64(1.0 - z) - -3.0)))) + Float64(Float64(12.507343278686905 / Float64(Float64(1.0 - z) - -4.0)) + Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) - -5.0)))) + Float64(Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0)) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0)))));
	else
		tmp = Float64(t_2 * Float64(Float64(t_6 * exp(-7.5)) * Float64(t_7 + Float64(263.383186962231 + Float64(z * Float64(436.896172553987 + Float64(z * 545.0353078425886)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = (z + -1.0) - -1.0;
	t_1 = sin((pi * z));
	t_2 = pi / t_1;
	t_3 = sqrt((pi * 2.0));
	t_4 = (1.0 - z) + -1.0;
	t_5 = t_4 + 7.0;
	t_6 = ((t_5 + 0.5) ^ (t_4 + 0.5)) * t_3;
	t_7 = 1.5056327351493116e-7 / (t_4 + 8.0);
	tmp = 0.0;
	if ((t_2 * ((t_6 * exp(((t_0 - 7.0) - 0.5))) * ((((((((0.9999999999998099 - (676.5203681218851 / (-1.0 + t_0))) + (-1259.1392167224028 / (2.0 - t_0))) + (771.3234287776531 / (t_4 + 3.0))) + (-176.6150291621406 / (t_4 + 4.0))) + (12.507343278686905 / (5.0 - t_0))) + (-0.13857109526572012 / (6.0 - t_0))) + (9.984369578019572e-6 / t_5)) + t_7))) <= 5e+302)
		tmp = ((pi * (t_3 * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(((z + -1.0) + -6.5))))) / t_1) * (((((-1259.1392167224028 / ((1.0 - z) - -1.0)) + (0.9999999999998099 + (676.5203681218851 / (1.0 - z)))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) + (-176.6150291621406 / ((1.0 - z) - -3.0)))) + ((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0)))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))));
	else
		tmp = t_2 * ((t_6 * exp(-7.5)) * (t_7 + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(z + -1.0), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(Pi / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(1.0 - z), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 + 7.0), $MachinePrecision]}, Block[{t$95$6 = N[(N[Power[N[(t$95$5 + 0.5), $MachinePrecision], N[(t$95$4 + 0.5), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision]}, Block[{t$95$7 = N[(1.5056327351493116e-7 / N[(t$95$4 + 8.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$2 * N[(N[(t$95$6 * N[Exp[N[(N[(t$95$0 - 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$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(t$95$4 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(t$95$4 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(5.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(6.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$5), $MachinePrecision]), $MachinePrecision] + t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+302], N[(N[(N[(Pi * 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[Exp[N[(N[(z + -1.0), $MachinePrecision] + -6.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(N[(N[(N[(-1259.1392167224028 / N[(N[(1.0 - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(N[(1.0 - z), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] - -4.0), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] - -5.0), $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], N[(t$95$2 * N[(N[(t$95$6 * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision] * N[(t$95$7 + N[(263.383186962231 + N[(z * N[(436.896172553987 + N[(z * 545.0353078425886), $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)))))) < 5e302

   1. Initial program 97.3%

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

   3. Simplified 99.0%

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

      1. associate-*l/ 99.2%

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

   6. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{\pi \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right)}{\sin \left(z \cdot \pi\right)}} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   if 5e302 < (*.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 1.2%

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

   3. Taylor expanded in z around 0 1.2%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(263.383186962231 + z \cdot \left(436.896172553987 + 545.0353078425886 \cdot z\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative 1.2%

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + \color{blue}{z \cdot 545.0353078425886}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. Simplified 1.2%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. Taylor expanded in z around 0 84.5%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot \color{blue}{e^{-7.5}}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\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)} \cdot \sqrt{\pi \cdot 2}\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^{+302}:\\ \;\;\;\;\frac{\pi \cdot \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(z + -1\right) + -6.5}\right)\right)}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\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)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\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)} \cdot \sqrt{\pi \cdot 2}\right) \cdot e^{-7.5}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) + -1\right) + 8} + \left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) + -1\\ t_1 := \sqrt{\pi \cdot 2}\\ t_2 := \sin \left(\pi \cdot z\right)\\ \mathbf{if}\;z \leq -1000:\\ \;\;\;\;\frac{\pi}{t\_2} \cdot \left(\left(\left({\left(\left(t\_0 + 7\right) + 0.5\right)}^{\left(t\_0 + 0.5\right)} \cdot t\_1\right) \cdot e^{-7.5}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8} + \left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\left(t\_1 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(z + -1\right) + -6.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} - \left(\frac{-0.13857109526572012}{z - 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{-1 + \left(z - 7\right)}\right)\right)\right) - \left(\left(\frac{771.3234287776531}{z - 3} + \left(\frac{-176.6150291621406}{z - 4} + \frac{-1259.1392167224028}{z - 2}\right)\right) - 0.9999999999998099\right)\right)\right)}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ (- 1.0 z) -1.0)) (t_1 (sqrt (* PI 2.0))) (t_2 (sin (* PI z))))
   (if (<= z -1000.0)
     (*
      (/ PI t_2)
      (*
       (* (* (pow (+ (+ t_0 7.0) 0.5) (+ t_0 0.5)) t_1) (exp -7.5))
       (+
        (/ 1.5056327351493116e-7 (+ t_0 8.0))
        (+
         263.383186962231
         (* z (+ 436.896172553987 (* z 545.0353078425886)))))))
     (*
      PI
      (/
       (*
        (* t_1 (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ (+ z -1.0) -6.5))))
        (+
         (/ 676.5203681218851 (- 1.0 z))
         (-
          (+
           (/ 12.507343278686905 (- 5.0 z))
           (-
            (/ 9.984369578019572e-6 (- 7.0 z))
            (+
             (/ -0.13857109526572012 (- z 6.0))
             (/ 1.5056327351493116e-7 (+ -1.0 (- z 7.0))))))
          (-
           (+
            (/ 771.3234287776531 (- z 3.0))
            (+
             (/ -176.6150291621406 (- z 4.0))
             (/ -1259.1392167224028 (- z 2.0))))
           0.9999999999998099))))
       t_2)))))

C

double code(double z) {
	double t_0 = (1.0 - z) + -1.0;
	double t_1 = sqrt((((double) M_PI) * 2.0));
	double t_2 = sin((((double) M_PI) * z));
	double tmp;
	if (z <= -1000.0) {
		tmp = (((double) M_PI) / t_2) * (((pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5)) * t_1) * exp(-7.5)) * ((1.5056327351493116e-7 / (t_0 + 8.0)) + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))));
	} else {
		tmp = ((double) M_PI) * (((t_1 * (pow((7.5 - z), (0.5 - z)) * exp(((z + -1.0) + -6.5)))) * ((676.5203681218851 / (1.0 - z)) + (((12.507343278686905 / (5.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) - ((-0.13857109526572012 / (z - 6.0)) + (1.5056327351493116e-7 / (-1.0 + (z - 7.0)))))) - (((771.3234287776531 / (z - 3.0)) + ((-176.6150291621406 / (z - 4.0)) + (-1259.1392167224028 / (z - 2.0)))) - 0.9999999999998099)))) / t_2);
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) + -1.0;
	double t_1 = Math.sqrt((Math.PI * 2.0));
	double t_2 = Math.sin((Math.PI * z));
	double tmp;
	if (z <= -1000.0) {
		tmp = (Math.PI / t_2) * (((Math.pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5)) * t_1) * Math.exp(-7.5)) * ((1.5056327351493116e-7 / (t_0 + 8.0)) + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))));
	} else {
		tmp = Math.PI * (((t_1 * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp(((z + -1.0) + -6.5)))) * ((676.5203681218851 / (1.0 - z)) + (((12.507343278686905 / (5.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) - ((-0.13857109526572012 / (z - 6.0)) + (1.5056327351493116e-7 / (-1.0 + (z - 7.0)))))) - (((771.3234287776531 / (z - 3.0)) + ((-176.6150291621406 / (z - 4.0)) + (-1259.1392167224028 / (z - 2.0)))) - 0.9999999999998099)))) / t_2);
	}
	return tmp;
}

Python

def code(z):
	t_0 = (1.0 - z) + -1.0
	t_1 = math.sqrt((math.pi * 2.0))
	t_2 = math.sin((math.pi * z))
	tmp = 0
	if z <= -1000.0:
		tmp = (math.pi / t_2) * (((math.pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5)) * t_1) * math.exp(-7.5)) * ((1.5056327351493116e-7 / (t_0 + 8.0)) + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))))
	else:
		tmp = math.pi * (((t_1 * (math.pow((7.5 - z), (0.5 - z)) * math.exp(((z + -1.0) + -6.5)))) * ((676.5203681218851 / (1.0 - z)) + (((12.507343278686905 / (5.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) - ((-0.13857109526572012 / (z - 6.0)) + (1.5056327351493116e-7 / (-1.0 + (z - 7.0)))))) - (((771.3234287776531 / (z - 3.0)) + ((-176.6150291621406 / (z - 4.0)) + (-1259.1392167224028 / (z - 2.0)))) - 0.9999999999998099)))) / t_2)
	return tmp

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) + -1.0)
	t_1 = sqrt(Float64(pi * 2.0))
	t_2 = sin(Float64(pi * z))
	tmp = 0.0
	if (z <= -1000.0)
		tmp = Float64(Float64(pi / t_2) * Float64(Float64(Float64((Float64(Float64(t_0 + 7.0) + 0.5) ^ Float64(t_0 + 0.5)) * t_1) * exp(-7.5)) * Float64(Float64(1.5056327351493116e-7 / Float64(t_0 + 8.0)) + Float64(263.383186962231 + Float64(z * Float64(436.896172553987 + Float64(z * 545.0353078425886)))))));
	else
		tmp = Float64(pi * Float64(Float64(Float64(t_1 * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(Float64(z + -1.0) + -6.5)))) * Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) - Float64(Float64(-0.13857109526572012 / Float64(z - 6.0)) + Float64(1.5056327351493116e-7 / Float64(-1.0 + Float64(z - 7.0)))))) - Float64(Float64(Float64(771.3234287776531 / Float64(z - 3.0)) + Float64(Float64(-176.6150291621406 / Float64(z - 4.0)) + Float64(-1259.1392167224028 / Float64(z - 2.0)))) - 0.9999999999998099)))) / t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = (1.0 - z) + -1.0;
	t_1 = sqrt((pi * 2.0));
	t_2 = sin((pi * z));
	tmp = 0.0;
	if (z <= -1000.0)
		tmp = (pi / t_2) * ((((((t_0 + 7.0) + 0.5) ^ (t_0 + 0.5)) * t_1) * exp(-7.5)) * ((1.5056327351493116e-7 / (t_0 + 8.0)) + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))));
	else
		tmp = pi * (((t_1 * (((7.5 - z) ^ (0.5 - z)) * exp(((z + -1.0) + -6.5)))) * ((676.5203681218851 / (1.0 - z)) + (((12.507343278686905 / (5.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) - ((-0.13857109526572012 / (z - 6.0)) + (1.5056327351493116e-7 / (-1.0 + (z - 7.0)))))) - (((771.3234287776531 / (z - 3.0)) + ((-176.6150291621406 / (z - 4.0)) + (-1259.1392167224028 / (z - 2.0)))) - 0.9999999999998099)))) / t_2);
	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[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1000.0], N[(N[(Pi / t$95$2), $MachinePrecision] * N[(N[(N[(N[Power[N[(N[(t$95$0 + 7.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision] * N[(N[(1.5056327351493116e-7 / N[(t$95$0 + 8.0), $MachinePrecision]), $MachinePrecision] + N[(263.383186962231 + N[(z * N[(436.896172553987 + N[(z * 545.0353078425886), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(N[(N[(t$95$1 * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(z + -1.0), $MachinePrecision] + -6.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.13857109526572012 / N[(z - 6.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(-1.0 + N[(z - 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(771.3234287776531 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-176.6150291621406 / N[(z - 4.0), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(z - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.9999999999998099), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -1e3

   1. Initial program 0.0%

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

   3. Taylor expanded in z around 0 0.0%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(263.383186962231 + z \cdot \left(436.896172553987 + 545.0353078425886 \cdot z\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + \color{blue}{z \cdot 545.0353078425886}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. Simplified 0.0%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. Taylor expanded in z around 0 100.0%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot \color{blue}{e^{-7.5}}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   if -1e3 < z

   1. Initial program 97.0%

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

   3. Simplified 97.1%

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

   6. Applied egg-rr 98.4%

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

   8. Simplified 98.8%

      \[\leadsto \color{blue}{\pi \cdot \frac{\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{1 + \left(7 - z\right)} + \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\right)\right)}{\sin \left(z \cdot \pi\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1000:\\ \;\;\;\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\left(\left(\left(\left(1 - z\right) + -1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) + -1\right) + 0.5\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot e^{-7.5}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) + -1\right) + 8} + \left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(z + -1\right) + -6.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} - \left(\frac{-0.13857109526572012}{z - 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{-1 + \left(z - 7\right)}\right)\right)\right) - \left(\left(\frac{771.3234287776531}{z - 3} + \left(\frac{-176.6150291621406}{z - 4} + \frac{-1259.1392167224028}{z - 2}\right)\right) - 0.9999999999998099\right)\right)\right)}{\sin \left(\pi \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) + -1\\ t_1 := \frac{\pi}{\sin \left(\pi \cdot z\right)}\\ t_2 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq -1000:\\ \;\;\;\;t\_1 \cdot \left(\left(\left({\left(\left(t\_0 + 7\right) + 0.5\right)}^{\left(t\_0 + 0.5\right)} \cdot t\_2\right) \cdot e^{-7.5}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8} + \left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot t\_2\right) \cdot \left(t\_1 \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) - \left(771.3234287776531 \cdot \frac{1}{z - 3} - 0.9999999999998099\right)\right)\right) - \left(\frac{-0.13857109526572012}{z - 6} + \left(\frac{-176.6150291621406}{z - 4} + \frac{12.507343278686905}{z - 5}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ (- 1.0 z) -1.0))
        (t_1 (/ PI (sin (* PI z))))
        (t_2 (sqrt (* PI 2.0))))
   (if (<= z -1000.0)
     (*
      t_1
      (*
       (* (* (pow (+ (+ t_0 7.0) 0.5) (+ t_0 0.5)) t_2) (exp -7.5))
       (+
        (/ 1.5056327351493116e-7 (+ t_0 8.0))
        (+
         263.383186962231
         (* z (+ 436.896172553987 (* z 545.0353078425886)))))))
     (*
      (* (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5))) t_2)
      (*
       t_1
       (-
        (+
         (+
          (/ 9.984369578019572e-6 (- 7.0 z))
          (/ 1.5056327351493116e-7 (- 8.0 z)))
         (-
          (+ (/ 676.5203681218851 (- 1.0 z)) (/ -1259.1392167224028 (- 2.0 z)))
          (- (* 771.3234287776531 (/ 1.0 (- z 3.0))) 0.9999999999998099)))
        (+
         (/ -0.13857109526572012 (- z 6.0))
         (+
          (/ -176.6150291621406 (- z 4.0))
          (/ 12.507343278686905 (- z 5.0))))))))))

C

double code(double z) {
	double t_0 = (1.0 - z) + -1.0;
	double t_1 = ((double) M_PI) / sin((((double) M_PI) * z));
	double t_2 = sqrt((((double) M_PI) * 2.0));
	double tmp;
	if (z <= -1000.0) {
		tmp = t_1 * (((pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5)) * t_2) * exp(-7.5)) * ((1.5056327351493116e-7 / (t_0 + 8.0)) + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))));
	} else {
		tmp = ((pow((7.5 - z), (0.5 - z)) * exp((z + -7.5))) * t_2) * (t_1 * ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z))) - ((771.3234287776531 * (1.0 / (z - 3.0))) - 0.9999999999998099))) - ((-0.13857109526572012 / (z - 6.0)) + ((-176.6150291621406 / (z - 4.0)) + (12.507343278686905 / (z - 5.0))))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) + -1.0;
	double t_1 = Math.PI / Math.sin((Math.PI * z));
	double t_2 = Math.sqrt((Math.PI * 2.0));
	double tmp;
	if (z <= -1000.0) {
		tmp = t_1 * (((Math.pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5)) * t_2) * Math.exp(-7.5)) * ((1.5056327351493116e-7 / (t_0 + 8.0)) + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))));
	} else {
		tmp = ((Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5))) * t_2) * (t_1 * ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z))) - ((771.3234287776531 * (1.0 / (z - 3.0))) - 0.9999999999998099))) - ((-0.13857109526572012 / (z - 6.0)) + ((-176.6150291621406 / (z - 4.0)) + (12.507343278686905 / (z - 5.0))))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = (1.0 - z) + -1.0
	t_1 = math.pi / math.sin((math.pi * z))
	t_2 = math.sqrt((math.pi * 2.0))
	tmp = 0
	if z <= -1000.0:
		tmp = t_1 * (((math.pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5)) * t_2) * math.exp(-7.5)) * ((1.5056327351493116e-7 / (t_0 + 8.0)) + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))))
	else:
		tmp = ((math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5))) * t_2) * (t_1 * ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z))) - ((771.3234287776531 * (1.0 / (z - 3.0))) - 0.9999999999998099))) - ((-0.13857109526572012 / (z - 6.0)) + ((-176.6150291621406 / (z - 4.0)) + (12.507343278686905 / (z - 5.0))))))
	return tmp

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) + -1.0)
	t_1 = Float64(pi / sin(Float64(pi * z)))
	t_2 = sqrt(Float64(pi * 2.0))
	tmp = 0.0
	if (z <= -1000.0)
		tmp = Float64(t_1 * Float64(Float64(Float64((Float64(Float64(t_0 + 7.0) + 0.5) ^ Float64(t_0 + 0.5)) * t_2) * exp(-7.5)) * Float64(Float64(1.5056327351493116e-7 / Float64(t_0 + 8.0)) + Float64(263.383186962231 + Float64(z * Float64(436.896172553987 + Float64(z * 545.0353078425886)))))));
	else
		tmp = Float64(Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5))) * t_2) * Float64(t_1 * Float64(Float64(Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))) + Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(-1259.1392167224028 / Float64(2.0 - z))) - Float64(Float64(771.3234287776531 * Float64(1.0 / Float64(z - 3.0))) - 0.9999999999998099))) - Float64(Float64(-0.13857109526572012 / Float64(z - 6.0)) + Float64(Float64(-176.6150291621406 / Float64(z - 4.0)) + Float64(12.507343278686905 / Float64(z - 5.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = (1.0 - z) + -1.0;
	t_1 = pi / sin((pi * z));
	t_2 = sqrt((pi * 2.0));
	tmp = 0.0;
	if (z <= -1000.0)
		tmp = t_1 * ((((((t_0 + 7.0) + 0.5) ^ (t_0 + 0.5)) * t_2) * exp(-7.5)) * ((1.5056327351493116e-7 / (t_0 + 8.0)) + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))));
	else
		tmp = ((((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5))) * t_2) * (t_1 * ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z))) - ((771.3234287776531 * (1.0 / (z - 3.0))) - 0.9999999999998099))) - ((-0.13857109526572012 / (z - 6.0)) + ((-176.6150291621406 / (z - 4.0)) + (12.507343278686905 / (z - 5.0))))));
	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[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1000.0], N[(t$95$1 * N[(N[(N[(N[Power[N[(N[(t$95$0 + 7.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision] * N[(N[(1.5056327351493116e-7 / N[(t$95$0 + 8.0), $MachinePrecision]), $MachinePrecision] + N[(263.383186962231 + N[(z * N[(436.896172553987 + N[(z * 545.0353078425886), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$1 * N[(N[(N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(771.3234287776531 * N[(1.0 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.9999999999998099), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.13857109526572012 / N[(z - 6.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-176.6150291621406 / N[(z - 4.0), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(z - 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -1e3

   1. Initial program 0.0%

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

   3. Taylor expanded in z around 0 0.0%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(263.383186962231 + z \cdot \left(436.896172553987 + 545.0353078425886 \cdot z\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + \color{blue}{z \cdot 545.0353078425886}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. Simplified 0.0%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. Taylor expanded in z around 0 100.0%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot \color{blue}{e^{-7.5}}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   if -1e3 < z

   1. Initial program 97.0%

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

   3. Simplified 97.2%

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

      1. div-inv 98.6%

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

   6. Applied egg-rr 98.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1000:\\ \;\;\;\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\left(\left(\left(\left(1 - z\right) + -1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) + -1\right) + 0.5\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot e^{-7.5}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) + -1\right) + 8} + \left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) - \left(771.3234287776531 \cdot \frac{1}{z - 3} - 0.9999999999998099\right)\right)\right) - \left(\frac{-0.13857109526572012}{z - 6} + \left(\frac{-176.6150291621406}{z - 4} + \frac{12.507343278686905}{z - 5}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) + -1\\ t_1 := \sqrt{\pi \cdot 2}\\ t_2 := \sin \left(\pi \cdot z\right)\\ \mathbf{if}\;z \leq -1000:\\ \;\;\;\;\frac{\pi}{t\_2} \cdot \left(\left(\left({\left(\left(t\_0 + 7\right) + 0.5\right)}^{\left(t\_0 + 0.5\right)} \cdot t\_1\right) \cdot e^{-7.5}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8} + \left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \frac{\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot t\_1}{t\_2}\right) \cdot \left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} - \left(\frac{771.3234287776531}{z - 3} + \left(\frac{676.5203681218851}{z + -1} - \left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} - \left(\frac{12.507343278686905}{z - 5} + \left(\frac{-176.6150291621406}{z - 4} - \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ (- 1.0 z) -1.0)) (t_1 (sqrt (* PI 2.0))) (t_2 (sin (* PI z))))
   (if (<= z -1000.0)
     (*
      (/ PI t_2)
      (*
       (* (* (pow (+ (+ t_0 7.0) 0.5) (+ t_0 0.5)) t_1) (exp -7.5))
       (+
        (/ 1.5056327351493116e-7 (+ t_0 8.0))
        (+
         263.383186962231
         (* z (+ 436.896172553987 (* z 545.0353078425886)))))))
     (*
      (* PI (/ (* (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5))) t_1) t_2))
      (+
       (-
        (/ 1.5056327351493116e-7 (- 8.0 z))
        (+
         (/ 771.3234287776531 (- z 3.0))
         (-
          (/ 676.5203681218851 (+ z -1.0))
          (+ 0.9999999999998099 (/ -1259.1392167224028 (- 2.0 z))))))
       (-
        (/ 9.984369578019572e-6 (- 7.0 z))
        (+
         (/ 12.507343278686905 (- z 5.0))
         (-
          (/ -176.6150291621406 (- z 4.0))
          (/ -0.13857109526572012 (- 6.0 z))))))))))

C

double code(double z) {
	double t_0 = (1.0 - z) + -1.0;
	double t_1 = sqrt((((double) M_PI) * 2.0));
	double t_2 = sin((((double) M_PI) * z));
	double tmp;
	if (z <= -1000.0) {
		tmp = (((double) M_PI) / t_2) * (((pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5)) * t_1) * exp(-7.5)) * ((1.5056327351493116e-7 / (t_0 + 8.0)) + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))));
	} else {
		tmp = (((double) M_PI) * (((pow((7.5 - z), (0.5 - z)) * exp((z + -7.5))) * t_1) / t_2)) * (((1.5056327351493116e-7 / (8.0 - z)) - ((771.3234287776531 / (z - 3.0)) + ((676.5203681218851 / (z + -1.0)) - (0.9999999999998099 + (-1259.1392167224028 / (2.0 - z)))))) + ((9.984369578019572e-6 / (7.0 - z)) - ((12.507343278686905 / (z - 5.0)) + ((-176.6150291621406 / (z - 4.0)) - (-0.13857109526572012 / (6.0 - z))))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) + -1.0;
	double t_1 = Math.sqrt((Math.PI * 2.0));
	double t_2 = Math.sin((Math.PI * z));
	double tmp;
	if (z <= -1000.0) {
		tmp = (Math.PI / t_2) * (((Math.pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5)) * t_1) * Math.exp(-7.5)) * ((1.5056327351493116e-7 / (t_0 + 8.0)) + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))));
	} else {
		tmp = (Math.PI * (((Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5))) * t_1) / t_2)) * (((1.5056327351493116e-7 / (8.0 - z)) - ((771.3234287776531 / (z - 3.0)) + ((676.5203681218851 / (z + -1.0)) - (0.9999999999998099 + (-1259.1392167224028 / (2.0 - z)))))) + ((9.984369578019572e-6 / (7.0 - z)) - ((12.507343278686905 / (z - 5.0)) + ((-176.6150291621406 / (z - 4.0)) - (-0.13857109526572012 / (6.0 - z))))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = (1.0 - z) + -1.0
	t_1 = math.sqrt((math.pi * 2.0))
	t_2 = math.sin((math.pi * z))
	tmp = 0
	if z <= -1000.0:
		tmp = (math.pi / t_2) * (((math.pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5)) * t_1) * math.exp(-7.5)) * ((1.5056327351493116e-7 / (t_0 + 8.0)) + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))))
	else:
		tmp = (math.pi * (((math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5))) * t_1) / t_2)) * (((1.5056327351493116e-7 / (8.0 - z)) - ((771.3234287776531 / (z - 3.0)) + ((676.5203681218851 / (z + -1.0)) - (0.9999999999998099 + (-1259.1392167224028 / (2.0 - z)))))) + ((9.984369578019572e-6 / (7.0 - z)) - ((12.507343278686905 / (z - 5.0)) + ((-176.6150291621406 / (z - 4.0)) - (-0.13857109526572012 / (6.0 - z))))))
	return tmp

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) + -1.0)
	t_1 = sqrt(Float64(pi * 2.0))
	t_2 = sin(Float64(pi * z))
	tmp = 0.0
	if (z <= -1000.0)
		tmp = Float64(Float64(pi / t_2) * Float64(Float64(Float64((Float64(Float64(t_0 + 7.0) + 0.5) ^ Float64(t_0 + 0.5)) * t_1) * exp(-7.5)) * Float64(Float64(1.5056327351493116e-7 / Float64(t_0 + 8.0)) + Float64(263.383186962231 + Float64(z * Float64(436.896172553987 + Float64(z * 545.0353078425886)))))));
	else
		tmp = Float64(Float64(pi * Float64(Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5))) * t_1) / t_2)) * Float64(Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) - Float64(Float64(771.3234287776531 / Float64(z - 3.0)) + Float64(Float64(676.5203681218851 / Float64(z + -1.0)) - Float64(0.9999999999998099 + Float64(-1259.1392167224028 / Float64(2.0 - z)))))) + Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) - Float64(Float64(12.507343278686905 / Float64(z - 5.0)) + Float64(Float64(-176.6150291621406 / Float64(z - 4.0)) - Float64(-0.13857109526572012 / Float64(6.0 - z)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = (1.0 - z) + -1.0;
	t_1 = sqrt((pi * 2.0));
	t_2 = sin((pi * z));
	tmp = 0.0;
	if (z <= -1000.0)
		tmp = (pi / t_2) * ((((((t_0 + 7.0) + 0.5) ^ (t_0 + 0.5)) * t_1) * exp(-7.5)) * ((1.5056327351493116e-7 / (t_0 + 8.0)) + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))));
	else
		tmp = (pi * (((((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5))) * t_1) / t_2)) * (((1.5056327351493116e-7 / (8.0 - z)) - ((771.3234287776531 / (z - 3.0)) + ((676.5203681218851 / (z + -1.0)) - (0.9999999999998099 + (-1259.1392167224028 / (2.0 - z)))))) + ((9.984369578019572e-6 / (7.0 - z)) - ((12.507343278686905 / (z - 5.0)) + ((-176.6150291621406 / (z - 4.0)) - (-0.13857109526572012 / (6.0 - z))))));
	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[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1000.0], N[(N[(Pi / t$95$2), $MachinePrecision] * N[(N[(N[(N[Power[N[(N[(t$95$0 + 7.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision] * N[(N[(1.5056327351493116e-7 / N[(t$95$0 + 8.0), $MachinePrecision]), $MachinePrecision] + N[(263.383186962231 + N[(z * N[(436.896172553987 + N[(z * 545.0353078425886), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * N[(N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] - N[(N[(771.3234287776531 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(676.5203681218851 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision] - N[(0.9999999999998099 + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] - N[(N[(12.507343278686905 / N[(z - 5.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-176.6150291621406 / N[(z - 4.0), $MachinePrecision]), $MachinePrecision] - N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -1e3

   1. Initial program 0.0%

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

   3. Taylor expanded in z around 0 0.0%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(263.383186962231 + z \cdot \left(436.896172553987 + 545.0353078425886 \cdot z\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + \color{blue}{z \cdot 545.0353078425886}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. Simplified 0.0%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. Taylor expanded in z around 0 100.0%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot \color{blue}{e^{-7.5}}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   if -1e3 < z

   1. Initial program 97.0%

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

   3. Simplified 97.2%

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

      1. add-exp-log 98.9%

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

      2. *-commutative 98.9%

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

      3. *-commutative 98.9%

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

   6. Applied egg-rr 98.9%

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

      1. pow1 98.9%

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

   8. Applied egg-rr 98.6%

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

   10. Simplified 98.4%

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

       1. associate-*r/ 98.2%

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

   12. Applied egg-rr 98.2%

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

       1. *-commutative 98.2%

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

       2. associate-/l* 98.5%

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

       3. *-commutative 98.5%

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

       4. *-commutative 98.5%

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

   14. Simplified 98.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1000:\\ \;\;\;\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\left(\left(\left(\left(1 - z\right) + -1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) + -1\right) + 0.5\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot e^{-7.5}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) + -1\right) + 8} + \left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \frac{\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} - \left(\frac{771.3234287776531}{z - 3} + \left(\frac{676.5203681218851}{z + -1} - \left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} - \left(\frac{12.507343278686905}{z - 5} + \left(\frac{-176.6150291621406}{z - 4} - \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) + -1\\ t_1 := \sqrt{\pi \cdot 2}\\ t_2 := \sin \left(\pi \cdot z\right)\\ \mathbf{if}\;z \leq -1000:\\ \;\;\;\;\frac{\pi}{t\_2} \cdot \left(\left(\left({\left(\left(t\_0 + 7\right) + 0.5\right)}^{\left(t\_0 + 0.5\right)} \cdot t\_1\right) \cdot e^{-7.5}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8} + \left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \left(t\_1 \cdot \frac{{\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}}{t\_2}\right)\right) \cdot \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\left(\frac{-1259.1392167224028}{2 - z} - \left(\frac{-176.6150291621406}{z - 4} + \frac{771.3234287776531}{z - 3}\right)\right) + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ (- 1.0 z) -1.0)) (t_1 (sqrt (* PI 2.0))) (t_2 (sin (* PI z))))
   (if (<= z -1000.0)
     (*
      (/ PI t_2)
      (*
       (* (* (pow (+ (+ t_0 7.0) 0.5) (+ t_0 0.5)) t_1) (exp -7.5))
       (+
        (/ 1.5056327351493116e-7 (+ t_0 8.0))
        (+
         263.383186962231
         (* z (+ 436.896172553987 (* z 545.0353078425886)))))))
     (*
      (* PI (* t_1 (/ (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5))) t_2)))
      (+
       (/ 9.984369578019572e-6 (- 7.0 z))
       (+
        (/ 1.5056327351493116e-7 (- 8.0 z))
        (+
         (-
          (/ -1259.1392167224028 (- 2.0 z))
          (+ (/ -176.6150291621406 (- z 4.0)) (/ 771.3234287776531 (- z 3.0))))
         (+
          (+ 0.9999999999998099 (/ 676.5203681218851 (- 1.0 z)))
          (+
           (/ -0.13857109526572012 (- 6.0 z))
           (/ 12.507343278686905 (- 5.0 z)))))))))))

C

double code(double z) {
	double t_0 = (1.0 - z) + -1.0;
	double t_1 = sqrt((((double) M_PI) * 2.0));
	double t_2 = sin((((double) M_PI) * z));
	double tmp;
	if (z <= -1000.0) {
		tmp = (((double) M_PI) / t_2) * (((pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5)) * t_1) * exp(-7.5)) * ((1.5056327351493116e-7 / (t_0 + 8.0)) + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))));
	} else {
		tmp = (((double) M_PI) * (t_1 * ((pow((7.5 - z), (0.5 - z)) * exp((z + -7.5))) / t_2))) * ((9.984369578019572e-6 / (7.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + (((-1259.1392167224028 / (2.0 - z)) - ((-176.6150291621406 / (z - 4.0)) + (771.3234287776531 / (z - 3.0)))) + ((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + ((-0.13857109526572012 / (6.0 - z)) + (12.507343278686905 / (5.0 - z)))))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) + -1.0;
	double t_1 = Math.sqrt((Math.PI * 2.0));
	double t_2 = Math.sin((Math.PI * z));
	double tmp;
	if (z <= -1000.0) {
		tmp = (Math.PI / t_2) * (((Math.pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5)) * t_1) * Math.exp(-7.5)) * ((1.5056327351493116e-7 / (t_0 + 8.0)) + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))));
	} else {
		tmp = (Math.PI * (t_1 * ((Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5))) / t_2))) * ((9.984369578019572e-6 / (7.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + (((-1259.1392167224028 / (2.0 - z)) - ((-176.6150291621406 / (z - 4.0)) + (771.3234287776531 / (z - 3.0)))) + ((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + ((-0.13857109526572012 / (6.0 - z)) + (12.507343278686905 / (5.0 - z)))))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = (1.0 - z) + -1.0
	t_1 = math.sqrt((math.pi * 2.0))
	t_2 = math.sin((math.pi * z))
	tmp = 0
	if z <= -1000.0:
		tmp = (math.pi / t_2) * (((math.pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5)) * t_1) * math.exp(-7.5)) * ((1.5056327351493116e-7 / (t_0 + 8.0)) + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))))
	else:
		tmp = (math.pi * (t_1 * ((math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5))) / t_2))) * ((9.984369578019572e-6 / (7.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + (((-1259.1392167224028 / (2.0 - z)) - ((-176.6150291621406 / (z - 4.0)) + (771.3234287776531 / (z - 3.0)))) + ((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + ((-0.13857109526572012 / (6.0 - z)) + (12.507343278686905 / (5.0 - z)))))))
	return tmp

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) + -1.0)
	t_1 = sqrt(Float64(pi * 2.0))
	t_2 = sin(Float64(pi * z))
	tmp = 0.0
	if (z <= -1000.0)
		tmp = Float64(Float64(pi / t_2) * Float64(Float64(Float64((Float64(Float64(t_0 + 7.0) + 0.5) ^ Float64(t_0 + 0.5)) * t_1) * exp(-7.5)) * Float64(Float64(1.5056327351493116e-7 / Float64(t_0 + 8.0)) + Float64(263.383186962231 + Float64(z * Float64(436.896172553987 + Float64(z * 545.0353078425886)))))));
	else
		tmp = Float64(Float64(pi * Float64(t_1 * Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5))) / t_2))) * Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) + Float64(Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) - Float64(Float64(-176.6150291621406 / Float64(z - 4.0)) + Float64(771.3234287776531 / Float64(z - 3.0)))) + Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 - z))) + Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(12.507343278686905 / Float64(5.0 - z))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = (1.0 - z) + -1.0;
	t_1 = sqrt((pi * 2.0));
	t_2 = sin((pi * z));
	tmp = 0.0;
	if (z <= -1000.0)
		tmp = (pi / t_2) * ((((((t_0 + 7.0) + 0.5) ^ (t_0 + 0.5)) * t_1) * exp(-7.5)) * ((1.5056327351493116e-7 / (t_0 + 8.0)) + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))));
	else
		tmp = (pi * (t_1 * ((((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5))) / t_2))) * ((9.984369578019572e-6 / (7.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + (((-1259.1392167224028 / (2.0 - z)) - ((-176.6150291621406 / (z - 4.0)) + (771.3234287776531 / (z - 3.0)))) + ((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + ((-0.13857109526572012 / (6.0 - z)) + (12.507343278686905 / (5.0 - z)))))));
	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[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1000.0], N[(N[(Pi / t$95$2), $MachinePrecision] * N[(N[(N[(N[Power[N[(N[(t$95$0 + 7.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision] * N[(N[(1.5056327351493116e-7 / N[(t$95$0 + 8.0), $MachinePrecision]), $MachinePrecision] + N[(263.383186962231 + N[(z * N[(436.896172553987 + N[(z * 545.0353078425886), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * N[(t$95$1 * N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] - N[(N[(-176.6150291621406 / N[(z - 4.0), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -1e3

   1. Initial program 0.0%

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

   3. Taylor expanded in z around 0 0.0%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(263.383186962231 + z \cdot \left(436.896172553987 + 545.0353078425886 \cdot z\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + \color{blue}{z \cdot 545.0353078425886}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. Simplified 0.0%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. Taylor expanded in z around 0 100.0%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot \color{blue}{e^{-7.5}}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   if -1e3 < z

   1. Initial program 97.0%

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

   3. Simplified 98.6%

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

      1. associate-*l/ 98.9%

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

   6. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\frac{\pi \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right)}{\sin \left(z \cdot \pi\right)}} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   8. Applied egg-rr 97.7%

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

   10. Simplified 98.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1000:\\ \;\;\;\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\left(\left(\left(\left(1 - z\right) + -1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) + -1\right) + 0.5\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot e^{-7.5}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) + -1\right) + 8} + \left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot \frac{{\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}}{\sin \left(\pi \cdot z\right)}\right)\right) \cdot \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\left(\frac{-1259.1392167224028}{2 - z} - \left(\frac{-176.6150291621406}{z - 4} + \frac{771.3234287776531}{z - 3}\right)\right) + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) + -1\\ t_1 := \sqrt{\pi \cdot 2}\\ t_2 := \sin \left(\pi \cdot z\right)\\ \mathbf{if}\;z \leq -0.7:\\ \;\;\;\;\frac{\pi}{t\_2} \cdot \left(\left(\left({\left(\left(t\_0 + 7\right) + 0.5\right)}^{\left(t\_0 + 0.5\right)} \cdot t\_1\right) \cdot e^{-7.5}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8} + \left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \left(t\_1 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(z + -1\right) + -6.5}\right)\right)}{t\_2} \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) + \left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ (- 1.0 z) -1.0)) (t_1 (sqrt (* PI 2.0))) (t_2 (sin (* PI z))))
   (if (<= z -0.7)
     (*
      (/ PI t_2)
      (*
       (* (* (pow (+ (+ t_0 7.0) 0.5) (+ t_0 0.5)) t_1) (exp -7.5))
       (+
        (/ 1.5056327351493116e-7 (+ t_0 8.0))
        (+
         263.383186962231
         (* z (+ 436.896172553987 (* z 545.0353078425886)))))))
     (*
      (/
       (*
        PI
        (*
         t_1
         (*
          (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5))
          (exp (+ (+ z -1.0) -6.5)))))
       t_2)
      (+
       (+
        (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))
        (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0)))
       (+
        263.3831855358925
        (*
         z
         (+
          436.8961723502244
          (* z (+ 545.0353078134797 (* z 606.6766809125655)))))))))))

C

double code(double z) {
	double t_0 = (1.0 - z) + -1.0;
	double t_1 = sqrt((((double) M_PI) * 2.0));
	double t_2 = sin((((double) M_PI) * z));
	double tmp;
	if (z <= -0.7) {
		tmp = (((double) M_PI) / t_2) * (((pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5)) * t_1) * exp(-7.5)) * ((1.5056327351493116e-7 / (t_0 + 8.0)) + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))));
	} else {
		tmp = ((((double) M_PI) * (t_1 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(((z + -1.0) + -6.5))))) / t_2) * (((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))) + (263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) + -1.0;
	double t_1 = Math.sqrt((Math.PI * 2.0));
	double t_2 = Math.sin((Math.PI * z));
	double tmp;
	if (z <= -0.7) {
		tmp = (Math.PI / t_2) * (((Math.pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5)) * t_1) * Math.exp(-7.5)) * ((1.5056327351493116e-7 / (t_0 + 8.0)) + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))));
	} else {
		tmp = ((Math.PI * (t_1 * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(((z + -1.0) + -6.5))))) / t_2) * (((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))) + (263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = (1.0 - z) + -1.0
	t_1 = math.sqrt((math.pi * 2.0))
	t_2 = math.sin((math.pi * z))
	tmp = 0
	if z <= -0.7:
		tmp = (math.pi / t_2) * (((math.pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5)) * t_1) * math.exp(-7.5)) * ((1.5056327351493116e-7 / (t_0 + 8.0)) + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))))
	else:
		tmp = ((math.pi * (t_1 * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(((z + -1.0) + -6.5))))) / t_2) * (((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))) + (263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))))
	return tmp

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) + -1.0)
	t_1 = sqrt(Float64(pi * 2.0))
	t_2 = sin(Float64(pi * z))
	tmp = 0.0
	if (z <= -0.7)
		tmp = Float64(Float64(pi / t_2) * Float64(Float64(Float64((Float64(Float64(t_0 + 7.0) + 0.5) ^ Float64(t_0 + 0.5)) * t_1) * exp(-7.5)) * Float64(Float64(1.5056327351493116e-7 / Float64(t_0 + 8.0)) + Float64(263.383186962231 + Float64(z * Float64(436.896172553987 + Float64(z * 545.0353078425886)))))));
	else
		tmp = Float64(Float64(Float64(pi * Float64(t_1 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(Float64(Float64(z + -1.0) + -6.5))))) / t_2) * Float64(Float64(Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0)) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0))) + Float64(263.3831855358925 + Float64(z * Float64(436.8961723502244 + Float64(z * Float64(545.0353078134797 + Float64(z * 606.6766809125655))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = (1.0 - z) + -1.0;
	t_1 = sqrt((pi * 2.0));
	t_2 = sin((pi * z));
	tmp = 0.0;
	if (z <= -0.7)
		tmp = (pi / t_2) * ((((((t_0 + 7.0) + 0.5) ^ (t_0 + 0.5)) * t_1) * exp(-7.5)) * ((1.5056327351493116e-7 / (t_0 + 8.0)) + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))));
	else
		tmp = ((pi * (t_1 * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(((z + -1.0) + -6.5))))) / t_2) * (((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))) + (263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))));
	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[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -0.7], N[(N[(Pi / t$95$2), $MachinePrecision] * N[(N[(N[(N[Power[N[(N[(t$95$0 + 7.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision] * N[(N[(1.5056327351493116e-7 / N[(t$95$0 + 8.0), $MachinePrecision]), $MachinePrecision] + N[(263.383186962231 + N[(z * N[(436.896172553987 + N[(z * 545.0353078425886), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * N[(t$95$1 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(z + -1.0), $MachinePrecision] + -6.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(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] + N[(263.3831855358925 + N[(z * N[(436.8961723502244 + N[(z * N[(545.0353078134797 + N[(z * 606.6766809125655), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.69999999999999996

   1. Initial program 31.9%

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

   3. Taylor expanded in z around 0 5.3%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(263.383186962231 + z \cdot \left(436.896172553987 + 545.0353078425886 \cdot z\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative 5.3%

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + \color{blue}{z \cdot 545.0353078425886}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. Simplified 5.3%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. Taylor expanded in z around 0 60.2%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot \color{blue}{e^{-7.5}}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   if -0.69999999999999996 < 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 99.0%

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

      1. associate-*l/ 99.3%

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

   6. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\frac{\pi \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right)}{\sin \left(z \cdot \pi\right)}} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   7. Taylor expanded in z around 0 98.9%

      \[\leadsto \frac{\pi \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right)}{\sin \left(z \cdot \pi\right)} \cdot \left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
   8. Step-by-step derivation

      1. *-commutative 98.9%

         \[\leadsto \frac{\pi \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right)}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + \color{blue}{z \cdot 606.6766809125655}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   9. Simplified 98.9%

      \[\leadsto \frac{\pi \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right)}{\sin \left(z \cdot \pi\right)} \cdot \left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.7:\\ \;\;\;\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\left(\left(\left(\left(1 - z\right) + -1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) + -1\right) + 0.5\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot e^{-7.5}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) + -1\right) + 8} + \left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(z + -1\right) + -6.5}\right)\right)}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) + \left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) + -1\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\left(\left(t\_0 + 7\right) + 0.5\right)}^{\left(t\_0 + 0.5\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot e^{-7.5}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8} + \left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ (- 1.0 z) -1.0)))
   (*
    (/ PI (sin (* PI z)))
    (*
     (* (* (pow (+ (+ t_0 7.0) 0.5) (+ t_0 0.5)) (sqrt (* PI 2.0))) (exp -7.5))
     (+
      (/ 1.5056327351493116e-7 (+ t_0 8.0))
      (+
       263.383186962231
       (* z (+ 436.896172553987 (* z 545.0353078425886)))))))))

C

double code(double z) {
	double t_0 = (1.0 - z) + -1.0;
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (((pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5)) * sqrt((((double) M_PI) * 2.0))) * exp(-7.5)) * ((1.5056327351493116e-7 / (t_0 + 8.0)) + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) + -1.0;
	return (Math.PI / Math.sin((Math.PI * z))) * (((Math.pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5)) * Math.sqrt((Math.PI * 2.0))) * Math.exp(-7.5)) * ((1.5056327351493116e-7 / (t_0 + 8.0)) + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))));
}

Python

def code(z):
	t_0 = (1.0 - z) + -1.0
	return (math.pi / math.sin((math.pi * z))) * (((math.pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5)) * math.sqrt((math.pi * 2.0))) * math.exp(-7.5)) * ((1.5056327351493116e-7 / (t_0 + 8.0)) + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) + -1.0)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64((Float64(Float64(t_0 + 7.0) + 0.5) ^ Float64(t_0 + 0.5)) * sqrt(Float64(pi * 2.0))) * exp(-7.5)) * Float64(Float64(1.5056327351493116e-7 / Float64(t_0 + 8.0)) + Float64(263.383186962231 + Float64(z * Float64(436.896172553987 + Float64(z * 545.0353078425886)))))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) + -1.0;
	tmp = (pi / sin((pi * z))) * ((((((t_0 + 7.0) + 0.5) ^ (t_0 + 0.5)) * sqrt((pi * 2.0))) * exp(-7.5)) * ((1.5056327351493116e-7 / (t_0 + 8.0)) + (263.383186962231 + (z * (436.896172553987 + (z * 545.0353078425886))))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] + -1.0), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(N[(t$95$0 + 7.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision] * N[(N[(1.5056327351493116e-7 / N[(t$95$0 + 8.0), $MachinePrecision]), $MachinePrecision] + N[(263.383186962231 + N[(z * N[(436.896172553987 + N[(z * 545.0353078425886), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 95.1%

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

3. Taylor expanded in z around 0 94.7%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(263.383186962231 + z \cdot \left(436.896172553987 + 545.0353078425886 \cdot z\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
4. Step-by-step derivation

   1. *-commutative 94.7%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + \color{blue}{z \cdot 545.0353078425886}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

5. Simplified 94.7%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Taylor expanded in z around 0 96.0%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot \color{blue}{e^{-7.5}}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

7. Final simplification 96.0%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\left(\left(\left(\left(1 - z\right) + -1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) + -1\right) + 0.5\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot e^{-7.5}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) + -1\right) + 8} + \left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot 545.0353078425886\right)\right)\right)\right) \]
8. Add Preprocessing

Alternative 10: 96.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.1%

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

3. Simplified 95.2%

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

5. Taylor expanded in z around 0 94.4%

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

6. Taylor expanded in z around 0 94.4%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\sqrt{7.5}} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \frac{263.3831869810514}{z} \]

7. Taylor expanded in z around inf 94.4%

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

   1. associate-*r* 95.1%

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

   2. sub-neg 95.1%

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

   3. remove-double-neg 95.1%

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

   4. distribute-neg-in 95.1%

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

   5. +-commutative 95.1%

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

   6. neg-mul-1 95.1%

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

   7. neg-mul-1 95.1%

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

   8. +-commutative 95.1%

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

   9. distribute-neg-in 95.1%

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

   10. remove-double-neg 95.1%

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

   11. metadata-eval 95.1%

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

   12. *-commutative 95.1%

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

9. Simplified 95.1%

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

10. Final simplification 95.1%

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

Alternative 11: 96.0% 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 96.7%

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

5. Taylor expanded in z around 0 94.4%

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

6. Taylor expanded in z around 0 94.5%

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

   1. *-un-lft-identity 94.5%

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

   2. associate-/l* 94.7%

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

   3. sqrt-unprod 94.7%

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

   4. metadata-eval 94.7%

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

8. Applied egg-rr 94.7%

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

   1. *-lft-identity 94.7%

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

10. Simplified 94.7%

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

11. Final simplification 94.7%

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

Alternative 12: 96.0% accurate, 1.7× speedup

Math

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

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(263.3831869810514 * Float64(exp(-7.5) * Float64(sqrt(pi) * Float64(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[(263.3831869810514 * N[(N[Exp[-7.5], $MachinePrecision] * N[(N[Sqrt[Pi], $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 96.7%

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

5. Taylor expanded in z around 0 94.4%

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

6. Taylor expanded in z around 0 94.5%

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

   1. associate-*l/ 94.4%

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

   2. sqrt-unprod 94.4%

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

   3. metadata-eval 94.4%

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

8. Applied egg-rr 94.4%

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

   1. associate-*l/ 94.5%

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

   2. associate-*r/ 94.7%

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

   3. associate-*l* 94.7%

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

10. Simplified 94.7%

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

11. Final simplification 94.7%

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

Reproduce

herbie shell --seed 2024113 
(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))))))