Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.3% → 99.2%

Time: 54.6s

Alternatives: 15

Speedup: 1.2×

• could not determine a ground truth

  1. z = -6.877728229032268e+258

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: 96.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{\sin \left(\pi \cdot z\right)}\\ t_1 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq -2000:\\ \;\;\;\;\left(t\_1 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \left(t\_0 \cdot \left(0.9999999999998099 - \frac{24.458333333348836}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(\frac{-176.6150291621406}{4 - z} - \left(\left(\left(\frac{676.5203681218851}{z + -1} + \left(\frac{-1259.1392167224028}{z - 2} + \frac{771.3234287776531}{z - 3}\right)\right) - 0.9999999999998099\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - 7} - \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} - \left(\frac{12.507343278686905}{z - 5} - \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\right)\right) \cdot \left(t\_1 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ PI (sin (* PI z)))) (t_1 (sqrt (* PI 2.0))))
   (if (<= z -2000.0)
     (*
      (* t_1 (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp -7.5)))
      (* t_0 (- 0.9999999999998099 (/ 24.458333333348836 z))))
     (*
      t_0
      (*
       (-
        (/ -176.6150291621406 (- 4.0 z))
        (+
         (-
          (+
           (/ 676.5203681218851 (+ z -1.0))
           (+
            (/ -1259.1392167224028 (- z 2.0))
            (/ 771.3234287776531 (- z 3.0))))
          0.9999999999998099)
         (-
          (/ 9.984369578019572e-6 (- z 7.0))
          (-
           (/ 1.5056327351493116e-7 (- 8.0 z))
           (-
            (/ 12.507343278686905 (- z 5.0))
            (/ -0.13857109526572012 (- 6.0 z)))))))
       (* t_1 (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5)))))))))

C

double code(double z) {
	double t_0 = ((double) M_PI) / sin((((double) M_PI) * z));
	double t_1 = sqrt((((double) M_PI) * 2.0));
	double tmp;
	if (z <= -2000.0) {
		tmp = (t_1 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)));
	} else {
		tmp = t_0 * (((-176.6150291621406 / (4.0 - z)) - ((((676.5203681218851 / (z + -1.0)) + ((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0)))) - 0.9999999999998099) + ((9.984369578019572e-6 / (z - 7.0)) - ((1.5056327351493116e-7 / (8.0 - z)) - ((12.507343278686905 / (z - 5.0)) - (-0.13857109526572012 / (6.0 - z))))))) * (t_1 * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5)))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.PI / Math.sin((Math.PI * z));
	double t_1 = Math.sqrt((Math.PI * 2.0));
	double tmp;
	if (z <= -2000.0) {
		tmp = (t_1 * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)));
	} else {
		tmp = t_0 * (((-176.6150291621406 / (4.0 - z)) - ((((676.5203681218851 / (z + -1.0)) + ((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0)))) - 0.9999999999998099) + ((9.984369578019572e-6 / (z - 7.0)) - ((1.5056327351493116e-7 / (8.0 - z)) - ((12.507343278686905 / (z - 5.0)) - (-0.13857109526572012 / (6.0 - z))))))) * (t_1 * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5)))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.pi / math.sin((math.pi * z))
	t_1 = math.sqrt((math.pi * 2.0))
	tmp = 0
	if z <= -2000.0:
		tmp = (t_1 * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)))
	else:
		tmp = t_0 * (((-176.6150291621406 / (4.0 - z)) - ((((676.5203681218851 / (z + -1.0)) + ((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0)))) - 0.9999999999998099) + ((9.984369578019572e-6 / (z - 7.0)) - ((1.5056327351493116e-7 / (8.0 - z)) - ((12.507343278686905 / (z - 5.0)) - (-0.13857109526572012 / (6.0 - z))))))) * (t_1 * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5)))))
	return tmp

Julia

function code(z)
	t_0 = Float64(pi / sin(Float64(pi * z)))
	t_1 = sqrt(Float64(pi * 2.0))
	tmp = 0.0
	if (z <= -2000.0)
		tmp = Float64(Float64(t_1 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(-7.5))) * Float64(t_0 * Float64(0.9999999999998099 - Float64(24.458333333348836 / z))));
	else
		tmp = Float64(t_0 * Float64(Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) - Float64(Float64(Float64(Float64(676.5203681218851 / Float64(z + -1.0)) + Float64(Float64(-1259.1392167224028 / Float64(z - 2.0)) + Float64(771.3234287776531 / Float64(z - 3.0)))) - 0.9999999999998099) + Float64(Float64(9.984369578019572e-6 / Float64(z - 7.0)) - Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) - Float64(Float64(12.507343278686905 / Float64(z - 5.0)) - Float64(-0.13857109526572012 / Float64(6.0 - z))))))) * Float64(t_1 * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = pi / sin((pi * z));
	t_1 = sqrt((pi * 2.0));
	tmp = 0.0;
	if (z <= -2000.0)
		tmp = (t_1 * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)));
	else
		tmp = t_0 * (((-176.6150291621406 / (4.0 - z)) - ((((676.5203681218851 / (z + -1.0)) + ((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0)))) - 0.9999999999998099) + ((9.984369578019572e-6 / (z - 7.0)) - ((1.5056327351493116e-7 / (8.0 - z)) - ((12.507343278686905 / (z - 5.0)) - (-0.13857109526572012 / (6.0 - z))))))) * (t_1 * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -2000.0], N[(N[(t$95$1 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(0.9999999999998099 - N[(24.458333333348836 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(676.5203681218851 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1259.1392167224028 / N[(z - 2.0), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.9999999999998099), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(z - 7.0), $MachinePrecision]), $MachinePrecision] - N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] - N[(N[(12.507343278686905 / N[(z - 5.0), $MachinePrecision]), $MachinePrecision] - N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < -2e3

   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 inf 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(0.9999999999998099 - 24.458333333348836 \cdot \frac{1}{z}\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 0.0%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(0.9999999999998099 - \color{blue}{\frac{24.458333333348836 \cdot 1}{z}}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

      2. metadata-eval 0.0%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(0.9999999999998099 - \frac{\color{blue}{24.458333333348836}}{z}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   7. Simplified 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(0.9999999999998099 - \frac{24.458333333348836}{z}\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. Taylor expanded in z around 0 100.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{e^{-7.5}}\right)\right) \cdot \left(\left(0.9999999999998099 - \frac{24.458333333348836}{z}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   if -2e3 < z

   1. Initial program 97.3%

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

   3. Simplified 97.1%

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

   6. Applied egg-rr 97.2%

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

   8. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right)} \]

   9. Taylor expanded in z around inf 99.2%

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

       1. exp-to-pow 99.2%

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

       2. sub-neg 99.2%

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

       3. metadata-eval 99.2%

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

       4. +-commutative 99.2%

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

   11. Simplified 99.2%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2000:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(0.9999999999998099 - \frac{24.458333333348836}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} - \left(\left(\left(\frac{676.5203681218851}{z + -1} + \left(\frac{-1259.1392167224028}{z - 2} + \frac{771.3234287776531}{z - 3}\right)\right) - 0.9999999999998099\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - 7} - \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} - \left(\frac{12.507343278686905}{z - 5} - \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\right)\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\log \left(\frac{-176.6150291621406}{4 - z} - \left(\left(\left(\frac{676.5203681218851}{z + -1} + \left(\frac{-1259.1392167224028}{z - 2} + \frac{771.3234287776531}{z - 3}\right)\right) - 0.9999999999998099\right) + \left(\left(\frac{12.507343278686905}{z - 5} - \frac{-0.13857109526572012}{6 - z}\right) - \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} - \frac{9.984369578019572 \cdot 10^{-6}}{z - 7}\right)\right)\right)\right) + \left(\log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) + \left(-1 + \left(z + -6.5\right)\right)\right)} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* PI z)))
  (exp
   (+
    (log
     (-
      (/ -176.6150291621406 (- 4.0 z))
      (+
       (-
        (+
         (/ 676.5203681218851 (+ z -1.0))
         (+ (/ -1259.1392167224028 (- z 2.0)) (/ 771.3234287776531 (- z 3.0))))
        0.9999999999998099)
       (-
        (- (/ 12.507343278686905 (- z 5.0)) (/ -0.13857109526572012 (- 6.0 z)))
        (-
         (/ 1.5056327351493116e-7 (- 8.0 z))
         (/ 9.984369578019572e-6 (- z 7.0)))))))
    (+
     (log (* (sqrt (* PI 2.0)) (pow (- 7.5 z) (- 0.5 z))))
     (+ -1.0 (+ z -6.5)))))))

C

double code(double z) {
	return (((double) M_PI) / sin((((double) M_PI) * z))) * exp((log(((-176.6150291621406 / (4.0 - z)) - ((((676.5203681218851 / (z + -1.0)) + ((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0)))) - 0.9999999999998099) + (((12.507343278686905 / (z - 5.0)) - (-0.13857109526572012 / (6.0 - z))) - ((1.5056327351493116e-7 / (8.0 - z)) - (9.984369578019572e-6 / (z - 7.0))))))) + (log((sqrt((((double) M_PI) * 2.0)) * pow((7.5 - z), (0.5 - z)))) + (-1.0 + (z + -6.5)))));
}

Java

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

Python

def code(z):
	return (math.pi / math.sin((math.pi * z))) * math.exp((math.log(((-176.6150291621406 / (4.0 - z)) - ((((676.5203681218851 / (z + -1.0)) + ((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0)))) - 0.9999999999998099) + (((12.507343278686905 / (z - 5.0)) - (-0.13857109526572012 / (6.0 - z))) - ((1.5056327351493116e-7 / (8.0 - z)) - (9.984369578019572e-6 / (z - 7.0))))))) + (math.log((math.sqrt((math.pi * 2.0)) * math.pow((7.5 - z), (0.5 - z)))) + (-1.0 + (z + -6.5)))))

Julia

function code(z)
	return Float64(Float64(pi / sin(Float64(pi * z))) * exp(Float64(log(Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) - Float64(Float64(Float64(Float64(676.5203681218851 / Float64(z + -1.0)) + Float64(Float64(-1259.1392167224028 / Float64(z - 2.0)) + Float64(771.3234287776531 / Float64(z - 3.0)))) - 0.9999999999998099) + Float64(Float64(Float64(12.507343278686905 / Float64(z - 5.0)) - Float64(-0.13857109526572012 / Float64(6.0 - z))) - Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) - Float64(9.984369578019572e-6 / Float64(z - 7.0))))))) + Float64(log(Float64(sqrt(Float64(pi * 2.0)) * (Float64(7.5 - z) ^ Float64(0.5 - z)))) + Float64(-1.0 + Float64(z + -6.5))))))
end

MATLAB

function tmp = code(z)
	tmp = (pi / sin((pi * z))) * exp((log(((-176.6150291621406 / (4.0 - z)) - ((((676.5203681218851 / (z + -1.0)) + ((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0)))) - 0.9999999999998099) + (((12.507343278686905 / (z - 5.0)) - (-0.13857109526572012 / (6.0 - z))) - ((1.5056327351493116e-7 / (8.0 - z)) - (9.984369578019572e-6 / (z - 7.0))))))) + (log((sqrt((pi * 2.0)) * ((7.5 - z) ^ (0.5 - z)))) + (-1.0 + (z + -6.5)))));
end

Wolfram

code[z_] := N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Log[N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(676.5203681218851 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1259.1392167224028 / N[(z - 2.0), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.9999999999998099), $MachinePrecision] + N[(N[(N[(12.507343278686905 / N[(z - 5.0), $MachinePrecision]), $MachinePrecision] - N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] - N[(9.984369578019572e-6 / N[(z - 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Log[N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-1.0 + N[(z + -6.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.4%

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

3. Simplified 95.2%

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

6. Applied egg-rr 95.3%

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

8. Simplified 97.3%

   \[\leadsto \color{blue}{\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right)} \]
9. Step-by-step derivation

   1. add-cbrt-cube 96.5%

      \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \color{blue}{\sqrt[3]{\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right) \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

   2. associate-+r+ 95.9%

      \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\left(\left(0.9999999999998099 + \color{blue}{\left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)}\right) \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right) \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

   3. associate-+r+ 96.5%

      \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \color{blue}{\left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)}\right)\right) \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

   4. associate-+r+ 96.5%

      \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) \cdot \left(0.9999999999998099 + \color{blue}{\left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)}\right)}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

10. Applied egg-rr 96.5%

    \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \color{blue}{\sqrt[3]{\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)}}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]
11. Step-by-step derivation

    1. associate-*l* 96.5%

       \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\color{blue}{\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

    2. associate-+r+ 95.4%

       \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\color{blue}{\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right)\right) + \frac{-1259.1392167224028}{2 - z}\right)} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

    3. +-commutative 95.4%

       \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\left(\left(0.9999999999998099 + \color{blue}{\left(\frac{771.3234287776531}{3 - z} + \frac{676.5203681218851}{1 - z}\right)}\right) + \frac{-1259.1392167224028}{2 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

12. Simplified 95.4%

    \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \color{blue}{\sqrt[3]{\left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{676.5203681218851}{1 - z}\right)\right) + \frac{-1259.1392167224028}{2 - z}\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{676.5203681218851}{1 - z}\right)\right) + \frac{-1259.1392167224028}{2 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{676.5203681218851}{1 - z}\right)\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)}}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

14. Applied egg-rr 98.4%

    \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \color{blue}{e^{\log \left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right) + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)\right) + \left(\log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) + \left(-1 + \left(z + -6.5\right)\right)\right)}} \]

15. Final simplification 98.4%

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\log \left(\frac{-176.6150291621406}{4 - z} - \left(\left(\left(\frac{676.5203681218851}{z + -1} + \left(\frac{-1259.1392167224028}{z - 2} + \frac{771.3234287776531}{z - 3}\right)\right) - 0.9999999999998099\right) + \left(\left(\frac{12.507343278686905}{z - 5} - \frac{-0.13857109526572012}{6 - z}\right) - \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} - \frac{9.984369578019572 \cdot 10^{-6}}{z - 7}\right)\right)\right)\right) + \left(\log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) + \left(-1 + \left(z + -6.5\right)\right)\right)} \]
16. Add Preprocessing

Alternative 3: 98.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 95.2%

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

6. Applied egg-rr 95.3%

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

8. Simplified 97.3%

   \[\leadsto \color{blue}{\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right)} \]
9. Step-by-step derivation

   1. add-cbrt-cube 96.5%

      \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \color{blue}{\sqrt[3]{\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right) \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

   2. associate-+r+ 95.9%

      \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\left(\left(0.9999999999998099 + \color{blue}{\left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)}\right) \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right) \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

   3. associate-+r+ 96.5%

      \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \color{blue}{\left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)}\right)\right) \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

   4. associate-+r+ 96.5%

      \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) \cdot \left(0.9999999999998099 + \color{blue}{\left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)}\right)}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

10. Applied egg-rr 96.5%

    \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \color{blue}{\sqrt[3]{\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)}}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]
11. Step-by-step derivation

    1. associate-*l* 96.5%

       \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\color{blue}{\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

    2. associate-+r+ 95.4%

       \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\color{blue}{\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right)\right) + \frac{-1259.1392167224028}{2 - z}\right)} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

    3. +-commutative 95.4%

       \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\left(\left(0.9999999999998099 + \color{blue}{\left(\frac{771.3234287776531}{3 - z} + \frac{676.5203681218851}{1 - z}\right)}\right) + \frac{-1259.1392167224028}{2 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

12. Simplified 95.4%

    \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \color{blue}{\sqrt[3]{\left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{676.5203681218851}{1 - z}\right)\right) + \frac{-1259.1392167224028}{2 - z}\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{676.5203681218851}{1 - z}\right)\right) + \frac{-1259.1392167224028}{2 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{676.5203681218851}{1 - z}\right)\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)}}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

14. Applied egg-rr 98.4%

    \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \color{blue}{e^{\log \left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right) + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)\right) + \left(\log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) + \left(-1 + \left(z + -6.5\right)\right)\right)}} \]

16. Applied egg-rr 47.4%

    \[\leadsto \color{blue}{e^{\left(\log \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right) + \left(-1 + \left(z + -6.5\right)\right)\right) + \log \left(\frac{\pi}{\sin \left(z \cdot \pi\right)}\right)}} \]

18. Simplified 98.4%

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

19. Final simplification 98.4%

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

Alternative 4: 98.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{\sin \left(\pi \cdot z\right)}\\ t_1 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq -2000:\\ \;\;\;\;\left(t\_1 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \left(t\_0 \cdot \left(0.9999999999998099 - \frac{24.458333333348836}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(t\_1 \cdot e^{z + -7.5}\right)\right) \cdot \left(t\_0 \cdot \left(\frac{-176.6150291621406}{4 - z} - \left(\left(\left(\frac{676.5203681218851}{z + -1} + \left(\frac{-1259.1392167224028}{z - 2} + \frac{771.3234287776531}{z - 3}\right)\right) - 0.9999999999998099\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - 7} - \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} - \left(\frac{12.507343278686905}{z - 5} - \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ PI (sin (* PI z)))) (t_1 (sqrt (* PI 2.0))))
   (if (<= z -2000.0)
     (*
      (* t_1 (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp -7.5)))
      (* t_0 (- 0.9999999999998099 (/ 24.458333333348836 z))))
     (*
      (* (pow (- 7.5 z) (- 0.5 z)) (* t_1 (exp (+ z -7.5))))
      (*
       t_0
       (-
        (/ -176.6150291621406 (- 4.0 z))
        (+
         (-
          (+
           (/ 676.5203681218851 (+ z -1.0))
           (+
            (/ -1259.1392167224028 (- z 2.0))
            (/ 771.3234287776531 (- z 3.0))))
          0.9999999999998099)
         (-
          (/ 9.984369578019572e-6 (- z 7.0))
          (-
           (/ 1.5056327351493116e-7 (- 8.0 z))
           (-
            (/ 12.507343278686905 (- z 5.0))
            (/ -0.13857109526572012 (- 6.0 z))))))))))))

C

double code(double z) {
	double t_0 = ((double) M_PI) / sin((((double) M_PI) * z));
	double t_1 = sqrt((((double) M_PI) * 2.0));
	double tmp;
	if (z <= -2000.0) {
		tmp = (t_1 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)));
	} else {
		tmp = (pow((7.5 - z), (0.5 - z)) * (t_1 * exp((z + -7.5)))) * (t_0 * ((-176.6150291621406 / (4.0 - z)) - ((((676.5203681218851 / (z + -1.0)) + ((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0)))) - 0.9999999999998099) + ((9.984369578019572e-6 / (z - 7.0)) - ((1.5056327351493116e-7 / (8.0 - z)) - ((12.507343278686905 / (z - 5.0)) - (-0.13857109526572012 / (6.0 - z))))))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.PI / Math.sin((Math.PI * z));
	double t_1 = Math.sqrt((Math.PI * 2.0));
	double tmp;
	if (z <= -2000.0) {
		tmp = (t_1 * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)));
	} else {
		tmp = (Math.pow((7.5 - z), (0.5 - z)) * (t_1 * Math.exp((z + -7.5)))) * (t_0 * ((-176.6150291621406 / (4.0 - z)) - ((((676.5203681218851 / (z + -1.0)) + ((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0)))) - 0.9999999999998099) + ((9.984369578019572e-6 / (z - 7.0)) - ((1.5056327351493116e-7 / (8.0 - z)) - ((12.507343278686905 / (z - 5.0)) - (-0.13857109526572012 / (6.0 - z))))))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.pi / math.sin((math.pi * z))
	t_1 = math.sqrt((math.pi * 2.0))
	tmp = 0
	if z <= -2000.0:
		tmp = (t_1 * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)))
	else:
		tmp = (math.pow((7.5 - z), (0.5 - z)) * (t_1 * math.exp((z + -7.5)))) * (t_0 * ((-176.6150291621406 / (4.0 - z)) - ((((676.5203681218851 / (z + -1.0)) + ((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0)))) - 0.9999999999998099) + ((9.984369578019572e-6 / (z - 7.0)) - ((1.5056327351493116e-7 / (8.0 - z)) - ((12.507343278686905 / (z - 5.0)) - (-0.13857109526572012 / (6.0 - z))))))))
	return tmp

Julia

function code(z)
	t_0 = Float64(pi / sin(Float64(pi * z)))
	t_1 = sqrt(Float64(pi * 2.0))
	tmp = 0.0
	if (z <= -2000.0)
		tmp = Float64(Float64(t_1 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(-7.5))) * Float64(t_0 * Float64(0.9999999999998099 - Float64(24.458333333348836 / z))));
	else
		tmp = Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * Float64(t_1 * exp(Float64(z + -7.5)))) * Float64(t_0 * Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) - Float64(Float64(Float64(Float64(676.5203681218851 / Float64(z + -1.0)) + Float64(Float64(-1259.1392167224028 / Float64(z - 2.0)) + Float64(771.3234287776531 / Float64(z - 3.0)))) - 0.9999999999998099) + Float64(Float64(9.984369578019572e-6 / Float64(z - 7.0)) - Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) - Float64(Float64(12.507343278686905 / Float64(z - 5.0)) - Float64(-0.13857109526572012 / Float64(6.0 - z)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = pi / sin((pi * z));
	t_1 = sqrt((pi * 2.0));
	tmp = 0.0;
	if (z <= -2000.0)
		tmp = (t_1 * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)));
	else
		tmp = (((7.5 - z) ^ (0.5 - z)) * (t_1 * exp((z + -7.5)))) * (t_0 * ((-176.6150291621406 / (4.0 - z)) - ((((676.5203681218851 / (z + -1.0)) + ((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0)))) - 0.9999999999998099) + ((9.984369578019572e-6 / (z - 7.0)) - ((1.5056327351493116e-7 / (8.0 - z)) - ((12.507343278686905 / (z - 5.0)) - (-0.13857109526572012 / (6.0 - z))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -2000.0], N[(N[(t$95$1 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(0.9999999999998099 - N[(24.458333333348836 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(676.5203681218851 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1259.1392167224028 / N[(z - 2.0), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.9999999999998099), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(z - 7.0), $MachinePrecision]), $MachinePrecision] - N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] - N[(N[(12.507343278686905 / N[(z - 5.0), $MachinePrecision]), $MachinePrecision] - N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < -2e3

   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 inf 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(0.9999999999998099 - 24.458333333348836 \cdot \frac{1}{z}\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 0.0%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(0.9999999999998099 - \color{blue}{\frac{24.458333333348836 \cdot 1}{z}}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

      2. metadata-eval 0.0%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(0.9999999999998099 - \frac{\color{blue}{24.458333333348836}}{z}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   7. Simplified 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(0.9999999999998099 - \frac{24.458333333348836}{z}\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. Taylor expanded in z around 0 100.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{e^{-7.5}}\right)\right) \cdot \left(\left(0.9999999999998099 - \frac{24.458333333348836}{z}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   if -2e3 < z

   1. Initial program 97.3%

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

   3. Simplified 97.3%

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

   6. Applied egg-rr 97.3%

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

   8. Simplified 98.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2000:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(0.9999999999998099 - \frac{24.458333333348836}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\frac{-176.6150291621406}{4 - z} - \left(\left(\left(\frac{676.5203681218851}{z + -1} + \left(\frac{-1259.1392167224028}{z - 2} + \frac{771.3234287776531}{z - 3}\right)\right) - 0.9999999999998099\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - 7} - \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} - \left(\frac{12.507343278686905}{z - 5} - \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{\sin \left(\pi \cdot z\right)}\\ t_1 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq -2000:\\ \;\;\;\;\left(t\_1 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \left(t\_0 \cdot \left(0.9999999999998099 - \frac{24.458333333348836}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(t\_1 \cdot e^{z + -7.5}\right)\right)\right) \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} - \frac{9.984369578019572 \cdot 10^{-6}}{z - 7}\right) - \left(\left(\frac{676.5203681218851}{z + -1} + \frac{-1259.1392167224028}{z - 2}\right) + \left(\frac{771.3234287776531}{z - 3} - 0.9999999999998099\right)\right)\right) - \left(\frac{-176.6150291621406}{z - 4} + \left(\frac{12.507343278686905}{z - 5} - \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ PI (sin (* PI z)))) (t_1 (sqrt (* PI 2.0))))
   (if (<= z -2000.0)
     (*
      (* t_1 (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp -7.5)))
      (* t_0 (- 0.9999999999998099 (/ 24.458333333348836 z))))
     (*
      (* t_0 (* (pow (- 7.5 z) (- 0.5 z)) (* t_1 (exp (+ z -7.5)))))
      (-
       (-
        (-
         (/ 1.5056327351493116e-7 (- 8.0 z))
         (/ 9.984369578019572e-6 (- z 7.0)))
        (+
         (+ (/ 676.5203681218851 (+ z -1.0)) (/ -1259.1392167224028 (- z 2.0)))
         (- (/ 771.3234287776531 (- z 3.0)) 0.9999999999998099)))
       (+
        (/ -176.6150291621406 (- z 4.0))
        (-
         (/ 12.507343278686905 (- z 5.0))
         (/ -0.13857109526572012 (- 6.0 z)))))))))

C

double code(double z) {
	double t_0 = ((double) M_PI) / sin((((double) M_PI) * z));
	double t_1 = sqrt((((double) M_PI) * 2.0));
	double tmp;
	if (z <= -2000.0) {
		tmp = (t_1 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)));
	} else {
		tmp = (t_0 * (pow((7.5 - z), (0.5 - z)) * (t_1 * exp((z + -7.5))))) * ((((1.5056327351493116e-7 / (8.0 - z)) - (9.984369578019572e-6 / (z - 7.0))) - (((676.5203681218851 / (z + -1.0)) + (-1259.1392167224028 / (z - 2.0))) + ((771.3234287776531 / (z - 3.0)) - 0.9999999999998099))) - ((-176.6150291621406 / (z - 4.0)) + ((12.507343278686905 / (z - 5.0)) - (-0.13857109526572012 / (6.0 - z)))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.PI / Math.sin((Math.PI * z));
	double t_1 = Math.sqrt((Math.PI * 2.0));
	double tmp;
	if (z <= -2000.0) {
		tmp = (t_1 * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)));
	} else {
		tmp = (t_0 * (Math.pow((7.5 - z), (0.5 - z)) * (t_1 * Math.exp((z + -7.5))))) * ((((1.5056327351493116e-7 / (8.0 - z)) - (9.984369578019572e-6 / (z - 7.0))) - (((676.5203681218851 / (z + -1.0)) + (-1259.1392167224028 / (z - 2.0))) + ((771.3234287776531 / (z - 3.0)) - 0.9999999999998099))) - ((-176.6150291621406 / (z - 4.0)) + ((12.507343278686905 / (z - 5.0)) - (-0.13857109526572012 / (6.0 - z)))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.pi / math.sin((math.pi * z))
	t_1 = math.sqrt((math.pi * 2.0))
	tmp = 0
	if z <= -2000.0:
		tmp = (t_1 * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)))
	else:
		tmp = (t_0 * (math.pow((7.5 - z), (0.5 - z)) * (t_1 * math.exp((z + -7.5))))) * ((((1.5056327351493116e-7 / (8.0 - z)) - (9.984369578019572e-6 / (z - 7.0))) - (((676.5203681218851 / (z + -1.0)) + (-1259.1392167224028 / (z - 2.0))) + ((771.3234287776531 / (z - 3.0)) - 0.9999999999998099))) - ((-176.6150291621406 / (z - 4.0)) + ((12.507343278686905 / (z - 5.0)) - (-0.13857109526572012 / (6.0 - z)))))
	return tmp

Julia

function code(z)
	t_0 = Float64(pi / sin(Float64(pi * z)))
	t_1 = sqrt(Float64(pi * 2.0))
	tmp = 0.0
	if (z <= -2000.0)
		tmp = Float64(Float64(t_1 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(-7.5))) * Float64(t_0 * Float64(0.9999999999998099 - Float64(24.458333333348836 / z))));
	else
		tmp = Float64(Float64(t_0 * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * Float64(t_1 * exp(Float64(z + -7.5))))) * Float64(Float64(Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) - Float64(9.984369578019572e-6 / Float64(z - 7.0))) - Float64(Float64(Float64(676.5203681218851 / Float64(z + -1.0)) + Float64(-1259.1392167224028 / Float64(z - 2.0))) + Float64(Float64(771.3234287776531 / Float64(z - 3.0)) - 0.9999999999998099))) - Float64(Float64(-176.6150291621406 / Float64(z - 4.0)) + Float64(Float64(12.507343278686905 / Float64(z - 5.0)) - Float64(-0.13857109526572012 / Float64(6.0 - z))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = pi / sin((pi * z));
	t_1 = sqrt((pi * 2.0));
	tmp = 0.0;
	if (z <= -2000.0)
		tmp = (t_1 * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)));
	else
		tmp = (t_0 * (((7.5 - z) ^ (0.5 - z)) * (t_1 * exp((z + -7.5))))) * ((((1.5056327351493116e-7 / (8.0 - z)) - (9.984369578019572e-6 / (z - 7.0))) - (((676.5203681218851 / (z + -1.0)) + (-1259.1392167224028 / (z - 2.0))) + ((771.3234287776531 / (z - 3.0)) - 0.9999999999998099))) - ((-176.6150291621406 / (z - 4.0)) + ((12.507343278686905 / (z - 5.0)) - (-0.13857109526572012 / (6.0 - z)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -2000.0], N[(N[(t$95$1 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(0.9999999999998099 - N[(24.458333333348836 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] - N[(9.984369578019572e-6 / N[(z - 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(676.5203681218851 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(z - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision] - 0.9999999999998099), $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[(6.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < -2e3

   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 inf 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(0.9999999999998099 - 24.458333333348836 \cdot \frac{1}{z}\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 0.0%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(0.9999999999998099 - \color{blue}{\frac{24.458333333348836 \cdot 1}{z}}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

      2. metadata-eval 0.0%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(0.9999999999998099 - \frac{\color{blue}{24.458333333348836}}{z}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   7. Simplified 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(0.9999999999998099 - \frac{24.458333333348836}{z}\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. Taylor expanded in z around 0 100.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{e^{-7.5}}\right)\right) \cdot \left(\left(0.9999999999998099 - \frac{24.458333333348836}{z}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   if -2e3 < z

   1. Initial program 97.3%

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

   3. Simplified 97.3%

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

   6. Applied egg-rr 97.3%

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

      1. unpow1 97.3%

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

      2. associate-*r* 97.6%

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

   8. Simplified 97.6%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2000:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(0.9999999999998099 - \frac{24.458333333348836}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{z + -7.5}\right)\right)\right) \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} - \frac{9.984369578019572 \cdot 10^{-6}}{z - 7}\right) - \left(\left(\frac{676.5203681218851}{z + -1} + \frac{-1259.1392167224028}{z - 2}\right) + \left(\frac{771.3234287776531}{z - 3} - 0.9999999999998099\right)\right)\right) - \left(\frac{-176.6150291621406}{z - 4} + \left(\frac{12.507343278686905}{z - 5} - \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 95.2%

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

6. Applied egg-rr 95.3%

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

8. Simplified 97.3%

   \[\leadsto \color{blue}{\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right)} \]

9. Taylor expanded in z around inf 97.3%

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

    1. exp-to-pow 97.3%

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

    2. sub-neg 97.3%

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

    3. metadata-eval 97.3%

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

    4. +-commutative 97.3%

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

11. Simplified 97.3%

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

    1. add-exp-log 95.8%

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

    2. *-commutative 95.8%

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

    3. log-prod 95.8%

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

    4. add-log-exp 97.8%

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

    5. +-commutative 97.8%

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

    6. log-pow 97.8%

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

13. Applied egg-rr 97.8%

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

14. Final simplification 97.8%

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

Alternative 7: 98.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{\sin \left(\pi \cdot z\right)}\\ t_1 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq -0.58:\\ \;\;\;\;\left(t\_1 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \left(t\_0 \cdot \left(0.9999999999998099 - \frac{24.458333333348836}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \left(t\_1 \cdot \left({\left(7.5 + \left(\left(1 - z\right) + -1\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{-0.5 + \left(-6 + \left(z + -1\right)\right)}\right)\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ PI (sin (* PI z)))) (t_1 (sqrt (* PI 2.0))))
   (if (<= z -0.58)
     (*
      (* t_1 (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp -7.5)))
      (* t_0 (- 0.9999999999998099 (/ 24.458333333348836 z))))
     (*
      (*
       t_0
       (*
        t_1
        (*
         (pow (+ 7.5 (+ (- 1.0 z) -1.0)) (- (- 1.0 z) 0.5))
         (exp (+ -0.5 (+ -6.0 (+ z -1.0)))))))
      (+
       (+
        263.3831855358925
        (*
         z
         (+
          436.8961723502244
          (* z (+ 545.0353078134797 (* z 606.6766809125655))))))
       (+
        (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))
        (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0))))))))

C

double code(double z) {
	double t_0 = ((double) M_PI) / sin((((double) M_PI) * z));
	double t_1 = sqrt((((double) M_PI) * 2.0));
	double tmp;
	if (z <= -0.58) {
		tmp = (t_1 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)));
	} else {
		tmp = (t_0 * (t_1 * (pow((7.5 + ((1.0 - z) + -1.0)), ((1.0 - z) - 0.5)) * exp((-0.5 + (-6.0 + (z + -1.0))))))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))) + ((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.PI / Math.sin((Math.PI * z));
	double t_1 = Math.sqrt((Math.PI * 2.0));
	double tmp;
	if (z <= -0.58) {
		tmp = (t_1 * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)));
	} else {
		tmp = (t_0 * (t_1 * (Math.pow((7.5 + ((1.0 - z) + -1.0)), ((1.0 - z) - 0.5)) * Math.exp((-0.5 + (-6.0 + (z + -1.0))))))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))) + ((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.pi / math.sin((math.pi * z))
	t_1 = math.sqrt((math.pi * 2.0))
	tmp = 0
	if z <= -0.58:
		tmp = (t_1 * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)))
	else:
		tmp = (t_0 * (t_1 * (math.pow((7.5 + ((1.0 - z) + -1.0)), ((1.0 - z) - 0.5)) * math.exp((-0.5 + (-6.0 + (z + -1.0))))))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))))
	return tmp

Julia

function code(z)
	t_0 = Float64(pi / sin(Float64(pi * z)))
	t_1 = sqrt(Float64(pi * 2.0))
	tmp = 0.0
	if (z <= -0.58)
		tmp = Float64(Float64(t_1 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(-7.5))) * Float64(t_0 * Float64(0.9999999999998099 - Float64(24.458333333348836 / z))));
	else
		tmp = Float64(Float64(t_0 * Float64(t_1 * Float64((Float64(7.5 + Float64(Float64(1.0 - z) + -1.0)) ^ Float64(Float64(1.0 - z) - 0.5)) * exp(Float64(-0.5 + Float64(-6.0 + Float64(z + -1.0))))))) * Float64(Float64(263.3831855358925 + Float64(z * Float64(436.8961723502244 + Float64(z * Float64(545.0353078134797 + Float64(z * 606.6766809125655)))))) + 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 = pi / sin((pi * z));
	t_1 = sqrt((pi * 2.0));
	tmp = 0.0;
	if (z <= -0.58)
		tmp = (t_1 * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)));
	else
		tmp = (t_0 * (t_1 * (((7.5 + ((1.0 - z) + -1.0)) ^ ((1.0 - z) - 0.5)) * exp((-0.5 + (-6.0 + (z + -1.0))))))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))) + ((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[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -0.58], N[(N[(t$95$1 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(0.9999999999998099 - N[(24.458333333348836 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(t$95$1 * N[(N[Power[N[(7.5 + N[(N[(1.0 - z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-0.5 + N[(-6.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(263.3831855358925 + N[(z * N[(436.8961723502244 + N[(z * N[(545.0353078134797 + N[(z * 606.6766809125655), $MachinePrecision]), $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 < -0.57999999999999996

   1. Initial program 34.7%

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

   3. Simplified 34.8%

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

   5. Taylor expanded in z around inf 8.1%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(0.9999999999998099 - 24.458333333348836 \cdot \frac{1}{z}\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 8.1%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(0.9999999999998099 - \color{blue}{\frac{24.458333333348836 \cdot 1}{z}}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

      2. metadata-eval 8.1%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(0.9999999999998099 - \frac{\color{blue}{24.458333333348836}}{z}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   7. Simplified 8.1%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(0.9999999999998099 - \frac{24.458333333348836}{z}\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. Taylor expanded in z around 0 67.8%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{e^{-7.5}}\right)\right) \cdot \left(\left(0.9999999999998099 - \frac{24.458333333348836}{z}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   if -0.57999999999999996 < z

   1. Initial program 97.3%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in z around 0 98.5%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 98.5%

         \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + \color{blue}{z \cdot 606.6766809125655}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   7. Simplified 98.5%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.58:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(0.9999999999998099 - \frac{24.458333333348836}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 + \left(\left(1 - z\right) + -1\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{-0.5 + \left(-6 + \left(z + -1\right)\right)}\right)\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{\sin \left(\pi \cdot z\right)}\\ t_1 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq -0.54:\\ \;\;\;\;\left(t\_1 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \left(t\_0 \cdot \left(0.9999999999998099 - \frac{24.458333333348836}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \left(t\_1 \cdot \left({\left(7.5 + \left(\left(1 - z\right) + -1\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{-0.5 + \left(-6 + \left(z + -1\right)\right)}\right)\right)\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) + \left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ PI (sin (* PI z)))) (t_1 (sqrt (* PI 2.0))))
   (if (<= z -0.54)
     (*
      (* t_1 (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp -7.5)))
      (* t_0 (- 0.9999999999998099 (/ 24.458333333348836 z))))
     (*
      (*
       t_0
       (*
        t_1
        (*
         (pow (+ 7.5 (+ (- 1.0 z) -1.0)) (- (- 1.0 z) 0.5))
         (exp (+ -0.5 (+ -6.0 (+ z -1.0)))))))
      (+
       (+
        (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))
        (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0)))
       (+
        263.3831855358925
        (* z (+ 436.8961723502244 (* z 545.0353078134797)))))))))

C

double code(double z) {
	double t_0 = ((double) M_PI) / sin((((double) M_PI) * z));
	double t_1 = sqrt((((double) M_PI) * 2.0));
	double tmp;
	if (z <= -0.54) {
		tmp = (t_1 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)));
	} else {
		tmp = (t_0 * (t_1 * (pow((7.5 + ((1.0 - z) + -1.0)), ((1.0 - z) - 0.5)) * exp((-0.5 + (-6.0 + (z + -1.0))))))) * (((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))) + (263.3831855358925 + (z * (436.8961723502244 + (z * 545.0353078134797)))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.PI / Math.sin((Math.PI * z));
	double t_1 = Math.sqrt((Math.PI * 2.0));
	double tmp;
	if (z <= -0.54) {
		tmp = (t_1 * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)));
	} else {
		tmp = (t_0 * (t_1 * (Math.pow((7.5 + ((1.0 - z) + -1.0)), ((1.0 - z) - 0.5)) * Math.exp((-0.5 + (-6.0 + (z + -1.0))))))) * (((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))) + (263.3831855358925 + (z * (436.8961723502244 + (z * 545.0353078134797)))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.pi / math.sin((math.pi * z))
	t_1 = math.sqrt((math.pi * 2.0))
	tmp = 0
	if z <= -0.54:
		tmp = (t_1 * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)))
	else:
		tmp = (t_0 * (t_1 * (math.pow((7.5 + ((1.0 - z) + -1.0)), ((1.0 - z) - 0.5)) * math.exp((-0.5 + (-6.0 + (z + -1.0))))))) * (((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))) + (263.3831855358925 + (z * (436.8961723502244 + (z * 545.0353078134797)))))
	return tmp

Julia

function code(z)
	t_0 = Float64(pi / sin(Float64(pi * z)))
	t_1 = sqrt(Float64(pi * 2.0))
	tmp = 0.0
	if (z <= -0.54)
		tmp = Float64(Float64(t_1 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(-7.5))) * Float64(t_0 * Float64(0.9999999999998099 - Float64(24.458333333348836 / z))));
	else
		tmp = Float64(Float64(t_0 * Float64(t_1 * Float64((Float64(7.5 + Float64(Float64(1.0 - z) + -1.0)) ^ Float64(Float64(1.0 - z) - 0.5)) * exp(Float64(-0.5 + Float64(-6.0 + Float64(z + -1.0))))))) * Float64(Float64(Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0)) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0))) + Float64(263.3831855358925 + Float64(z * Float64(436.8961723502244 + Float64(z * 545.0353078134797))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = pi / sin((pi * z));
	t_1 = sqrt((pi * 2.0));
	tmp = 0.0;
	if (z <= -0.54)
		tmp = (t_1 * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)));
	else
		tmp = (t_0 * (t_1 * (((7.5 + ((1.0 - z) + -1.0)) ^ ((1.0 - z) - 0.5)) * exp((-0.5 + (-6.0 + (z + -1.0))))))) * (((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))) + (263.3831855358925 + (z * (436.8961723502244 + (z * 545.0353078134797)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -0.54], N[(N[(t$95$1 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(0.9999999999998099 - N[(24.458333333348836 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(t$95$1 * N[(N[Power[N[(7.5 + N[(N[(1.0 - z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-0.5 + N[(-6.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(263.3831855358925 + N[(z * N[(436.8961723502244 + N[(z * 545.0353078134797), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.54000000000000004

   1. Initial program 34.7%

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

   3. Simplified 34.8%

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

   5. Taylor expanded in z around inf 8.1%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(0.9999999999998099 - 24.458333333348836 \cdot \frac{1}{z}\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 8.1%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(0.9999999999998099 - \color{blue}{\frac{24.458333333348836 \cdot 1}{z}}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

      2. metadata-eval 8.1%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(0.9999999999998099 - \frac{\color{blue}{24.458333333348836}}{z}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   7. Simplified 8.1%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(0.9999999999998099 - \frac{24.458333333348836}{z}\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. Taylor expanded in z around 0 67.8%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{e^{-7.5}}\right)\right) \cdot \left(\left(0.9999999999998099 - \frac{24.458333333348836}{z}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   if -0.54000000000000004 < z

   1. Initial program 97.3%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in z around 0 98.3%

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

      1. *-commutative 98.3%

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

   7. Simplified 98.3%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.54:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(0.9999999999998099 - \frac{24.458333333348836}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 + \left(\left(1 - z\right) + -1\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{-0.5 + \left(-6 + \left(z + -1\right)\right)}\right)\right)\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) + \left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{\sin \left(\pi \cdot z\right)}\\ t_1 := \sqrt{\pi \cdot 2}\\ t_2 := {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\\ \mathbf{if}\;z \leq -0.58:\\ \;\;\;\;\left(t\_1 \cdot \left(t\_2 \cdot e^{-7.5}\right)\right) \cdot \left(t\_0 \cdot \left(0.9999999999998099 - \frac{24.458333333348836}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot \left(t\_2 \cdot e^{\left(z + -1\right) - 6.5}\right)\right) \cdot \left(t\_0 \cdot \left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + z \cdot 606.656776085461\right)\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ PI (sin (* PI z))))
        (t_1 (sqrt (* PI 2.0)))
        (t_2 (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5))))
   (if (<= z -0.58)
     (*
      (* t_1 (* t_2 (exp -7.5)))
      (* t_0 (- 0.9999999999998099 (/ 24.458333333348836 z))))
     (*
      (* t_1 (* t_2 (exp (- (+ z -1.0) 6.5))))
      (*
       t_0
       (+
        (+
         260.9048120626994
         (*
          z
          (+
           436.3997278161676
           (* z (+ 544.9358906000987 (* z 606.656776085461))))))
        (+
         2.4783749183520145
         (* z (+ 0.49644474017195733 (* z 0.09941724278406093))))))))))

C

double code(double z) {
	double t_0 = ((double) M_PI) / sin((((double) M_PI) * z));
	double t_1 = sqrt((((double) M_PI) * 2.0));
	double t_2 = pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5));
	double tmp;
	if (z <= -0.58) {
		tmp = (t_1 * (t_2 * exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)));
	} else {
		tmp = (t_1 * (t_2 * exp(((z + -1.0) - 6.5)))) * (t_0 * ((260.9048120626994 + (z * (436.3997278161676 + (z * (544.9358906000987 + (z * 606.656776085461)))))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * 0.09941724278406093))))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.PI / Math.sin((Math.PI * z));
	double t_1 = Math.sqrt((Math.PI * 2.0));
	double t_2 = Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5));
	double tmp;
	if (z <= -0.58) {
		tmp = (t_1 * (t_2 * Math.exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)));
	} else {
		tmp = (t_1 * (t_2 * Math.exp(((z + -1.0) - 6.5)))) * (t_0 * ((260.9048120626994 + (z * (436.3997278161676 + (z * (544.9358906000987 + (z * 606.656776085461)))))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * 0.09941724278406093))))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.pi / math.sin((math.pi * z))
	t_1 = math.sqrt((math.pi * 2.0))
	t_2 = math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5))
	tmp = 0
	if z <= -0.58:
		tmp = (t_1 * (t_2 * math.exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)))
	else:
		tmp = (t_1 * (t_2 * math.exp(((z + -1.0) - 6.5)))) * (t_0 * ((260.9048120626994 + (z * (436.3997278161676 + (z * (544.9358906000987 + (z * 606.656776085461)))))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * 0.09941724278406093))))))
	return tmp

Julia

function code(z)
	t_0 = Float64(pi / sin(Float64(pi * z)))
	t_1 = sqrt(Float64(pi * 2.0))
	t_2 = Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)
	tmp = 0.0
	if (z <= -0.58)
		tmp = Float64(Float64(t_1 * Float64(t_2 * exp(-7.5))) * Float64(t_0 * Float64(0.9999999999998099 - Float64(24.458333333348836 / z))));
	else
		tmp = Float64(Float64(t_1 * Float64(t_2 * exp(Float64(Float64(z + -1.0) - 6.5)))) * Float64(t_0 * Float64(Float64(260.9048120626994 + Float64(z * Float64(436.3997278161676 + Float64(z * Float64(544.9358906000987 + Float64(z * 606.656776085461)))))) + Float64(2.4783749183520145 + Float64(z * Float64(0.49644474017195733 + Float64(z * 0.09941724278406093)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = pi / sin((pi * z));
	t_1 = sqrt((pi * 2.0));
	t_2 = ((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5);
	tmp = 0.0;
	if (z <= -0.58)
		tmp = (t_1 * (t_2 * exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)));
	else
		tmp = (t_1 * (t_2 * exp(((z + -1.0) - 6.5)))) * (t_0 * ((260.9048120626994 + (z * (436.3997278161676 + (z * (544.9358906000987 + (z * 606.656776085461)))))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * 0.09941724278406093))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -0.58], N[(N[(t$95$1 * N[(t$95$2 * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(0.9999999999998099 - N[(24.458333333348836 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(t$95$2 * N[Exp[N[(N[(z + -1.0), $MachinePrecision] - 6.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[(260.9048120626994 + N[(z * N[(436.3997278161676 + N[(z * N[(544.9358906000987 + N[(z * 606.656776085461), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.4783749183520145 + N[(z * N[(0.49644474017195733 + N[(z * 0.09941724278406093), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.57999999999999996

   1. Initial program 34.7%

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

   3. Simplified 34.8%

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

   5. Taylor expanded in z around inf 8.1%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(0.9999999999998099 - 24.458333333348836 \cdot \frac{1}{z}\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 8.1%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(0.9999999999998099 - \color{blue}{\frac{24.458333333348836 \cdot 1}{z}}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

      2. metadata-eval 8.1%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(0.9999999999998099 - \frac{\color{blue}{24.458333333348836}}{z}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   7. Simplified 8.1%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(0.9999999999998099 - \frac{24.458333333348836}{z}\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. Taylor expanded in z around 0 67.8%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{e^{-7.5}}\right)\right) \cdot \left(\left(0.9999999999998099 - \frac{24.458333333348836}{z}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   if -0.57999999999999996 < z

   1. Initial program 97.3%

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

   3. Simplified 97.2%

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

   5. Taylor expanded in z around 0 96.8%

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

      1. *-commutative 96.8%

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

   7. Simplified 96.8%

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

   8. Taylor expanded in z around 0 98.2%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + 606.656776085461 \cdot z\right)\right)\right)} + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative 98.2%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + \color{blue}{z \cdot 606.656776085461}\right)\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   10. Simplified 98.2%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + z \cdot 606.656776085461\right)\right)\right)} + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.58:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(0.9999999999998099 - \frac{24.458333333348836}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(z + -1\right) - 6.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + z \cdot 606.656776085461\right)\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{\sin \left(\pi \cdot z\right)}\\ t_1 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq -0.58:\\ \;\;\;\;\left(t\_1 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \left(t\_0 \cdot \left(0.9999999999998099 - \frac{24.458333333348836}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + z \cdot 606.6766809167608\right)\right)\right) \cdot \left(t\_1 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-6.5 + \left(z + -1\right)}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ PI (sin (* PI z)))) (t_1 (sqrt (* PI 2.0))))
   (if (<= z -0.58)
     (*
      (* t_1 (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp -7.5)))
      (* t_0 (- 0.9999999999998099 (/ 24.458333333348836 z))))
     (*
      t_0
      (*
       (+
        263.3831869810514
        (*
         z
         (+
          436.8961725563396
          (* z (+ 545.0353078428827 (* z 606.6766809167608))))))
       (* t_1 (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ -6.5 (+ z -1.0))))))))))

C

double code(double z) {
	double t_0 = ((double) M_PI) / sin((((double) M_PI) * z));
	double t_1 = sqrt((((double) M_PI) * 2.0));
	double tmp;
	if (z <= -0.58) {
		tmp = (t_1 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)));
	} else {
		tmp = t_0 * ((263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (z * 606.6766809167608)))))) * (t_1 * (pow((7.5 - z), (0.5 - z)) * exp((-6.5 + (z + -1.0))))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.PI / Math.sin((Math.PI * z));
	double t_1 = Math.sqrt((Math.PI * 2.0));
	double tmp;
	if (z <= -0.58) {
		tmp = (t_1 * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)));
	} else {
		tmp = t_0 * ((263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (z * 606.6766809167608)))))) * (t_1 * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((-6.5 + (z + -1.0))))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.pi / math.sin((math.pi * z))
	t_1 = math.sqrt((math.pi * 2.0))
	tmp = 0
	if z <= -0.58:
		tmp = (t_1 * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)))
	else:
		tmp = t_0 * ((263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (z * 606.6766809167608)))))) * (t_1 * (math.pow((7.5 - z), (0.5 - z)) * math.exp((-6.5 + (z + -1.0))))))
	return tmp

Julia

function code(z)
	t_0 = Float64(pi / sin(Float64(pi * z)))
	t_1 = sqrt(Float64(pi * 2.0))
	tmp = 0.0
	if (z <= -0.58)
		tmp = Float64(Float64(t_1 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(-7.5))) * Float64(t_0 * Float64(0.9999999999998099 - Float64(24.458333333348836 / z))));
	else
		tmp = Float64(t_0 * Float64(Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * Float64(545.0353078428827 + Float64(z * 606.6766809167608)))))) * Float64(t_1 * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(-6.5 + Float64(z + -1.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = pi / sin((pi * z));
	t_1 = sqrt((pi * 2.0));
	tmp = 0.0;
	if (z <= -0.58)
		tmp = (t_1 * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)));
	else
		tmp = t_0 * ((263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (z * 606.6766809167608)))))) * (t_1 * (((7.5 - z) ^ (0.5 - z)) * exp((-6.5 + (z + -1.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -0.58], N[(N[(t$95$1 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(0.9999999999998099 - N[(24.458333333348836 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * N[(545.0353078428827 + N[(z * 606.6766809167608), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-6.5 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.57999999999999996

   1. Initial program 34.7%

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

   3. Simplified 34.8%

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

   5. Taylor expanded in z around inf 8.1%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(0.9999999999998099 - 24.458333333348836 \cdot \frac{1}{z}\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 8.1%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(0.9999999999998099 - \color{blue}{\frac{24.458333333348836 \cdot 1}{z}}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

      2. metadata-eval 8.1%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(0.9999999999998099 - \frac{\color{blue}{24.458333333348836}}{z}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   7. Simplified 8.1%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(0.9999999999998099 - \frac{24.458333333348836}{z}\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. Taylor expanded in z around 0 67.8%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{e^{-7.5}}\right)\right) \cdot \left(\left(0.9999999999998099 - \frac{24.458333333348836}{z}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   if -0.57999999999999996 < z

   1. Initial program 97.3%

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

   3. Simplified 97.2%

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

   6. Applied egg-rr 97.2%

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

   8. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. add-cbrt-cube 98.5%

         \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \color{blue}{\sqrt[3]{\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right) \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

      2. associate-+r+ 97.8%

         \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\left(\left(0.9999999999998099 + \color{blue}{\left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)}\right) \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right) \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

      3. associate-+r+ 98.5%

         \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \color{blue}{\left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)}\right)\right) \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

      4. associate-+r+ 98.5%

         \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) \cdot \left(0.9999999999998099 + \color{blue}{\left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)}\right)}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

   10. Applied egg-rr 98.5%

       \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \color{blue}{\sqrt[3]{\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)}}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]
   11. Step-by-step derivation

       1. associate-*l* 98.5%

          \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\color{blue}{\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

       2. associate-+r+ 97.3%

          \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\color{blue}{\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right)\right) + \frac{-1259.1392167224028}{2 - z}\right)} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

       3. +-commutative 97.3%

          \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\left(\left(0.9999999999998099 + \color{blue}{\left(\frac{771.3234287776531}{3 - z} + \frac{676.5203681218851}{1 - z}\right)}\right) + \frac{-1259.1392167224028}{2 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

   12. Simplified 97.3%

       \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \color{blue}{\sqrt[3]{\left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{676.5203681218851}{1 - z}\right)\right) + \frac{-1259.1392167224028}{2 - z}\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{676.5203681218851}{1 - z}\right)\right) + \frac{-1259.1392167224028}{2 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{676.5203681218851}{1 - z}\right)\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)}}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

   13. Taylor expanded in z around 0 98.0%

       \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]
   14. Step-by-step derivation

       1. *-commutative 98.0%

          \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + \color{blue}{z \cdot 606.6766809167608}\right)\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

   15. Simplified 98.0%

       \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + z \cdot 606.6766809167608\right)\right)\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.58:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(0.9999999999998099 - \frac{24.458333333348836}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + z \cdot 606.6766809167608\right)\right)\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-6.5 + \left(z + -1\right)}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{\sin \left(\pi \cdot z\right)}\\ t_1 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq -0.54:\\ \;\;\;\;\left(t\_1 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \left(t\_0 \cdot \left(0.9999999999998099 - \frac{24.458333333348836}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(t\_1 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-6.5 + \left(z + -1\right)}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ PI (sin (* PI z)))) (t_1 (sqrt (* PI 2.0))))
   (if (<= z -0.54)
     (*
      (* t_1 (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp -7.5)))
      (* t_0 (- 0.9999999999998099 (/ 24.458333333348836 z))))
     (*
      t_0
      (*
       (* t_1 (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ -6.5 (+ z -1.0)))))
       (+
        263.3831869810514
        (* z (+ 436.8961725563396 (* z 545.0353078428827)))))))))

C

double code(double z) {
	double t_0 = ((double) M_PI) / sin((((double) M_PI) * z));
	double t_1 = sqrt((((double) M_PI) * 2.0));
	double tmp;
	if (z <= -0.54) {
		tmp = (t_1 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)));
	} else {
		tmp = t_0 * ((t_1 * (pow((7.5 - z), (0.5 - z)) * exp((-6.5 + (z + -1.0))))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.PI / Math.sin((Math.PI * z));
	double t_1 = Math.sqrt((Math.PI * 2.0));
	double tmp;
	if (z <= -0.54) {
		tmp = (t_1 * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)));
	} else {
		tmp = t_0 * ((t_1 * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((-6.5 + (z + -1.0))))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.pi / math.sin((math.pi * z))
	t_1 = math.sqrt((math.pi * 2.0))
	tmp = 0
	if z <= -0.54:
		tmp = (t_1 * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)))
	else:
		tmp = t_0 * ((t_1 * (math.pow((7.5 - z), (0.5 - z)) * math.exp((-6.5 + (z + -1.0))))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))))
	return tmp

Julia

function code(z)
	t_0 = Float64(pi / sin(Float64(pi * z)))
	t_1 = sqrt(Float64(pi * 2.0))
	tmp = 0.0
	if (z <= -0.54)
		tmp = Float64(Float64(t_1 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(-7.5))) * Float64(t_0 * Float64(0.9999999999998099 - Float64(24.458333333348836 / z))));
	else
		tmp = Float64(t_0 * Float64(Float64(t_1 * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(-6.5 + Float64(z + -1.0))))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * 545.0353078428827))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = pi / sin((pi * z));
	t_1 = sqrt((pi * 2.0));
	tmp = 0.0;
	if (z <= -0.54)
		tmp = (t_1 * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(-7.5))) * (t_0 * (0.9999999999998099 - (24.458333333348836 / z)));
	else
		tmp = t_0 * ((t_1 * (((7.5 - z) ^ (0.5 - z)) * exp((-6.5 + (z + -1.0))))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -0.54], N[(N[(t$95$1 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(0.9999999999998099 - N[(24.458333333348836 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(t$95$1 * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-6.5 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * 545.0353078428827), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.54000000000000004

   1. Initial program 34.7%

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

   3. Simplified 34.8%

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

   5. Taylor expanded in z around inf 8.1%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(0.9999999999998099 - 24.458333333348836 \cdot \frac{1}{z}\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 8.1%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(0.9999999999998099 - \color{blue}{\frac{24.458333333348836 \cdot 1}{z}}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

      2. metadata-eval 8.1%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(0.9999999999998099 - \frac{\color{blue}{24.458333333348836}}{z}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   7. Simplified 8.1%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(0.9999999999998099 - \frac{24.458333333348836}{z}\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. Taylor expanded in z around 0 67.8%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{e^{-7.5}}\right)\right) \cdot \left(\left(0.9999999999998099 - \frac{24.458333333348836}{z}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   if -0.54000000000000004 < z

   1. Initial program 97.3%

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

   3. Simplified 97.2%

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

   6. Applied egg-rr 97.2%

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

   8. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. add-cbrt-cube 98.5%

         \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \color{blue}{\sqrt[3]{\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right) \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

      2. associate-+r+ 97.8%

         \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\left(\left(0.9999999999998099 + \color{blue}{\left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)}\right) \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right) \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

      3. associate-+r+ 98.5%

         \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \color{blue}{\left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)}\right)\right) \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

      4. associate-+r+ 98.5%

         \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) \cdot \left(0.9999999999998099 + \color{blue}{\left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)}\right)}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

   10. Applied egg-rr 98.5%

       \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \color{blue}{\sqrt[3]{\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)}}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]
   11. Step-by-step derivation

       1. associate-*l* 98.5%

          \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\color{blue}{\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

       2. associate-+r+ 97.3%

          \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\color{blue}{\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right)\right) + \frac{-1259.1392167224028}{2 - z}\right)} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

       3. +-commutative 97.3%

          \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\left(\left(0.9999999999998099 + \color{blue}{\left(\frac{771.3234287776531}{3 - z} + \frac{676.5203681218851}{1 - z}\right)}\right) + \frac{-1259.1392167224028}{2 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

   12. Simplified 97.3%

       \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \color{blue}{\sqrt[3]{\left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{676.5203681218851}{1 - z}\right)\right) + \frac{-1259.1392167224028}{2 - z}\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{676.5203681218851}{1 - z}\right)\right) + \frac{-1259.1392167224028}{2 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{676.5203681218851}{1 - z}\right)\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)}}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

   13. Taylor expanded in z around 0 97.8%

       \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]
   14. Step-by-step derivation

       1. *-commutative 97.8%

          \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(263.3831869810514 + z \cdot \left(436.8961725563396 + \color{blue}{z \cdot 545.0353078428827}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

   15. Simplified 97.8%

       \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.54:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(0.9999999999998099 - \frac{24.458333333348836}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-6.5 + \left(z + -1\right)}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.55:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(0.9999999999998099 - \frac{24.458333333348836}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;263.3831869810514 \cdot \frac{\left(e^{-7.5} \cdot \sqrt{\pi}\right) \cdot \left(\sqrt{7.5} \cdot \sqrt{2}\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (if (<= z -0.55)
   (*
    (*
     (sqrt (* PI 2.0))
     (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp -7.5)))
    (* (/ PI (sin (* PI z))) (- 0.9999999999998099 (/ 24.458333333348836 z))))
   (*
    263.3831869810514
    (/ (* (* (exp -7.5) (sqrt PI)) (* (sqrt 7.5) (sqrt 2.0))) z))))

C

double code(double z) {
	double tmp;
	if (z <= -0.55) {
		tmp = (sqrt((((double) M_PI) * 2.0)) * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(-7.5))) * ((((double) M_PI) / sin((((double) M_PI) * z))) * (0.9999999999998099 - (24.458333333348836 / z)));
	} else {
		tmp = 263.3831869810514 * (((exp(-7.5) * sqrt(((double) M_PI))) * (sqrt(7.5) * sqrt(2.0))) / z);
	}
	return tmp;
}

Java

public static double code(double z) {
	double tmp;
	if (z <= -0.55) {
		tmp = (Math.sqrt((Math.PI * 2.0)) * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(-7.5))) * ((Math.PI / Math.sin((Math.PI * z))) * (0.9999999999998099 - (24.458333333348836 / z)));
	} else {
		tmp = 263.3831869810514 * (((Math.exp(-7.5) * Math.sqrt(Math.PI)) * (Math.sqrt(7.5) * Math.sqrt(2.0))) / z);
	}
	return tmp;
}

Python

def code(z):
	tmp = 0
	if z <= -0.55:
		tmp = (math.sqrt((math.pi * 2.0)) * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(-7.5))) * ((math.pi / math.sin((math.pi * z))) * (0.9999999999998099 - (24.458333333348836 / z)))
	else:
		tmp = 263.3831869810514 * (((math.exp(-7.5) * math.sqrt(math.pi)) * (math.sqrt(7.5) * math.sqrt(2.0))) / z)
	return tmp

Julia

function code(z)
	tmp = 0.0
	if (z <= -0.55)
		tmp = Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(-7.5))) * Float64(Float64(pi / sin(Float64(pi * z))) * Float64(0.9999999999998099 - Float64(24.458333333348836 / z))));
	else
		tmp = Float64(263.3831869810514 * Float64(Float64(Float64(exp(-7.5) * sqrt(pi)) * Float64(sqrt(7.5) * sqrt(2.0))) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	tmp = 0.0;
	if (z <= -0.55)
		tmp = (sqrt((pi * 2.0)) * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(-7.5))) * ((pi / sin((pi * z))) * (0.9999999999998099 - (24.458333333348836 / z)));
	else
		tmp = 263.3831869810514 * (((exp(-7.5) * sqrt(pi)) * (sqrt(7.5) * sqrt(2.0))) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := If[LessEqual[z, -0.55], 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[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.9999999999998099 - N[(24.458333333348836 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(263.3831869810514 * N[(N[(N[(N[Exp[-7.5], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[7.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.55000000000000004

   1. Initial program 34.7%

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

   3. Simplified 34.8%

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

   5. Taylor expanded in z around inf 8.1%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(0.9999999999998099 - 24.458333333348836 \cdot \frac{1}{z}\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 8.1%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(0.9999999999998099 - \color{blue}{\frac{24.458333333348836 \cdot 1}{z}}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

      2. metadata-eval 8.1%

         \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(0.9999999999998099 - \frac{\color{blue}{24.458333333348836}}{z}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   7. Simplified 8.1%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(0.9999999999998099 - \frac{24.458333333348836}{z}\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. Taylor expanded in z around 0 67.8%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{e^{-7.5}}\right)\right) \cdot \left(\left(0.9999999999998099 - \frac{24.458333333348836}{z}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   if -0.55000000000000004 < z

   1. Initial program 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in z around 0 96.8%

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

      1. associate-*l/ 96.7%

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

      2. *-commutative 96.7%

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

      3. associate-*r* 97.4%

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

      4. *-commutative 97.4%

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

   7. Simplified 97.4%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.55:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(0.9999999999998099 - \frac{24.458333333348836}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;263.3831869810514 \cdot \frac{\left(e^{-7.5} \cdot \sqrt{\pi}\right) \cdot \left(\sqrt{7.5} \cdot \sqrt{2}\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 95.7% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (sqrt (* PI 2.0))
   (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp (- (+ z -1.0) 6.5))))
  (/ (+ 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)) * exp(((z + -1.0) - 6.5)))) * ((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)) * Math.exp(((z + -1.0) - 6.5)))) * ((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)) * math.exp(((z + -1.0) - 6.5)))) * ((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)) * exp(Float64(Float64(z + -1.0) - 6.5)))) * 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)) * exp(((z + -1.0) - 6.5)))) * ((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[Exp[N[(N[(z + -1.0), $MachinePrecision] - 6.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(263.3831869810514 + N[(z * 436.8961725563396), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.4%

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

3. Simplified 95.2%

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

5. Taylor expanded in z around 0 94.5%

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

   1. *-commutative 94.5%

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

7. Simplified 94.5%

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

8. Final simplification 94.5%

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

Alternative 14: 95.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 95.2%

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

5. Taylor expanded in z around 0 93.4%

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

6. Taylor expanded in z around 0 93.9%

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

   1. associate-/l* 94.1%

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

   2. *-commutative 94.1%

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

8. Simplified 94.1%

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

   1. pow1 94.1%

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

   2. associate-*l* 94.1%

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

   3. sqrt-unprod 94.1%

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

   4. metadata-eval 94.1%

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

10. Applied egg-rr 94.1%

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

    1. unpow1 94.1%

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

    2. *-commutative 94.1%

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

12. Simplified 94.1%

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

Alternative 15: 95.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{z} \end{array} \]

FPCore

(FPCore (z) :precision binary64 (* (/ PI (sin (* PI z))) (exp z)))

C

double code(double z) {
	return (((double) M_PI) / sin((((double) M_PI) * z))) * exp(z);
}

Java

public static double code(double z) {
	return (Math.PI / Math.sin((Math.PI * z))) * Math.exp(z);
}

Python

def code(z):
	return (math.pi / math.sin((math.pi * z))) * math.exp(z)

Julia

function code(z)
	return Float64(Float64(pi / sin(Float64(pi * z))) * exp(z))
end

MATLAB

function tmp = code(z)
	tmp = (pi / sin((pi * z))) * exp(z);
end

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 95.2%

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

6. Applied egg-rr 95.3%

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

8. Simplified 97.3%

   \[\leadsto \color{blue}{\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right)} \]
9. Step-by-step derivation

   1. add-cbrt-cube 96.5%

      \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \color{blue}{\sqrt[3]{\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right) \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

   2. associate-+r+ 95.9%

      \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\left(\left(0.9999999999998099 + \color{blue}{\left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)}\right) \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right) \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

   3. associate-+r+ 96.5%

      \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \color{blue}{\left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)}\right)\right) \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

   4. associate-+r+ 96.5%

      \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) \cdot \left(0.9999999999998099 + \color{blue}{\left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)}\right)}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

10. Applied egg-rr 96.5%

    \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \color{blue}{\sqrt[3]{\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)}}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]
11. Step-by-step derivation

    1. associate-*l* 96.5%

       \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\color{blue}{\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

    2. associate-+r+ 95.4%

       \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\color{blue}{\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right)\right) + \frac{-1259.1392167224028}{2 - z}\right)} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

    3. +-commutative 95.4%

       \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \sqrt[3]{\left(\left(0.9999999999998099 + \color{blue}{\left(\frac{771.3234287776531}{3 - z} + \frac{676.5203681218851}{1 - z}\right)}\right) + \frac{-1259.1392167224028}{2 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

12. Simplified 95.4%

    \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \color{blue}{\sqrt[3]{\left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{676.5203681218851}{1 - z}\right)\right) + \frac{-1259.1392167224028}{2 - z}\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{676.5203681218851}{1 - z}\right)\right) + \frac{-1259.1392167224028}{2 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{676.5203681218851}{1 - z}\right)\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)}}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)\right) \]

14. Applied egg-rr 98.4%

    \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \color{blue}{e^{\log \left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right) + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)\right) + \left(\log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) + \left(-1 + \left(z + -6.5\right)\right)\right)}} \]

15. Taylor expanded in z around inf 93.6%

    \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot e^{\color{blue}{z}} \]

16. Final simplification 93.6%

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{z} \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024137 
(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))))))