Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.4% → 98.8%

Time: 55.0s

Alternatives: 12

Speedup: 1.2×

• could not determine a ground truth

  1. z = -1.3507888155531594e+244

Specification

\[z \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\\ t_1 := \left(1 - z\right) + -1\\ t_2 := t\_1 + 7\\ t_3 := \sin \left(\pi \cdot z\right)\\ t_4 := \frac{\pi}{t\_3}\\ t_5 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;t\_4 \cdot \left(\left(\left({\left(t\_2 + 0.5\right)}^{\left(t\_1 + 0.5\right)} \cdot t\_5\right) \cdot e^{\left(\left(\left(z + -1\right) - -1\right) - 7\right) - 0.5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 + t\_1}\right) + \frac{-1259.1392167224028}{2 + t\_1}\right) + \frac{771.3234287776531}{t\_1 + 3}\right) + \frac{-176.6150291621406}{t\_1 + 4}\right) + \frac{12.507343278686905}{t\_1 + 5}\right) + \frac{-0.13857109526572012}{t\_1 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_2}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_1 + 8}\right)\right) \leq 4.6 \cdot 10^{+289}:\\ \;\;\;\;\pi \cdot \frac{\left(t\_5 \cdot \left(t\_0 \cdot e^{-6.5 - \left(1 - z\right)}\right)\right) \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{1 + \left(1 - z\right)} - \left(\frac{771.3234287776531}{\left(z + -1\right) - 2} + \frac{-176.6150291621406}{\left(z + -1\right) - 3}\right)\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} - \left(\frac{-0.13857109526572012}{-5 - \left(1 - z\right)} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) - 7} - \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6}\right)\right)\right)\right)}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_5 \cdot \left(t\_0 \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right)\right) \cdot \left(t\_4 \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)))
        (t_1 (+ (- 1.0 z) -1.0))
        (t_2 (+ t_1 7.0))
        (t_3 (sin (* PI z)))
        (t_4 (/ PI t_3))
        (t_5 (sqrt (* PI 2.0))))
   (if (<=
        (*
         t_4
         (*
          (*
           (* (pow (+ t_2 0.5) (+ t_1 0.5)) t_5)
           (exp (- (- (- (+ z -1.0) -1.0) 7.0) 0.5)))
          (+
           (+
            (+
             (+
              (+
               (+
                (+
                 (+ 0.9999999999998099 (/ 676.5203681218851 (+ 1.0 t_1)))
                 (/ -1259.1392167224028 (+ 2.0 t_1)))
                (/ 771.3234287776531 (+ t_1 3.0)))
               (/ -176.6150291621406 (+ t_1 4.0)))
              (/ 12.507343278686905 (+ t_1 5.0)))
             (/ -0.13857109526572012 (+ t_1 6.0)))
            (/ 9.984369578019572e-6 t_2))
           (/ 1.5056327351493116e-7 (+ t_1 8.0)))))
        4.6e+289)
     (*
      PI
      (/
       (*
        (* t_5 (* t_0 (exp (- -6.5 (- 1.0 z)))))
        (+
         (+
          (+ 0.9999999999998099 (/ 676.5203681218851 (- 1.0 z)))
          (-
           (/ -1259.1392167224028 (+ 1.0 (- 1.0 z)))
           (+
            (/ 771.3234287776531 (- (+ z -1.0) 2.0))
            (/ -176.6150291621406 (- (+ z -1.0) 3.0)))))
         (-
          (/ 12.507343278686905 (+ (- 1.0 z) 4.0))
          (+
           (/ -0.13857109526572012 (- -5.0 (- 1.0 z)))
           (-
            (/ 1.5056327351493116e-7 (- (+ z -1.0) 7.0))
            (/ 9.984369578019572e-6 (+ (- 1.0 z) 6.0)))))))
       t_3))
     (*
      (* t_5 (* t_0 (* (+ z 1.0) (exp -7.5))))
      (* t_4 (+ 263.3831869810514 (* z 436.8961725563396)))))))

C

double code(double z) {
	double t_0 = pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5));
	double t_1 = (1.0 - z) + -1.0;
	double t_2 = t_1 + 7.0;
	double t_3 = sin((((double) M_PI) * z));
	double t_4 = ((double) M_PI) / t_3;
	double t_5 = sqrt((((double) M_PI) * 2.0));
	double tmp;
	if ((t_4 * (((pow((t_2 + 0.5), (t_1 + 0.5)) * t_5) * exp(((((z + -1.0) - -1.0) - 7.0) - 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / (1.0 + t_1))) + (-1259.1392167224028 / (2.0 + t_1))) + (771.3234287776531 / (t_1 + 3.0))) + (-176.6150291621406 / (t_1 + 4.0))) + (12.507343278686905 / (t_1 + 5.0))) + (-0.13857109526572012 / (t_1 + 6.0))) + (9.984369578019572e-6 / t_2)) + (1.5056327351493116e-7 / (t_1 + 8.0))))) <= 4.6e+289) {
		tmp = ((double) M_PI) * (((t_5 * (t_0 * exp((-6.5 - (1.0 - z))))) * (((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + ((-1259.1392167224028 / (1.0 + (1.0 - z))) - ((771.3234287776531 / ((z + -1.0) - 2.0)) + (-176.6150291621406 / ((z + -1.0) - 3.0))))) + ((12.507343278686905 / ((1.0 - z) + 4.0)) - ((-0.13857109526572012 / (-5.0 - (1.0 - z))) + ((1.5056327351493116e-7 / ((z + -1.0) - 7.0)) - (9.984369578019572e-6 / ((1.0 - z) + 6.0))))))) / t_3);
	} else {
		tmp = (t_5 * (t_0 * ((z + 1.0) * exp(-7.5)))) * (t_4 * (263.3831869810514 + (z * 436.8961725563396)));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5));
	double t_1 = (1.0 - z) + -1.0;
	double t_2 = t_1 + 7.0;
	double t_3 = Math.sin((Math.PI * z));
	double t_4 = Math.PI / t_3;
	double t_5 = Math.sqrt((Math.PI * 2.0));
	double tmp;
	if ((t_4 * (((Math.pow((t_2 + 0.5), (t_1 + 0.5)) * t_5) * Math.exp(((((z + -1.0) - -1.0) - 7.0) - 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / (1.0 + t_1))) + (-1259.1392167224028 / (2.0 + t_1))) + (771.3234287776531 / (t_1 + 3.0))) + (-176.6150291621406 / (t_1 + 4.0))) + (12.507343278686905 / (t_1 + 5.0))) + (-0.13857109526572012 / (t_1 + 6.0))) + (9.984369578019572e-6 / t_2)) + (1.5056327351493116e-7 / (t_1 + 8.0))))) <= 4.6e+289) {
		tmp = Math.PI * (((t_5 * (t_0 * Math.exp((-6.5 - (1.0 - z))))) * (((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + ((-1259.1392167224028 / (1.0 + (1.0 - z))) - ((771.3234287776531 / ((z + -1.0) - 2.0)) + (-176.6150291621406 / ((z + -1.0) - 3.0))))) + ((12.507343278686905 / ((1.0 - z) + 4.0)) - ((-0.13857109526572012 / (-5.0 - (1.0 - z))) + ((1.5056327351493116e-7 / ((z + -1.0) - 7.0)) - (9.984369578019572e-6 / ((1.0 - z) + 6.0))))))) / t_3);
	} else {
		tmp = (t_5 * (t_0 * ((z + 1.0) * Math.exp(-7.5)))) * (t_4 * (263.3831869810514 + (z * 436.8961725563396)));
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5))
	t_1 = (1.0 - z) + -1.0
	t_2 = t_1 + 7.0
	t_3 = math.sin((math.pi * z))
	t_4 = math.pi / t_3
	t_5 = math.sqrt((math.pi * 2.0))
	tmp = 0
	if (t_4 * (((math.pow((t_2 + 0.5), (t_1 + 0.5)) * t_5) * math.exp(((((z + -1.0) - -1.0) - 7.0) - 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / (1.0 + t_1))) + (-1259.1392167224028 / (2.0 + t_1))) + (771.3234287776531 / (t_1 + 3.0))) + (-176.6150291621406 / (t_1 + 4.0))) + (12.507343278686905 / (t_1 + 5.0))) + (-0.13857109526572012 / (t_1 + 6.0))) + (9.984369578019572e-6 / t_2)) + (1.5056327351493116e-7 / (t_1 + 8.0))))) <= 4.6e+289:
		tmp = math.pi * (((t_5 * (t_0 * math.exp((-6.5 - (1.0 - z))))) * (((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + ((-1259.1392167224028 / (1.0 + (1.0 - z))) - ((771.3234287776531 / ((z + -1.0) - 2.0)) + (-176.6150291621406 / ((z + -1.0) - 3.0))))) + ((12.507343278686905 / ((1.0 - z) + 4.0)) - ((-0.13857109526572012 / (-5.0 - (1.0 - z))) + ((1.5056327351493116e-7 / ((z + -1.0) - 7.0)) - (9.984369578019572e-6 / ((1.0 - z) + 6.0))))))) / t_3)
	else:
		tmp = (t_5 * (t_0 * ((z + 1.0) * math.exp(-7.5)))) * (t_4 * (263.3831869810514 + (z * 436.8961725563396)))
	return tmp

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)
	t_1 = Float64(Float64(1.0 - z) + -1.0)
	t_2 = Float64(t_1 + 7.0)
	t_3 = sin(Float64(pi * z))
	t_4 = Float64(pi / t_3)
	t_5 = sqrt(Float64(pi * 2.0))
	tmp = 0.0
	if (Float64(t_4 * Float64(Float64(Float64((Float64(t_2 + 0.5) ^ Float64(t_1 + 0.5)) * t_5) * exp(Float64(Float64(Float64(Float64(z + -1.0) - -1.0) - 7.0) - 0.5))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 + t_1))) + Float64(-1259.1392167224028 / Float64(2.0 + t_1))) + Float64(771.3234287776531 / Float64(t_1 + 3.0))) + Float64(-176.6150291621406 / Float64(t_1 + 4.0))) + Float64(12.507343278686905 / Float64(t_1 + 5.0))) + Float64(-0.13857109526572012 / Float64(t_1 + 6.0))) + Float64(9.984369578019572e-6 / t_2)) + Float64(1.5056327351493116e-7 / Float64(t_1 + 8.0))))) <= 4.6e+289)
		tmp = Float64(pi * Float64(Float64(Float64(t_5 * Float64(t_0 * exp(Float64(-6.5 - Float64(1.0 - z))))) * Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 - z))) + Float64(Float64(-1259.1392167224028 / Float64(1.0 + Float64(1.0 - z))) - Float64(Float64(771.3234287776531 / Float64(Float64(z + -1.0) - 2.0)) + Float64(-176.6150291621406 / Float64(Float64(z + -1.0) - 3.0))))) + Float64(Float64(12.507343278686905 / Float64(Float64(1.0 - z) + 4.0)) - Float64(Float64(-0.13857109526572012 / Float64(-5.0 - Float64(1.0 - z))) + Float64(Float64(1.5056327351493116e-7 / Float64(Float64(z + -1.0) - 7.0)) - Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) + 6.0))))))) / t_3));
	else
		tmp = Float64(Float64(t_5 * Float64(t_0 * Float64(Float64(z + 1.0) * exp(-7.5)))) * Float64(t_4 * Float64(263.3831869810514 + Float64(z * 436.8961725563396))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = ((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5);
	t_1 = (1.0 - z) + -1.0;
	t_2 = t_1 + 7.0;
	t_3 = sin((pi * z));
	t_4 = pi / t_3;
	t_5 = sqrt((pi * 2.0));
	tmp = 0.0;
	if ((t_4 * (((((t_2 + 0.5) ^ (t_1 + 0.5)) * t_5) * exp(((((z + -1.0) - -1.0) - 7.0) - 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / (1.0 + t_1))) + (-1259.1392167224028 / (2.0 + t_1))) + (771.3234287776531 / (t_1 + 3.0))) + (-176.6150291621406 / (t_1 + 4.0))) + (12.507343278686905 / (t_1 + 5.0))) + (-0.13857109526572012 / (t_1 + 6.0))) + (9.984369578019572e-6 / t_2)) + (1.5056327351493116e-7 / (t_1 + 8.0))))) <= 4.6e+289)
		tmp = pi * (((t_5 * (t_0 * exp((-6.5 - (1.0 - z))))) * (((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + ((-1259.1392167224028 / (1.0 + (1.0 - z))) - ((771.3234287776531 / ((z + -1.0) - 2.0)) + (-176.6150291621406 / ((z + -1.0) - 3.0))))) + ((12.507343278686905 / ((1.0 - z) + 4.0)) - ((-0.13857109526572012 / (-5.0 - (1.0 - z))) + ((1.5056327351493116e-7 / ((z + -1.0) - 7.0)) - (9.984369578019572e-6 / ((1.0 - z) + 6.0))))))) / t_3);
	else
		tmp = (t_5 * (t_0 * ((z + 1.0) * exp(-7.5)))) * (t_4 * (263.3831869810514 + (z * 436.8961725563396)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - z), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + 7.0), $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(Pi / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$4 * N[(N[(N[(N[Power[N[(t$95$2 + 0.5), $MachinePrecision], N[(t$95$1 + 0.5), $MachinePrecision]], $MachinePrecision] * t$95$5), $MachinePrecision] * N[Exp[N[(N[(N[(N[(z + -1.0), $MachinePrecision] - -1.0), $MachinePrecision] - 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$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(t$95$1 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(t$95$1 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(t$95$1 + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(t$95$1 + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$2), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(t$95$1 + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.6e+289], N[(Pi * N[(N[(N[(t$95$5 * N[(t$95$0 * N[Exp[N[(-6.5 - N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1259.1392167224028 / N[(1.0 + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(771.3234287776531 / N[(N[(z + -1.0), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(z + -1.0), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.13857109526572012 / N[(-5.0 - N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5056327351493116e-7 / N[(N[(z + -1.0), $MachinePrecision] - 7.0), $MachinePrecision]), $MachinePrecision] - N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$5 * N[(t$95$0 * N[(N[(z + 1.0), $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$4 * N[(263.3831869810514 + N[(z * 436.8961725563396), $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)))))) < 4.5999999999999998e289

   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 97.2%

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

   6. Applied egg-rr 98.8%

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

   8. Simplified 99.3%

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

   if 4.5999999999999998e289 < (*.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 53.2%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in z around 0 54.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(263.3831869810514 + 436.8961725563396 \cdot z\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative 54.0%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(263.3831869810514 + \color{blue}{z \cdot 436.8961725563396}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   7. Simplified 54.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(263.3831869810514 + z \cdot 436.8961725563396\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. Taylor expanded in z around 0 99.4%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(e^{-7.5} + z \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   9. Step-by-step derivation

      1. distribute-rgt1-in 99.4%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(\left(z + 1\right) \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   10. Simplified 99.4%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(\left(z + 1\right) \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\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(1 - z\right) + -1\right)}\right) + \frac{-1259.1392167224028}{2 + \left(\left(1 - z\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}{\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) \leq 4.6 \cdot 10^{+289}:\\ \;\;\;\;\pi \cdot \frac{\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^{-6.5 - \left(1 - z\right)}\right)\right) \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{1 + \left(1 - z\right)} - \left(\frac{771.3234287776531}{\left(z + -1\right) - 2} + \frac{-176.6150291621406}{\left(z + -1\right) - 3}\right)\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} - \left(\frac{-0.13857109526572012}{-5 - \left(1 - z\right)} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) - 7} - \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6}\right)\right)\right)\right)}{\sin \left(\pi \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ t_1 := \sin \left(\pi \cdot z\right)\\ \mathbf{if}\;z \leq -1000:\\ \;\;\;\;\left(t\_0 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right)\right) \cdot \left(\frac{\pi}{t\_1} \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \frac{{\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(t\_0 \cdot e^{z + -7.5}\right)}{t\_1}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{1 + \left(1 - z\right)}\right) - \left(\frac{771.3234287776531}{-2 - \left(1 - z\right)} + \frac{-176.6150291621406}{-3 - \left(1 - z\right)}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} - \frac{-0.13857109526572012}{-5 - \left(1 - z\right)}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 2.0))) (t_1 (sin (* PI z))))
   (if (<= z -1000.0)
     (*
      (*
       t_0
       (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (* (+ z 1.0) (exp -7.5))))
      (* (/ PI t_1) (+ 263.3831869810514 (* z 436.8961725563396))))
     (*
      (* PI (/ (* (pow (- 7.5 z) (- 0.5 z)) (* t_0 (exp (+ z -7.5)))) t_1))
      (+
       (+
        (-
         (+
          (+ 0.9999999999998099 (/ 676.5203681218851 (- 1.0 z)))
          (/ -1259.1392167224028 (+ 1.0 (- 1.0 z))))
         (+
          (/ 771.3234287776531 (- -2.0 (- 1.0 z)))
          (/ -176.6150291621406 (- -3.0 (- 1.0 z)))))
        (-
         (/ 12.507343278686905 (- (- 1.0 z) -4.0))
         (/ -0.13857109526572012 (- -5.0 (- 1.0 z)))))
       (+
        (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))
        (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0))))))))

C

double code(double z) {
	double t_0 = sqrt((((double) M_PI) * 2.0));
	double t_1 = sin((((double) M_PI) * z));
	double tmp;
	if (z <= -1000.0) {
		tmp = (t_0 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * ((z + 1.0) * exp(-7.5)))) * ((((double) M_PI) / t_1) * (263.3831869810514 + (z * 436.8961725563396)));
	} else {
		tmp = (((double) M_PI) * ((pow((7.5 - z), (0.5 - z)) * (t_0 * exp((z + -7.5)))) / t_1)) * (((((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + (-1259.1392167224028 / (1.0 + (1.0 - z)))) - ((771.3234287776531 / (-2.0 - (1.0 - z))) + (-176.6150291621406 / (-3.0 - (1.0 - z))))) + ((12.507343278686905 / ((1.0 - z) - -4.0)) - (-0.13857109526572012 / (-5.0 - (1.0 - z))))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.sqrt((Math.PI * 2.0));
	double t_1 = Math.sin((Math.PI * z));
	double tmp;
	if (z <= -1000.0) {
		tmp = (t_0 * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * ((z + 1.0) * Math.exp(-7.5)))) * ((Math.PI / t_1) * (263.3831869810514 + (z * 436.8961725563396)));
	} else {
		tmp = (Math.PI * ((Math.pow((7.5 - z), (0.5 - z)) * (t_0 * Math.exp((z + -7.5)))) / t_1)) * (((((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + (-1259.1392167224028 / (1.0 + (1.0 - z)))) - ((771.3234287776531 / (-2.0 - (1.0 - z))) + (-176.6150291621406 / (-3.0 - (1.0 - z))))) + ((12.507343278686905 / ((1.0 - z) - -4.0)) - (-0.13857109526572012 / (-5.0 - (1.0 - z))))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.sqrt((math.pi * 2.0))
	t_1 = math.sin((math.pi * z))
	tmp = 0
	if z <= -1000.0:
		tmp = (t_0 * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * ((z + 1.0) * math.exp(-7.5)))) * ((math.pi / t_1) * (263.3831869810514 + (z * 436.8961725563396)))
	else:
		tmp = (math.pi * ((math.pow((7.5 - z), (0.5 - z)) * (t_0 * math.exp((z + -7.5)))) / t_1)) * (((((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + (-1259.1392167224028 / (1.0 + (1.0 - z)))) - ((771.3234287776531 / (-2.0 - (1.0 - z))) + (-176.6150291621406 / (-3.0 - (1.0 - z))))) + ((12.507343278686905 / ((1.0 - z) - -4.0)) - (-0.13857109526572012 / (-5.0 - (1.0 - z))))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))))
	return tmp

Julia

function code(z)
	t_0 = sqrt(Float64(pi * 2.0))
	t_1 = sin(Float64(pi * z))
	tmp = 0.0
	if (z <= -1000.0)
		tmp = Float64(Float64(t_0 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * Float64(Float64(z + 1.0) * exp(-7.5)))) * Float64(Float64(pi / t_1) * Float64(263.3831869810514 + Float64(z * 436.8961725563396))));
	else
		tmp = Float64(Float64(pi * Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * Float64(t_0 * exp(Float64(z + -7.5)))) / t_1)) * Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 - z))) + Float64(-1259.1392167224028 / Float64(1.0 + Float64(1.0 - z)))) - Float64(Float64(771.3234287776531 / Float64(-2.0 - Float64(1.0 - z))) + Float64(-176.6150291621406 / Float64(-3.0 - Float64(1.0 - z))))) + Float64(Float64(12.507343278686905 / Float64(Float64(1.0 - z) - -4.0)) - Float64(-0.13857109526572012 / Float64(-5.0 - Float64(1.0 - z))))) + Float64(Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0)) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = sqrt((pi * 2.0));
	t_1 = sin((pi * z));
	tmp = 0.0;
	if (z <= -1000.0)
		tmp = (t_0 * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * ((z + 1.0) * exp(-7.5)))) * ((pi / t_1) * (263.3831869810514 + (z * 436.8961725563396)));
	else
		tmp = (pi * ((((7.5 - z) ^ (0.5 - z)) * (t_0 * exp((z + -7.5)))) / t_1)) * (((((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + (-1259.1392167224028 / (1.0 + (1.0 - z)))) - ((771.3234287776531 / (-2.0 - (1.0 - z))) + (-176.6150291621406 / (-3.0 - (1.0 - z))))) + ((12.507343278686905 / ((1.0 - z) - -4.0)) - (-0.13857109526572012 / (-5.0 - (1.0 - z))))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1000.0], N[(N[(t$95$0 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[(z + 1.0), $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / t$95$1), $MachinePrecision] * N[(263.3831869810514 + N[(z * 436.8961725563396), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(1.0 + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(771.3234287776531 / N[(-2.0 - N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(-3.0 - N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] - -4.0), $MachinePrecision]), $MachinePrecision] - N[(-0.13857109526572012 / N[(-5.0 - N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < -1e3

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in z around 0 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(263.3831869810514 + 436.8961725563396 \cdot z\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(263.3831869810514 + \color{blue}{z \cdot 436.8961725563396}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   7. Simplified 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(263.3831869810514 + z \cdot 436.8961725563396\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. Taylor expanded in z around 0 100.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(e^{-7.5} + z \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   9. Step-by-step derivation

      1. distribute-rgt1-in 100.0%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(\left(z + 1\right) \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   10. Simplified 100.0%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(\left(z + 1\right) \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   if -1e3 < 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/ 99.2%

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

   6. Applied egg-rr 99.2%

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

   8. Simplified 99.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1000:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \frac{{\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{z + -7.5}\right)}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{1 + \left(1 - z\right)}\right) - \left(\frac{771.3234287776531}{-2 - \left(1 - z\right)} + \frac{-176.6150291621406}{-3 - \left(1 - z\right)}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} - \frac{-0.13857109526572012}{-5 - \left(1 - z\right)}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ t_1 := \sin \left(\pi \cdot z\right)\\ \mathbf{if}\;z \leq -1000:\\ \;\;\;\;\left(t\_0 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right)\right) \cdot \left(\frac{\pi}{t\_1} \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \frac{{\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(t\_0 \cdot e^{z + -7.5}\right)}{t\_1}\right) \cdot \left(\left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) - \left(\frac{771.3234287776531}{z - 3} + \left(\frac{-176.6150291621406}{z - 4} + \left(\frac{12.507343278686905}{\left(z + -1\right) - 4} + \frac{-0.13857109526572012}{-1 + \left(z - 5\right)}\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 2.0))) (t_1 (sin (* PI z))))
   (if (<= z -1000.0)
     (*
      (*
       t_0
       (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (* (+ z 1.0) (exp -7.5))))
      (* (/ PI t_1) (+ 263.3831869810514 (* z 436.8961725563396))))
     (*
      (* PI (/ (* (pow (- 7.5 z) (- 0.5 z)) (* t_0 (exp (+ z -7.5)))) t_1))
      (+
       (+
        (/ 676.5203681218851 (- 1.0 z))
        (+ 0.9999999999998099 (/ -1259.1392167224028 (- 2.0 z))))
       (-
        (+
         (/ 1.5056327351493116e-7 (+ (- 1.0 z) 7.0))
         (/ 9.984369578019572e-6 (- 7.0 z)))
        (+
         (/ 771.3234287776531 (- z 3.0))
         (+
          (/ -176.6150291621406 (- z 4.0))
          (+
           (/ 12.507343278686905 (- (+ z -1.0) 4.0))
           (/ -0.13857109526572012 (+ -1.0 (- z 5.0))))))))))))

C

double code(double z) {
	double t_0 = sqrt((((double) M_PI) * 2.0));
	double t_1 = sin((((double) M_PI) * z));
	double tmp;
	if (z <= -1000.0) {
		tmp = (t_0 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * ((z + 1.0) * exp(-7.5)))) * ((((double) M_PI) / t_1) * (263.3831869810514 + (z * 436.8961725563396)));
	} else {
		tmp = (((double) M_PI) * ((pow((7.5 - z), (0.5 - z)) * (t_0 * exp((z + -7.5)))) / t_1)) * (((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + (-1259.1392167224028 / (2.0 - z)))) + (((1.5056327351493116e-7 / ((1.0 - z) + 7.0)) + (9.984369578019572e-6 / (7.0 - z))) - ((771.3234287776531 / (z - 3.0)) + ((-176.6150291621406 / (z - 4.0)) + ((12.507343278686905 / ((z + -1.0) - 4.0)) + (-0.13857109526572012 / (-1.0 + (z - 5.0))))))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.sqrt((Math.PI * 2.0));
	double t_1 = Math.sin((Math.PI * z));
	double tmp;
	if (z <= -1000.0) {
		tmp = (t_0 * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * ((z + 1.0) * Math.exp(-7.5)))) * ((Math.PI / t_1) * (263.3831869810514 + (z * 436.8961725563396)));
	} else {
		tmp = (Math.PI * ((Math.pow((7.5 - z), (0.5 - z)) * (t_0 * Math.exp((z + -7.5)))) / t_1)) * (((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + (-1259.1392167224028 / (2.0 - z)))) + (((1.5056327351493116e-7 / ((1.0 - z) + 7.0)) + (9.984369578019572e-6 / (7.0 - z))) - ((771.3234287776531 / (z - 3.0)) + ((-176.6150291621406 / (z - 4.0)) + ((12.507343278686905 / ((z + -1.0) - 4.0)) + (-0.13857109526572012 / (-1.0 + (z - 5.0))))))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.sqrt((math.pi * 2.0))
	t_1 = math.sin((math.pi * z))
	tmp = 0
	if z <= -1000.0:
		tmp = (t_0 * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * ((z + 1.0) * math.exp(-7.5)))) * ((math.pi / t_1) * (263.3831869810514 + (z * 436.8961725563396)))
	else:
		tmp = (math.pi * ((math.pow((7.5 - z), (0.5 - z)) * (t_0 * math.exp((z + -7.5)))) / t_1)) * (((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + (-1259.1392167224028 / (2.0 - z)))) + (((1.5056327351493116e-7 / ((1.0 - z) + 7.0)) + (9.984369578019572e-6 / (7.0 - z))) - ((771.3234287776531 / (z - 3.0)) + ((-176.6150291621406 / (z - 4.0)) + ((12.507343278686905 / ((z + -1.0) - 4.0)) + (-0.13857109526572012 / (-1.0 + (z - 5.0))))))))
	return tmp

Julia

function code(z)
	t_0 = sqrt(Float64(pi * 2.0))
	t_1 = sin(Float64(pi * z))
	tmp = 0.0
	if (z <= -1000.0)
		tmp = Float64(Float64(t_0 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * Float64(Float64(z + 1.0) * exp(-7.5)))) * Float64(Float64(pi / t_1) * Float64(263.3831869810514 + Float64(z * 436.8961725563396))));
	else
		tmp = Float64(Float64(pi * Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * Float64(t_0 * exp(Float64(z + -7.5)))) / t_1)) * Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(0.9999999999998099 + Float64(-1259.1392167224028 / Float64(2.0 - z)))) + Float64(Float64(Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) + 7.0)) + Float64(9.984369578019572e-6 / Float64(7.0 - z))) - Float64(Float64(771.3234287776531 / Float64(z - 3.0)) + Float64(Float64(-176.6150291621406 / Float64(z - 4.0)) + Float64(Float64(12.507343278686905 / Float64(Float64(z + -1.0) - 4.0)) + Float64(-0.13857109526572012 / Float64(-1.0 + Float64(z - 5.0)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = sqrt((pi * 2.0));
	t_1 = sin((pi * z));
	tmp = 0.0;
	if (z <= -1000.0)
		tmp = (t_0 * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * ((z + 1.0) * exp(-7.5)))) * ((pi / t_1) * (263.3831869810514 + (z * 436.8961725563396)));
	else
		tmp = (pi * ((((7.5 - z) ^ (0.5 - z)) * (t_0 * exp((z + -7.5)))) / t_1)) * (((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + (-1259.1392167224028 / (2.0 - z)))) + (((1.5056327351493116e-7 / ((1.0 - z) + 7.0)) + (9.984369578019572e-6 / (7.0 - z))) - ((771.3234287776531 / (z - 3.0)) + ((-176.6150291621406 / (z - 4.0)) + ((12.507343278686905 / ((z + -1.0) - 4.0)) + (-0.13857109526572012 / (-1.0 + (z - 5.0))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1000.0], N[(N[(t$95$0 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[(z + 1.0), $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / t$95$1), $MachinePrecision] * N[(263.3831869810514 + N[(z * 436.8961725563396), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] + 7.0), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(771.3234287776531 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-176.6150291621406 / N[(z - 4.0), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(N[(z + -1.0), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(-1.0 + N[(z - 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < -1e3

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in z around 0 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(263.3831869810514 + 436.8961725563396 \cdot z\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(263.3831869810514 + \color{blue}{z \cdot 436.8961725563396}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   7. Simplified 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(263.3831869810514 + z \cdot 436.8961725563396\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. Taylor expanded in z around 0 100.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(e^{-7.5} + z \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   9. Step-by-step derivation

      1. distribute-rgt1-in 100.0%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(\left(z + 1\right) \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   10. Simplified 100.0%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(\left(z + 1\right) \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   if -1e3 < 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 97.2%

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

   6. Applied egg-rr 99.2%

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

   8. Simplified 98.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1000:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \frac{{\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{z + -7.5}\right)}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(\left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) - \left(\frac{771.3234287776531}{z - 3} + \left(\frac{-176.6150291621406}{z - 4} + \left(\frac{12.507343278686905}{\left(z + -1\right) - 4} + \frac{-0.13857109526572012}{-1 + \left(z - 5\right)}\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ t_1 := \frac{\pi}{\sin \left(\pi \cdot z\right)}\\ \mathbf{if}\;z \leq -1000:\\ \;\;\;\;\left(t\_0 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right)\right) \cdot \left(t\_1 \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(t\_0 \cdot e^{z + -7.5}\right)\right) \cdot \left(t\_1 \cdot \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} - \left(\left(\frac{771.3234287776531}{z - 3} - \left(0.9999999999998099 - \left(\frac{676.5203681218851}{z + -1} - \frac{-1259.1392167224028}{2 - z}\right)\right)\right) + \left(\frac{-176.6150291621406}{z - 4} + \left(\frac{12.507343278686905}{z - 5} + \frac{-0.13857109526572012}{z - 6}\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 2.0))) (t_1 (/ PI (sin (* PI z)))))
   (if (<= z -1000.0)
     (*
      (*
       t_0
       (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (* (+ z 1.0) (exp -7.5))))
      (* t_1 (+ 263.3831869810514 (* z 436.8961725563396))))
     (*
      (* (pow (- 7.5 z) (- 0.5 z)) (* t_0 (exp (+ z -7.5))))
      (*
       t_1
       (+
        (/ 9.984369578019572e-6 (- 7.0 z))
        (-
         (/ 1.5056327351493116e-7 (- 8.0 z))
         (+
          (-
           (/ 771.3234287776531 (- z 3.0))
           (-
            0.9999999999998099
            (-
             (/ 676.5203681218851 (+ z -1.0))
             (/ -1259.1392167224028 (- 2.0 z)))))
          (+
           (/ -176.6150291621406 (- z 4.0))
           (+
            (/ 12.507343278686905 (- z 5.0))
            (/ -0.13857109526572012 (- z 6.0))))))))))))

C

double code(double z) {
	double t_0 = sqrt((((double) M_PI) * 2.0));
	double t_1 = ((double) M_PI) / sin((((double) M_PI) * z));
	double tmp;
	if (z <= -1000.0) {
		tmp = (t_0 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * ((z + 1.0) * exp(-7.5)))) * (t_1 * (263.3831869810514 + (z * 436.8961725563396)));
	} else {
		tmp = (pow((7.5 - z), (0.5 - z)) * (t_0 * exp((z + -7.5)))) * (t_1 * ((9.984369578019572e-6 / (7.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) - (((771.3234287776531 / (z - 3.0)) - (0.9999999999998099 - ((676.5203681218851 / (z + -1.0)) - (-1259.1392167224028 / (2.0 - z))))) + ((-176.6150291621406 / (z - 4.0)) + ((12.507343278686905 / (z - 5.0)) + (-0.13857109526572012 / (z - 6.0))))))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.sqrt((Math.PI * 2.0));
	double t_1 = Math.PI / Math.sin((Math.PI * z));
	double tmp;
	if (z <= -1000.0) {
		tmp = (t_0 * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * ((z + 1.0) * Math.exp(-7.5)))) * (t_1 * (263.3831869810514 + (z * 436.8961725563396)));
	} else {
		tmp = (Math.pow((7.5 - z), (0.5 - z)) * (t_0 * Math.exp((z + -7.5)))) * (t_1 * ((9.984369578019572e-6 / (7.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) - (((771.3234287776531 / (z - 3.0)) - (0.9999999999998099 - ((676.5203681218851 / (z + -1.0)) - (-1259.1392167224028 / (2.0 - z))))) + ((-176.6150291621406 / (z - 4.0)) + ((12.507343278686905 / (z - 5.0)) + (-0.13857109526572012 / (z - 6.0))))))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.sqrt((math.pi * 2.0))
	t_1 = math.pi / math.sin((math.pi * z))
	tmp = 0
	if z <= -1000.0:
		tmp = (t_0 * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * ((z + 1.0) * math.exp(-7.5)))) * (t_1 * (263.3831869810514 + (z * 436.8961725563396)))
	else:
		tmp = (math.pow((7.5 - z), (0.5 - z)) * (t_0 * math.exp((z + -7.5)))) * (t_1 * ((9.984369578019572e-6 / (7.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) - (((771.3234287776531 / (z - 3.0)) - (0.9999999999998099 - ((676.5203681218851 / (z + -1.0)) - (-1259.1392167224028 / (2.0 - z))))) + ((-176.6150291621406 / (z - 4.0)) + ((12.507343278686905 / (z - 5.0)) + (-0.13857109526572012 / (z - 6.0))))))))
	return tmp

Julia

function code(z)
	t_0 = sqrt(Float64(pi * 2.0))
	t_1 = Float64(pi / sin(Float64(pi * z)))
	tmp = 0.0
	if (z <= -1000.0)
		tmp = Float64(Float64(t_0 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * Float64(Float64(z + 1.0) * exp(-7.5)))) * Float64(t_1 * Float64(263.3831869810514 + Float64(z * 436.8961725563396))));
	else
		tmp = Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * Float64(t_0 * exp(Float64(z + -7.5)))) * Float64(t_1 * Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) - Float64(Float64(Float64(771.3234287776531 / Float64(z - 3.0)) - Float64(0.9999999999998099 - Float64(Float64(676.5203681218851 / Float64(z + -1.0)) - Float64(-1259.1392167224028 / Float64(2.0 - z))))) + Float64(Float64(-176.6150291621406 / Float64(z - 4.0)) + Float64(Float64(12.507343278686905 / Float64(z - 5.0)) + Float64(-0.13857109526572012 / Float64(z - 6.0)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = sqrt((pi * 2.0));
	t_1 = pi / sin((pi * z));
	tmp = 0.0;
	if (z <= -1000.0)
		tmp = (t_0 * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * ((z + 1.0) * exp(-7.5)))) * (t_1 * (263.3831869810514 + (z * 436.8961725563396)));
	else
		tmp = (((7.5 - z) ^ (0.5 - z)) * (t_0 * exp((z + -7.5)))) * (t_1 * ((9.984369578019572e-6 / (7.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) - (((771.3234287776531 / (z - 3.0)) - (0.9999999999998099 - ((676.5203681218851 / (z + -1.0)) - (-1259.1392167224028 / (2.0 - z))))) + ((-176.6150291621406 / (z - 4.0)) + ((12.507343278686905 / (z - 5.0)) + (-0.13857109526572012 / (z - 6.0))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1000.0], N[(N[(t$95$0 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[(z + 1.0), $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(263.3831869810514 + N[(z * 436.8961725563396), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * 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[(771.3234287776531 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision] - N[(0.9999999999998099 - N[(N[(676.5203681218851 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision] - N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-176.6150291621406 / N[(z - 4.0), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(z - 5.0), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(z - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < -1e3

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in z around 0 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(263.3831869810514 + 436.8961725563396 \cdot z\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(263.3831869810514 + \color{blue}{z \cdot 436.8961725563396}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   7. Simplified 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(263.3831869810514 + z \cdot 436.8961725563396\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. Taylor expanded in z around 0 100.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(e^{-7.5} + z \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   9. Step-by-step derivation

      1. distribute-rgt1-in 100.0%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(\left(z + 1\right) \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   10. Simplified 100.0%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(\left(z + 1\right) \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   if -1e3 < 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 97.4%

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

      1. pow1 97.4%

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

   6. Applied egg-rr 97.7%

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

   8. Simplified 98.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1000:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} - \left(\left(\frac{771.3234287776531}{z - 3} - \left(0.9999999999998099 - \left(\frac{676.5203681218851}{z + -1} - \frac{-1259.1392167224028}{2 - z}\right)\right)\right) + \left(\frac{-176.6150291621406}{z - 4} + \left(\frac{12.507343278686905}{z - 5} + \frac{-0.13857109526572012}{z - 6}\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (sqrt (* PI 2.0))
   (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (* (+ z 1.0) (exp -7.5))))
  (* (/ PI (sin (* PI z))) (+ 263.3831869810514 (* z 436.8961725563396)))))

C

double code(double z) {
	return (sqrt((((double) M_PI) * 2.0)) * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * ((z + 1.0) * exp(-7.5)))) * ((((double) M_PI) / sin((((double) M_PI) * z))) * (263.3831869810514 + (z * 436.8961725563396)));
}

Java

public static double code(double z) {
	return (Math.sqrt((Math.PI * 2.0)) * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * ((z + 1.0) * Math.exp(-7.5)))) * ((Math.PI / Math.sin((Math.PI * z))) * (263.3831869810514 + (z * 436.8961725563396)));
}

Python

def code(z):
	return (math.sqrt((math.pi * 2.0)) * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * ((z + 1.0) * math.exp(-7.5)))) * ((math.pi / math.sin((math.pi * z))) * (263.3831869810514 + (z * 436.8961725563396)))

Julia

function code(z)
	return Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * Float64(Float64(z + 1.0) * exp(-7.5)))) * Float64(Float64(pi / sin(Float64(pi * z))) * Float64(263.3831869810514 + Float64(z * 436.8961725563396))))
end

MATLAB

function tmp = code(z)
	tmp = (sqrt((pi * 2.0)) * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * ((z + 1.0) * exp(-7.5)))) * ((pi / sin((pi * z))) * (263.3831869810514 + (z * 436.8961725563396)));
end

Wolfram

code[z_] := N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[(z + 1.0), $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * 436.8961725563396), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.5%

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

3. Simplified 95.3%

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

5. Taylor expanded in z around 0 94.7%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(263.3831869810514 + 436.8961725563396 \cdot z\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
6. Step-by-step derivation

   1. *-commutative 94.7%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(263.3831869810514 + \color{blue}{z \cdot 436.8961725563396}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

7. Simplified 94.7%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(263.3831869810514 + z \cdot 436.8961725563396\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

8. Taylor expanded in z around 0 97.1%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(e^{-7.5} + z \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
9. Step-by-step derivation

   1. distribute-rgt1-in 97.1%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(\left(z + 1\right) \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

10. Simplified 97.1%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(\left(z + 1\right) \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

11. Final simplification 97.1%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right)\right) \]
12. Add Preprocessing

Alternative 6: 96.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right)\right) \cdot \frac{263.3831869810514 + z \cdot 436.8961725563396}{z} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (sqrt (* PI 2.0))
   (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (* (+ z 1.0) (exp -7.5))))
  (/ (+ 263.3831869810514 (* z 436.8961725563396)) z)))

C

double code(double z) {
	return (sqrt((((double) M_PI) * 2.0)) * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * ((z + 1.0) * exp(-7.5)))) * ((263.3831869810514 + (z * 436.8961725563396)) / z);
}

Java

public static double code(double z) {
	return (Math.sqrt((Math.PI * 2.0)) * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * ((z + 1.0) * Math.exp(-7.5)))) * ((263.3831869810514 + (z * 436.8961725563396)) / z);
}

Python

def code(z):
	return (math.sqrt((math.pi * 2.0)) * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * ((z + 1.0) * math.exp(-7.5)))) * ((263.3831869810514 + (z * 436.8961725563396)) / z)

Julia

function code(z)
	return Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * Float64(Float64(z + 1.0) * exp(-7.5)))) * Float64(Float64(263.3831869810514 + Float64(z * 436.8961725563396)) / z))
end

MATLAB

function tmp = code(z)
	tmp = (sqrt((pi * 2.0)) * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * ((z + 1.0) * exp(-7.5)))) * ((263.3831869810514 + (z * 436.8961725563396)) / z);
end

Wolfram

code[z_] := N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[(z + 1.0), $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(263.3831869810514 + N[(z * 436.8961725563396), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.5%

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

3. Simplified 95.3%

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

5. Taylor expanded in z around 0 94.7%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\frac{263.3831869810514 + 436.8961725563396 \cdot z}{z}} \]
6. Step-by-step derivation

   1. *-commutative 94.7%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \frac{263.3831869810514 + \color{blue}{z \cdot 436.8961725563396}}{z} \]

7. Simplified 94.7%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\frac{263.3831869810514 + z \cdot 436.8961725563396}{z}} \]

8. Taylor expanded in z around 0 96.0%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(e^{-7.5} + z \cdot e^{-7.5}\right)}\right)\right) \cdot \frac{263.3831869810514 + z \cdot 436.8961725563396}{z} \]
9. Step-by-step derivation

   1. distribute-rgt1-in 97.1%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(\left(z + 1\right) \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

10. Simplified 96.0%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(\left(z + 1\right) \cdot e^{-7.5}\right)}\right)\right) \cdot \frac{263.3831869810514 + z \cdot 436.8961725563396}{z} \]
11. Add Preprocessing

Alternative 7: 95.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.5%

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

3. Simplified 95.3%

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

5. Taylor expanded in z around 0 94.7%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\frac{263.3831869810514 + 436.8961725563396 \cdot z}{z}} \]
6. Step-by-step derivation

   1. *-commutative 94.7%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \frac{263.3831869810514 + \color{blue}{z \cdot 436.8961725563396}}{z} \]

7. Simplified 94.7%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\frac{263.3831869810514 + z \cdot 436.8961725563396}{z}} \]
8. Step-by-step derivation

   1. *-commutative 94.7%

      \[\leadsto \left(\sqrt{\color{blue}{2 \cdot \pi}} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \frac{263.3831869810514 + z \cdot 436.8961725563396}{z} \]

   2. distribute-neg-in 94.7%

      \[\leadsto \left(\sqrt{2 \cdot \pi} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\color{blue}{\left(-\left(1 - z\right)\right) + \left(-6.5\right)}}\right)\right) \cdot \frac{263.3831869810514 + z \cdot 436.8961725563396}{z} \]

   3. metadata-eval 94.7%

      \[\leadsto \left(\sqrt{2 \cdot \pi} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(-\left(1 - z\right)\right) + \color{blue}{-6.5}}\right)\right) \cdot \frac{263.3831869810514 + z \cdot 436.8961725563396}{z} \]

   4. pow1 94.7%

      \[\leadsto \color{blue}{{\left(\sqrt{2 \cdot \pi} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right)}^{1}} \cdot \frac{263.3831869810514 + z \cdot 436.8961725563396}{z} \]

9. Applied egg-rr 94.7%

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

    1. unpow1 94.7%

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

    2. *-commutative 94.7%

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

    3. +-commutative 94.7%

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

    4. sub-neg 94.7%

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

    5. +-commutative 94.7%

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

    6. sub-neg 94.7%

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

11. Simplified 94.7%

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

12. Final simplification 94.7%

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

Alternative 8: 95.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.5%

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

3. Simplified 95.3%

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

5. Taylor expanded in z around 0 94.7%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\frac{263.3831869810514 + 436.8961725563396 \cdot z}{z}} \]
6. Step-by-step derivation

   1. *-commutative 94.7%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \frac{263.3831869810514 + \color{blue}{z \cdot 436.8961725563396}}{z} \]

7. Simplified 94.7%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\frac{263.3831869810514 + z \cdot 436.8961725563396}{z}} \]
8. Step-by-step derivation

   1. *-commutative 94.7%

      \[\leadsto \left(\sqrt{\color{blue}{2 \cdot \pi}} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \frac{263.3831869810514 + z \cdot 436.8961725563396}{z} \]

   2. distribute-neg-in 94.7%

      \[\leadsto \left(\sqrt{2 \cdot \pi} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\color{blue}{\left(-\left(1 - z\right)\right) + \left(-6.5\right)}}\right)\right) \cdot \frac{263.3831869810514 + z \cdot 436.8961725563396}{z} \]

   3. metadata-eval 94.7%

      \[\leadsto \left(\sqrt{2 \cdot \pi} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(-\left(1 - z\right)\right) + \color{blue}{-6.5}}\right)\right) \cdot \frac{263.3831869810514 + z \cdot 436.8961725563396}{z} \]

   4. pow1 94.7%

      \[\leadsto \color{blue}{{\left(\sqrt{2 \cdot \pi} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right)}^{1}} \cdot \frac{263.3831869810514 + z \cdot 436.8961725563396}{z} \]

9. Applied egg-rr 94.7%

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

    1. unpow1 94.7%

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

    2. *-commutative 94.7%

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

    3. +-commutative 94.7%

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

    4. associate-+r- 94.7%

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

    5. metadata-eval 94.7%

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

    6. +-commutative 94.7%

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

    7. sub-neg 94.7%

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

    8. +-commutative 94.7%

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

    9. distribute-neg-in 94.7%

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

    10. metadata-eval 94.7%

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

    11. remove-double-neg 94.7%

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

11. Simplified 94.7%

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

12. Final simplification 94.7%

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

Alternative 9: 96.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.5%

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

3. Simplified 95.3%

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

5. Taylor expanded in z around 0 93.9%

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

6. Taylor expanded in z around 0 93.7%

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

   1. associate-*r/ 93.6%

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

   2. associate-*r* 93.6%

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

   3. pow1/2 93.6%

      \[\leadsto \frac{\left(\left(\color{blue}{{\left(\pi \cdot 2\right)}^{0.5}} \cdot \sqrt{7.5}\right) \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot 263.3831869810514}{z} \]

   4. pow1/2 93.6%

      \[\leadsto \frac{\left(\left({\left(\pi \cdot 2\right)}^{0.5} \cdot \color{blue}{{7.5}^{0.5}}\right) \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot 263.3831869810514}{z} \]

   5. pow-prod-down 93.6%

      \[\leadsto \frac{\left(\color{blue}{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot 263.3831869810514}{z} \]

   6. distribute-neg-in 93.6%

      \[\leadsto \frac{\left({\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5} \cdot e^{\color{blue}{\left(-\left(1 - z\right)\right) + \left(-6.5\right)}}\right) \cdot 263.3831869810514}{z} \]

   7. metadata-eval 93.6%

      \[\leadsto \frac{\left({\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5} \cdot e^{\left(-\left(1 - z\right)\right) + \color{blue}{-6.5}}\right) \cdot 263.3831869810514}{z} \]

8. Applied egg-rr 93.6%

   \[\leadsto \color{blue}{\frac{\left({\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right) \cdot 263.3831869810514}{z}} \]
9. Step-by-step derivation

   1. associate-*r/ 93.7%

      \[\leadsto \color{blue}{\left({\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right) \cdot \frac{263.3831869810514}{z}} \]

   2. associate-*l* 93.6%

      \[\leadsto \color{blue}{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5} \cdot \left(e^{\left(-\left(1 - z\right)\right) + -6.5} \cdot \frac{263.3831869810514}{z}\right)} \]

   3. unpow1/2 93.6%

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

   4. associate-*l* 93.6%

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

   5. metadata-eval 93.6%

      \[\leadsto \sqrt{\pi \cdot \color{blue}{15}} \cdot \left(e^{\left(-\left(1 - z\right)\right) + -6.5} \cdot \frac{263.3831869810514}{z}\right) \]

   6. *-commutative 93.6%

      \[\leadsto \sqrt{\pi \cdot 15} \cdot \color{blue}{\left(\frac{263.3831869810514}{z} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)} \]

   7. metadata-eval 93.6%

      \[\leadsto \sqrt{\pi \cdot 15} \cdot \left(\frac{263.3831869810514}{z} \cdot e^{\left(-\left(1 - z\right)\right) + \color{blue}{\left(-6.5\right)}}\right) \]

   8. distribute-neg-in 93.6%

      \[\leadsto \sqrt{\pi \cdot 15} \cdot \left(\frac{263.3831869810514}{z} \cdot e^{\color{blue}{-\left(\left(1 - z\right) + 6.5\right)}}\right) \]

   9. neg-mul-1 93.6%

      \[\leadsto \sqrt{\pi \cdot 15} \cdot \left(\frac{263.3831869810514}{z} \cdot e^{\color{blue}{-1 \cdot \left(\left(1 - z\right) + 6.5\right)}}\right) \]

   10. +-commutative 93.6%

       \[\leadsto \sqrt{\pi \cdot 15} \cdot \left(\frac{263.3831869810514}{z} \cdot e^{-1 \cdot \color{blue}{\left(6.5 + \left(1 - z\right)\right)}}\right) \]

   11. sub-neg 93.6%

       \[\leadsto \sqrt{\pi \cdot 15} \cdot \left(\frac{263.3831869810514}{z} \cdot e^{-1 \cdot \left(6.5 + \color{blue}{\left(1 + \left(-z\right)\right)}\right)}\right) \]

   12. associate-+r+ 93.6%

       \[\leadsto \sqrt{\pi \cdot 15} \cdot \left(\frac{263.3831869810514}{z} \cdot e^{-1 \cdot \color{blue}{\left(\left(6.5 + 1\right) + \left(-z\right)\right)}}\right) \]

   13. metadata-eval 93.6%

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

   14. mul-1-neg 93.6%

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

   15. neg-mul-1 93.6%

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

   16. mul-1-neg 93.6%

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

   17. +-commutative 93.6%

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

   18. distribute-neg-in 93.6%

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

10. Simplified 93.6%

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

11. Taylor expanded in z around inf 94.1%

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

    1. associate-*l/ 94.0%

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

    2. associate-*l/ 94.1%

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

    3. sub-neg 94.1%

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

    4. metadata-eval 94.1%

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

    5. prod-exp 94.1%

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

    6. *-commutative 94.1%

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

    7. remove-double-neg 94.1%

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

    8. exp-sum 94.1%

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

    9. metadata-eval 94.1%

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

    10. distribute-neg-in 94.1%

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

    11. mul-1-neg 94.1%

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

13. Simplified 94.3%

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

14. Final simplification 94.3%

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

Alternative 10: 95.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.5%

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

3. Simplified 95.3%

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

5. Taylor expanded in z around 0 93.9%

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

6. Taylor expanded in z around 0 93.7%

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

   1. associate-*r/ 93.6%

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

   2. associate-*r* 93.6%

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

   3. pow1/2 93.6%

      \[\leadsto \frac{\left(\left(\color{blue}{{\left(\pi \cdot 2\right)}^{0.5}} \cdot \sqrt{7.5}\right) \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot 263.3831869810514}{z} \]

   4. pow1/2 93.6%

      \[\leadsto \frac{\left(\left({\left(\pi \cdot 2\right)}^{0.5} \cdot \color{blue}{{7.5}^{0.5}}\right) \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot 263.3831869810514}{z} \]

   5. pow-prod-down 93.6%

      \[\leadsto \frac{\left(\color{blue}{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot 263.3831869810514}{z} \]

   6. distribute-neg-in 93.6%

      \[\leadsto \frac{\left({\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5} \cdot e^{\color{blue}{\left(-\left(1 - z\right)\right) + \left(-6.5\right)}}\right) \cdot 263.3831869810514}{z} \]

   7. metadata-eval 93.6%

      \[\leadsto \frac{\left({\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5} \cdot e^{\left(-\left(1 - z\right)\right) + \color{blue}{-6.5}}\right) \cdot 263.3831869810514}{z} \]

8. Applied egg-rr 93.6%

   \[\leadsto \color{blue}{\frac{\left({\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right) \cdot 263.3831869810514}{z}} \]
9. Step-by-step derivation

   1. associate-*r/ 93.7%

      \[\leadsto \color{blue}{\left({\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right) \cdot \frac{263.3831869810514}{z}} \]

   2. associate-*l* 93.6%

      \[\leadsto \color{blue}{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5} \cdot \left(e^{\left(-\left(1 - z\right)\right) + -6.5} \cdot \frac{263.3831869810514}{z}\right)} \]

   3. unpow1/2 93.6%

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

   4. associate-*l* 93.6%

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

   5. metadata-eval 93.6%

      \[\leadsto \sqrt{\pi \cdot \color{blue}{15}} \cdot \left(e^{\left(-\left(1 - z\right)\right) + -6.5} \cdot \frac{263.3831869810514}{z}\right) \]

   6. *-commutative 93.6%

      \[\leadsto \sqrt{\pi \cdot 15} \cdot \color{blue}{\left(\frac{263.3831869810514}{z} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)} \]

   7. metadata-eval 93.6%

      \[\leadsto \sqrt{\pi \cdot 15} \cdot \left(\frac{263.3831869810514}{z} \cdot e^{\left(-\left(1 - z\right)\right) + \color{blue}{\left(-6.5\right)}}\right) \]

   8. distribute-neg-in 93.6%

      \[\leadsto \sqrt{\pi \cdot 15} \cdot \left(\frac{263.3831869810514}{z} \cdot e^{\color{blue}{-\left(\left(1 - z\right) + 6.5\right)}}\right) \]

   9. neg-mul-1 93.6%

      \[\leadsto \sqrt{\pi \cdot 15} \cdot \left(\frac{263.3831869810514}{z} \cdot e^{\color{blue}{-1 \cdot \left(\left(1 - z\right) + 6.5\right)}}\right) \]

   10. +-commutative 93.6%

       \[\leadsto \sqrt{\pi \cdot 15} \cdot \left(\frac{263.3831869810514}{z} \cdot e^{-1 \cdot \color{blue}{\left(6.5 + \left(1 - z\right)\right)}}\right) \]

   11. sub-neg 93.6%

       \[\leadsto \sqrt{\pi \cdot 15} \cdot \left(\frac{263.3831869810514}{z} \cdot e^{-1 \cdot \left(6.5 + \color{blue}{\left(1 + \left(-z\right)\right)}\right)}\right) \]

   12. associate-+r+ 93.6%

       \[\leadsto \sqrt{\pi \cdot 15} \cdot \left(\frac{263.3831869810514}{z} \cdot e^{-1 \cdot \color{blue}{\left(\left(6.5 + 1\right) + \left(-z\right)\right)}}\right) \]

   13. metadata-eval 93.6%

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

   14. mul-1-neg 93.6%

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

   15. neg-mul-1 93.6%

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

   16. mul-1-neg 93.6%

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

   17. +-commutative 93.6%

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

   18. distribute-neg-in 93.6%

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

10. Simplified 93.6%

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

11. Taylor expanded in z around 0 94.1%

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

    1. associate-*l/ 94.0%

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

    2. associate-/l* 94.1%

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

13. Simplified 94.1%

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

Alternative 11: 95.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.5%

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

3. Simplified 95.3%

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

5. Taylor expanded in z around 0 93.9%

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

6. Taylor expanded in z around 0 93.7%

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

   1. associate-*r/ 93.6%

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

   2. associate-*r* 93.6%

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

   3. pow1/2 93.6%

      \[\leadsto \frac{\left(\left(\color{blue}{{\left(\pi \cdot 2\right)}^{0.5}} \cdot \sqrt{7.5}\right) \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot 263.3831869810514}{z} \]

   4. pow1/2 93.6%

      \[\leadsto \frac{\left(\left({\left(\pi \cdot 2\right)}^{0.5} \cdot \color{blue}{{7.5}^{0.5}}\right) \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot 263.3831869810514}{z} \]

   5. pow-prod-down 93.6%

      \[\leadsto \frac{\left(\color{blue}{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot 263.3831869810514}{z} \]

   6. distribute-neg-in 93.6%

      \[\leadsto \frac{\left({\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5} \cdot e^{\color{blue}{\left(-\left(1 - z\right)\right) + \left(-6.5\right)}}\right) \cdot 263.3831869810514}{z} \]

   7. metadata-eval 93.6%

      \[\leadsto \frac{\left({\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5} \cdot e^{\left(-\left(1 - z\right)\right) + \color{blue}{-6.5}}\right) \cdot 263.3831869810514}{z} \]

8. Applied egg-rr 93.6%

   \[\leadsto \color{blue}{\frac{\left({\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right) \cdot 263.3831869810514}{z}} \]
9. Step-by-step derivation

   1. associate-*r/ 93.7%

      \[\leadsto \color{blue}{\left({\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right) \cdot \frac{263.3831869810514}{z}} \]

   2. associate-*l* 93.6%

      \[\leadsto \color{blue}{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5} \cdot \left(e^{\left(-\left(1 - z\right)\right) + -6.5} \cdot \frac{263.3831869810514}{z}\right)} \]

   3. unpow1/2 93.6%

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

   4. associate-*l* 93.6%

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

   5. metadata-eval 93.6%

      \[\leadsto \sqrt{\pi \cdot \color{blue}{15}} \cdot \left(e^{\left(-\left(1 - z\right)\right) + -6.5} \cdot \frac{263.3831869810514}{z}\right) \]

   6. *-commutative 93.6%

      \[\leadsto \sqrt{\pi \cdot 15} \cdot \color{blue}{\left(\frac{263.3831869810514}{z} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)} \]

   7. metadata-eval 93.6%

      \[\leadsto \sqrt{\pi \cdot 15} \cdot \left(\frac{263.3831869810514}{z} \cdot e^{\left(-\left(1 - z\right)\right) + \color{blue}{\left(-6.5\right)}}\right) \]

   8. distribute-neg-in 93.6%

      \[\leadsto \sqrt{\pi \cdot 15} \cdot \left(\frac{263.3831869810514}{z} \cdot e^{\color{blue}{-\left(\left(1 - z\right) + 6.5\right)}}\right) \]

   9. neg-mul-1 93.6%

      \[\leadsto \sqrt{\pi \cdot 15} \cdot \left(\frac{263.3831869810514}{z} \cdot e^{\color{blue}{-1 \cdot \left(\left(1 - z\right) + 6.5\right)}}\right) \]

   10. +-commutative 93.6%

       \[\leadsto \sqrt{\pi \cdot 15} \cdot \left(\frac{263.3831869810514}{z} \cdot e^{-1 \cdot \color{blue}{\left(6.5 + \left(1 - z\right)\right)}}\right) \]

   11. sub-neg 93.6%

       \[\leadsto \sqrt{\pi \cdot 15} \cdot \left(\frac{263.3831869810514}{z} \cdot e^{-1 \cdot \left(6.5 + \color{blue}{\left(1 + \left(-z\right)\right)}\right)}\right) \]

   12. associate-+r+ 93.6%

       \[\leadsto \sqrt{\pi \cdot 15} \cdot \left(\frac{263.3831869810514}{z} \cdot e^{-1 \cdot \color{blue}{\left(\left(6.5 + 1\right) + \left(-z\right)\right)}}\right) \]

   13. metadata-eval 93.6%

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

   14. mul-1-neg 93.6%

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

   15. neg-mul-1 93.6%

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

   16. mul-1-neg 93.6%

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

   17. +-commutative 93.6%

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

   18. distribute-neg-in 93.6%

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

10. Simplified 93.6%

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

11. Taylor expanded in z around inf 93.9%

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

Alternative 12: 95.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.5%

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

3. Simplified 95.3%

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

5. Taylor expanded in z around 0 93.9%

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

6. Taylor expanded in z around 0 93.7%

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

   1. associate-*r/ 93.6%

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

   2. associate-*r* 93.6%

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

   3. pow1/2 93.6%

      \[\leadsto \frac{\left(\left(\color{blue}{{\left(\pi \cdot 2\right)}^{0.5}} \cdot \sqrt{7.5}\right) \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot 263.3831869810514}{z} \]

   4. pow1/2 93.6%

      \[\leadsto \frac{\left(\left({\left(\pi \cdot 2\right)}^{0.5} \cdot \color{blue}{{7.5}^{0.5}}\right) \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot 263.3831869810514}{z} \]

   5. pow-prod-down 93.6%

      \[\leadsto \frac{\left(\color{blue}{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5}} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot 263.3831869810514}{z} \]

   6. distribute-neg-in 93.6%

      \[\leadsto \frac{\left({\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5} \cdot e^{\color{blue}{\left(-\left(1 - z\right)\right) + \left(-6.5\right)}}\right) \cdot 263.3831869810514}{z} \]

   7. metadata-eval 93.6%

      \[\leadsto \frac{\left({\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5} \cdot e^{\left(-\left(1 - z\right)\right) + \color{blue}{-6.5}}\right) \cdot 263.3831869810514}{z} \]

8. Applied egg-rr 93.6%

   \[\leadsto \color{blue}{\frac{\left({\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right) \cdot 263.3831869810514}{z}} \]
9. Step-by-step derivation

   1. associate-*r/ 93.7%

      \[\leadsto \color{blue}{\left({\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right) \cdot \frac{263.3831869810514}{z}} \]

   2. associate-*l* 93.6%

      \[\leadsto \color{blue}{{\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5} \cdot \left(e^{\left(-\left(1 - z\right)\right) + -6.5} \cdot \frac{263.3831869810514}{z}\right)} \]

   3. unpow1/2 93.6%

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

   4. associate-*l* 93.6%

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

   5. metadata-eval 93.6%

      \[\leadsto \sqrt{\pi \cdot \color{blue}{15}} \cdot \left(e^{\left(-\left(1 - z\right)\right) + -6.5} \cdot \frac{263.3831869810514}{z}\right) \]

   6. *-commutative 93.6%

      \[\leadsto \sqrt{\pi \cdot 15} \cdot \color{blue}{\left(\frac{263.3831869810514}{z} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)} \]

   7. metadata-eval 93.6%

      \[\leadsto \sqrt{\pi \cdot 15} \cdot \left(\frac{263.3831869810514}{z} \cdot e^{\left(-\left(1 - z\right)\right) + \color{blue}{\left(-6.5\right)}}\right) \]

   8. distribute-neg-in 93.6%

      \[\leadsto \sqrt{\pi \cdot 15} \cdot \left(\frac{263.3831869810514}{z} \cdot e^{\color{blue}{-\left(\left(1 - z\right) + 6.5\right)}}\right) \]

   9. neg-mul-1 93.6%

      \[\leadsto \sqrt{\pi \cdot 15} \cdot \left(\frac{263.3831869810514}{z} \cdot e^{\color{blue}{-1 \cdot \left(\left(1 - z\right) + 6.5\right)}}\right) \]

   10. +-commutative 93.6%

       \[\leadsto \sqrt{\pi \cdot 15} \cdot \left(\frac{263.3831869810514}{z} \cdot e^{-1 \cdot \color{blue}{\left(6.5 + \left(1 - z\right)\right)}}\right) \]

   11. sub-neg 93.6%

       \[\leadsto \sqrt{\pi \cdot 15} \cdot \left(\frac{263.3831869810514}{z} \cdot e^{-1 \cdot \left(6.5 + \color{blue}{\left(1 + \left(-z\right)\right)}\right)}\right) \]

   12. associate-+r+ 93.6%

       \[\leadsto \sqrt{\pi \cdot 15} \cdot \left(\frac{263.3831869810514}{z} \cdot e^{-1 \cdot \color{blue}{\left(\left(6.5 + 1\right) + \left(-z\right)\right)}}\right) \]

   13. metadata-eval 93.6%

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

   14. mul-1-neg 93.6%

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

   15. neg-mul-1 93.6%

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

   16. mul-1-neg 93.6%

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

   17. +-commutative 93.6%

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

   18. distribute-neg-in 93.6%

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

10. Simplified 93.6%

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

11. Taylor expanded in z around 0 93.9%

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

Reproduce

herbie shell --seed 2024131 
(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))))))