Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.5% → 98.6%

Time: 1.5min

Alternatives: 15

Speedup: 1.7×

• could not determine a ground truth

  1. z = -4.338559795482999e+47

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: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{676.5203681218851}{1 - z}\\ t_1 := \frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\\ t_2 := \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\\ t_3 := \frac{-0.13857109526572012}{6 - z}\\ t_4 := \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\\ t_5 := e^{z + -7.5} \cdot \frac{\sqrt{\pi \cdot 2} \cdot \left(\pi \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)}{\sin \left(\pi \cdot z\right)}\\ t_6 := \frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{-16}:\\ \;\;\;\;t_5 \cdot \left(\left(t_2 + t_4\right) + \left(t_1 + \left(t_3 + \left(t_0 + t_6\right)\right)\right)\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(263.3831869810514 \cdot e^{-7.5}\right) \cdot \sqrt{\pi}}{\frac{z}{\sqrt{15}}}\\ \mathbf{else}:\\ \;\;\;\;t_5 \cdot \left(\left(t_2 + \left(t_3 + t_4\right)\right) + \left(t_0 + \left(t_1 + t_6\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ 676.5203681218851 (- 1.0 z)))
        (t_1
         (+ (/ -176.6150291621406 (- 4.0 z)) (/ 12.507343278686905 (- 5.0 z))))
        (t_2 (/ 9.984369578019572e-6 (- 7.0 z)))
        (t_3 (/ -0.13857109526572012 (- 6.0 z)))
        (t_4 (/ 1.5056327351493116e-7 (- 8.0 z)))
        (t_5
         (*
          (exp (+ z -7.5))
          (/
           (* (sqrt (* PI 2.0)) (* PI (pow (- 7.5 z) (- 0.5 z))))
           (sin (* PI z)))))
        (t_6
         (+
          (/ -1259.1392167224028 (- 2.0 z))
          (+ 0.9999999999998099 (/ 771.3234287776531 (- 3.0 z))))))
   (if (<= z -6.5e-16)
     (* t_5 (+ (+ t_2 t_4) (+ t_1 (+ t_3 (+ t_0 t_6)))))
     (if (<= z 4e-18)
       (/ (* (* 263.3831869810514 (exp -7.5)) (sqrt PI)) (/ z (sqrt 15.0)))
       (* t_5 (+ (+ t_2 (+ t_3 t_4)) (+ t_0 (+ t_1 t_6))))))))

C

double code(double z) {
	double t_0 = 676.5203681218851 / (1.0 - z);
	double t_1 = (-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z));
	double t_2 = 9.984369578019572e-6 / (7.0 - z);
	double t_3 = -0.13857109526572012 / (6.0 - z);
	double t_4 = 1.5056327351493116e-7 / (8.0 - z);
	double t_5 = exp((z + -7.5)) * ((sqrt((((double) M_PI) * 2.0)) * (((double) M_PI) * pow((7.5 - z), (0.5 - z)))) / sin((((double) M_PI) * z)));
	double t_6 = (-1259.1392167224028 / (2.0 - z)) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z)));
	double tmp;
	if (z <= -6.5e-16) {
		tmp = t_5 * ((t_2 + t_4) + (t_1 + (t_3 + (t_0 + t_6))));
	} else if (z <= 4e-18) {
		tmp = ((263.3831869810514 * exp(-7.5)) * sqrt(((double) M_PI))) / (z / sqrt(15.0));
	} else {
		tmp = t_5 * ((t_2 + (t_3 + t_4)) + (t_0 + (t_1 + t_6)));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = 676.5203681218851 / (1.0 - z);
	double t_1 = (-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z));
	double t_2 = 9.984369578019572e-6 / (7.0 - z);
	double t_3 = -0.13857109526572012 / (6.0 - z);
	double t_4 = 1.5056327351493116e-7 / (8.0 - z);
	double t_5 = Math.exp((z + -7.5)) * ((Math.sqrt((Math.PI * 2.0)) * (Math.PI * Math.pow((7.5 - z), (0.5 - z)))) / Math.sin((Math.PI * z)));
	double t_6 = (-1259.1392167224028 / (2.0 - z)) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z)));
	double tmp;
	if (z <= -6.5e-16) {
		tmp = t_5 * ((t_2 + t_4) + (t_1 + (t_3 + (t_0 + t_6))));
	} else if (z <= 4e-18) {
		tmp = ((263.3831869810514 * Math.exp(-7.5)) * Math.sqrt(Math.PI)) / (z / Math.sqrt(15.0));
	} else {
		tmp = t_5 * ((t_2 + (t_3 + t_4)) + (t_0 + (t_1 + t_6)));
	}
	return tmp;
}

Python

def code(z):
	t_0 = 676.5203681218851 / (1.0 - z)
	t_1 = (-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))
	t_2 = 9.984369578019572e-6 / (7.0 - z)
	t_3 = -0.13857109526572012 / (6.0 - z)
	t_4 = 1.5056327351493116e-7 / (8.0 - z)
	t_5 = math.exp((z + -7.5)) * ((math.sqrt((math.pi * 2.0)) * (math.pi * math.pow((7.5 - z), (0.5 - z)))) / math.sin((math.pi * z)))
	t_6 = (-1259.1392167224028 / (2.0 - z)) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z)))
	tmp = 0
	if z <= -6.5e-16:
		tmp = t_5 * ((t_2 + t_4) + (t_1 + (t_3 + (t_0 + t_6))))
	elif z <= 4e-18:
		tmp = ((263.3831869810514 * math.exp(-7.5)) * math.sqrt(math.pi)) / (z / math.sqrt(15.0))
	else:
		tmp = t_5 * ((t_2 + (t_3 + t_4)) + (t_0 + (t_1 + t_6)))
	return tmp

Julia

function code(z)
	t_0 = Float64(676.5203681218851 / Float64(1.0 - z))
	t_1 = Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(12.507343278686905 / Float64(5.0 - z)))
	t_2 = Float64(9.984369578019572e-6 / Float64(7.0 - z))
	t_3 = Float64(-0.13857109526572012 / Float64(6.0 - z))
	t_4 = Float64(1.5056327351493116e-7 / Float64(8.0 - z))
	t_5 = Float64(exp(Float64(z + -7.5)) * Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64(pi * (Float64(7.5 - z) ^ Float64(0.5 - z)))) / sin(Float64(pi * z))))
	t_6 = Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) + Float64(0.9999999999998099 + Float64(771.3234287776531 / Float64(3.0 - z))))
	tmp = 0.0
	if (z <= -6.5e-16)
		tmp = Float64(t_5 * Float64(Float64(t_2 + t_4) + Float64(t_1 + Float64(t_3 + Float64(t_0 + t_6)))));
	elseif (z <= 4e-18)
		tmp = Float64(Float64(Float64(263.3831869810514 * exp(-7.5)) * sqrt(pi)) / Float64(z / sqrt(15.0)));
	else
		tmp = Float64(t_5 * Float64(Float64(t_2 + Float64(t_3 + t_4)) + Float64(t_0 + Float64(t_1 + t_6))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = 676.5203681218851 / (1.0 - z);
	t_1 = (-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z));
	t_2 = 9.984369578019572e-6 / (7.0 - z);
	t_3 = -0.13857109526572012 / (6.0 - z);
	t_4 = 1.5056327351493116e-7 / (8.0 - z);
	t_5 = exp((z + -7.5)) * ((sqrt((pi * 2.0)) * (pi * ((7.5 - z) ^ (0.5 - z)))) / sin((pi * z)));
	t_6 = (-1259.1392167224028 / (2.0 - z)) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z)));
	tmp = 0.0;
	if (z <= -6.5e-16)
		tmp = t_5 * ((t_2 + t_4) + (t_1 + (t_3 + (t_0 + t_6))));
	elseif (z <= 4e-18)
		tmp = ((263.3831869810514 * exp(-7.5)) * sqrt(pi)) / (z / sqrt(15.0));
	else
		tmp = t_5 * ((t_2 + (t_3 + t_4)) + (t_0 + (t_1 + t_6)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(Pi * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.5e-16], N[(t$95$5 * N[(N[(t$95$2 + t$95$4), $MachinePrecision] + N[(t$95$1 + N[(t$95$3 + N[(t$95$0 + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4e-18], N[(N[(N[(263.3831869810514 * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] / N[(z / N[Sqrt[15.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$5 * N[(N[(t$95$2 + N[(t$95$3 + t$95$4), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 + N[(t$95$1 + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.50000000000000011e-16

   1. Initial program 97.9%

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

   3. Simplified 97.8%

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

      1. expm1-log1p-u 97.0%

         \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right) + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

      2. expm1-udef 97.0%

         \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right) + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)} - 1\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

      3. associate-+l+ 97.1%

         \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)}\right)} - 1\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

      4. associate-+l+ 97.1%

         \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(e^{\mathsf{log1p}\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \color{blue}{\left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)}\right)\right)} - 1\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

   5. Applied egg-rr 97.1%

      \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)} - 1\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]
   6. Step-by-step derivation

      1. expm1-def 97.1%

         \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

      2. expm1-log1p 97.9%

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

      3. associate-+r+ 97.9%

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

   7. Simplified 97.9%

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

   if -6.50000000000000011e-16 < z < 4.0000000000000003e-18

   1. Initial program 97.4%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in z around 0 98.0%

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

   5. Taylor expanded in z around 0 98.1%

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

   6. Taylor expanded in z around 0 99.0%

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

      1. associate-*r* 98.9%

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

      2. associate-/l* 99.1%

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

      3. associate-*r/ 99.1%

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

   8. Simplified 99.1%

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

      1. associate-*l/ 99.6%

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

      2. sqrt-unprod 99.6%

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

      3. metadata-eval 99.6%

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

   10. Applied egg-rr 99.6%

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

   if 4.0000000000000003e-18 < z

   1. Initial program 97.6%

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

   3. Simplified 98.2%

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

      1. expm1-log1p-u 97.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right) + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)} \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

      2. expm1-udef 97.9%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right) + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)} - 1\right)} \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

   5. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)} - 1\right)} \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

   7. Simplified 98.8%

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

   9. Applied egg-rr 98.4%

      \[\leadsto \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{771.3234287776531}{3 - z} + 0.9999999999998099\right)\right) + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)}\right)\right)} - 1}}{\sin \left(\pi \cdot z\right)}\right) \]
   10. Step-by-step derivation

       1. expm1-def 98.4%

          \[\leadsto \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{771.3234287776531}{3 - z} + 0.9999999999998099\right)\right) + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)}\right)\right)\right)}}{\sin \left(\pi \cdot z\right)}\right) \]

       2. expm1-log1p 98.5%

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

       3. *-commutative 98.5%

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

       4. fma-udef 98.5%

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

       5. neg-mul-1 98.5%

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

       6. +-commutative 98.5%

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

       7. sub-neg 98.5%

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

       8. exp-to-pow 98.4%

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

       9. associate-*l* 98.9%

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

       10. *-commutative 98.9%

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

       11. exp-to-pow 98.8%

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

   11. Simplified 98.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-16}:\\ \;\;\;\;\left(e^{z + -7.5} \cdot \frac{\sqrt{\pi \cdot 2} \cdot \left(\pi \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(263.3831869810514 \cdot e^{-7.5}\right) \cdot \sqrt{\pi}}{\frac{z}{\sqrt{15}}}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{z + -7.5} \cdot \frac{\sqrt{\pi \cdot 2} \cdot \left(\pi \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{-0.13857109526572012}{6 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) + \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 2: 98.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

3. Simplified 98.9%

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

5. Applied egg-rr 99.3%

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

6. Final simplification 99.3%

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

Alternative 3: 98.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + {\left({\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)}^{3}\right)}^{0.3333333333333333}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\sqrt{\pi \cdot 2} \cdot \left(\pi \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)}{\sin \left(\pi \cdot z\right)}\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (+
   (+
    (+ (/ -176.6150291621406 (- 4.0 z)) (/ 12.507343278686905 (- 5.0 z)))
    (pow
     (pow
      (+
       (/ -0.13857109526572012 (- 6.0 z))
       (+
        (/ 676.5203681218851 (- 1.0 z))
        (+
         0.9999999999998099
         (+
          (/ 771.3234287776531 (- 3.0 z))
          (/ -1259.1392167224028 (- 2.0 z))))))
      3.0)
     0.3333333333333333))
   (+ (/ 9.984369578019572e-6 (- 7.0 z)) (/ 1.5056327351493116e-7 (- 8.0 z))))
  (*
   (exp (+ z -7.5))
   (/ (* (sqrt (* PI 2.0)) (* PI (pow (- 7.5 z) (- 0.5 z)))) (sin (* PI z))))))

C

double code(double z) {
	return ((((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + pow(pow(((-0.13857109526572012 / (6.0 - z)) + ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + ((771.3234287776531 / (3.0 - z)) + (-1259.1392167224028 / (2.0 - z)))))), 3.0), 0.3333333333333333)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) * (exp((z + -7.5)) * ((sqrt((((double) M_PI) * 2.0)) * (((double) M_PI) * pow((7.5 - z), (0.5 - z)))) / sin((((double) M_PI) * z))));
}

Java

public static double code(double z) {
	return ((((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + Math.pow(Math.pow(((-0.13857109526572012 / (6.0 - z)) + ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + ((771.3234287776531 / (3.0 - z)) + (-1259.1392167224028 / (2.0 - z)))))), 3.0), 0.3333333333333333)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) * (Math.exp((z + -7.5)) * ((Math.sqrt((Math.PI * 2.0)) * (Math.PI * Math.pow((7.5 - z), (0.5 - z)))) / Math.sin((Math.PI * z))));
}

Python

def code(z):
	return ((((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + math.pow(math.pow(((-0.13857109526572012 / (6.0 - z)) + ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + ((771.3234287776531 / (3.0 - z)) + (-1259.1392167224028 / (2.0 - z)))))), 3.0), 0.3333333333333333)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) * (math.exp((z + -7.5)) * ((math.sqrt((math.pi * 2.0)) * (math.pi * math.pow((7.5 - z), (0.5 - z)))) / math.sin((math.pi * z))))

Julia

function code(z)
	return Float64(Float64(Float64(Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(12.507343278686905 / Float64(5.0 - z))) + ((Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(0.9999999999998099 + Float64(Float64(771.3234287776531 / Float64(3.0 - z)) + Float64(-1259.1392167224028 / Float64(2.0 - z)))))) ^ 3.0) ^ 0.3333333333333333)) + Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z)))) * Float64(exp(Float64(z + -7.5)) * Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64(pi * (Float64(7.5 - z) ^ Float64(0.5 - z)))) / sin(Float64(pi * z)))))
end

MATLAB

function tmp = code(z)
	tmp = ((((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + ((((-0.13857109526572012 / (6.0 - z)) + ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + ((771.3234287776531 / (3.0 - z)) + (-1259.1392167224028 / (2.0 - z)))))) ^ 3.0) ^ 0.3333333333333333)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) * (exp((z + -7.5)) * ((sqrt((pi * 2.0)) * (pi * ((7.5 - z) ^ (0.5 - z)))) / sin((pi * z))));
end

Wolfram

code[z_] := N[(N[(N[(N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Power[N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(Pi * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.4%

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

3. Simplified 96.6%

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

   1. add-cbrt-cube 96.6%

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

   2. pow1/3 98.2%

      \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \color{blue}{{\left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right) + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right) + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right) + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}^{0.3333333333333333}}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

5. Applied egg-rr 99.2%

   \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \color{blue}{{\left({\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)}^{3}\right)}^{0.3333333333333333}}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

6. Final simplification 99.2%

   \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + {\left({\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)}^{3}\right)}^{0.3333333333333333}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\sqrt{\pi \cdot 2} \cdot \left(\pi \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)}{\sin \left(\pi \cdot z\right)}\right) \]

Alternative 4: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

3. Simplified 98.9%

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

4. Final simplification 98.9%

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

Alternative 5: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{\left(z + -1\right) - 6.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{1 - \left(z + -2\right)}\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 + \left(-1 - z\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 + \left(-1 - z\right)}\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (pow (+ (- 1.0 z) 6.5) (- 1.0 (+ z 0.5)))
   (* (sqrt (* PI 2.0)) (exp (- (+ z -1.0) 6.5))))
  (*
   (/ PI (sin (* PI z)))
   (+
    (+
     0.9999999999998099
     (+
      (/ 676.5203681218851 (- 1.0 z))
      (+
       (+
        (/ -1259.1392167224028 (- 2.0 z))
        (/ 771.3234287776531 (- 1.0 (+ z -2.0))))
       (/ -176.6150291621406 (+ (- 1.0 z) 3.0)))))
    (+
     (+ (/ 12.507343278686905 (- 5.0 z)) (/ -0.13857109526572012 (- 6.0 z)))
     (+
      (/ 9.984369578019572e-6 (+ 8.0 (- -1.0 z)))
      (/ 1.5056327351493116e-7 (+ 9.0 (- -1.0 z)))))))))

C

double code(double z) {
	return (pow(((1.0 - z) + 6.5), (1.0 - (z + 0.5))) * (sqrt((((double) M_PI) * 2.0)) * exp(((z + -1.0) - 6.5)))) * ((((double) M_PI) / sin((((double) M_PI) * z))) * ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (1.0 - (z + -2.0)))) + (-176.6150291621406 / ((1.0 - z) + 3.0))))) + (((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z))) + ((9.984369578019572e-6 / (8.0 + (-1.0 - z))) + (1.5056327351493116e-7 / (9.0 + (-1.0 - z)))))));
}

Java

public static double code(double z) {
	return (Math.pow(((1.0 - z) + 6.5), (1.0 - (z + 0.5))) * (Math.sqrt((Math.PI * 2.0)) * Math.exp(((z + -1.0) - 6.5)))) * ((Math.PI / Math.sin((Math.PI * z))) * ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (1.0 - (z + -2.0)))) + (-176.6150291621406 / ((1.0 - z) + 3.0))))) + (((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z))) + ((9.984369578019572e-6 / (8.0 + (-1.0 - z))) + (1.5056327351493116e-7 / (9.0 + (-1.0 - z)))))));
}

Python

def code(z):
	return (math.pow(((1.0 - z) + 6.5), (1.0 - (z + 0.5))) * (math.sqrt((math.pi * 2.0)) * math.exp(((z + -1.0) - 6.5)))) * ((math.pi / math.sin((math.pi * z))) * ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (1.0 - (z + -2.0)))) + (-176.6150291621406 / ((1.0 - z) + 3.0))))) + (((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z))) + ((9.984369578019572e-6 / (8.0 + (-1.0 - z))) + (1.5056327351493116e-7 / (9.0 + (-1.0 - z)))))))

Julia

function code(z)
	return Float64(Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(1.0 - Float64(z + 0.5))) * Float64(sqrt(Float64(pi * 2.0)) * exp(Float64(Float64(z + -1.0) - 6.5)))) * Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(0.9999999999998099 + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) + Float64(771.3234287776531 / Float64(1.0 - Float64(z + -2.0)))) + Float64(-176.6150291621406 / Float64(Float64(1.0 - z) + 3.0))))) + Float64(Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(-0.13857109526572012 / Float64(6.0 - z))) + Float64(Float64(9.984369578019572e-6 / Float64(8.0 + Float64(-1.0 - z))) + Float64(1.5056327351493116e-7 / Float64(9.0 + Float64(-1.0 - z))))))))
end

MATLAB

function tmp = code(z)
	tmp = ((((1.0 - z) + 6.5) ^ (1.0 - (z + 0.5))) * (sqrt((pi * 2.0)) * exp(((z + -1.0) - 6.5)))) * ((pi / sin((pi * z))) * ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (1.0 - (z + -2.0)))) + (-176.6150291621406 / ((1.0 - z) + 3.0))))) + (((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z))) + ((9.984369578019572e-6 / (8.0 + (-1.0 - z))) + (1.5056327351493116e-7 / (9.0 + (-1.0 - z)))))));
end

Wolfram

code[z_] := N[(N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(1.0 - N[(z + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(z + -1.0), $MachinePrecision] - 6.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.9999999999998099 + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(1.0 - N[(z + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(8.0 + N[(-1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(9.0 + N[(-1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.4%

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

3. Simplified 98.3%

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

   1. expm1-log1p-u 97.0%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right)\right)} + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   2. expm1-udef 97.0%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right)} - 1\right)} + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

5. Applied egg-rr 97.0%

   \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(e^{\mathsf{log1p}\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-1259.1392167224028}{3 - \left(1 + z\right)} + \frac{771.3234287776531}{1 - \left(z + -2\right)}\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right)} - 1\right)} + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
6. Step-by-step derivation

   1. expm1-def 97.0%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-1259.1392167224028}{3 - \left(1 + z\right)} + \frac{771.3234287776531}{1 - \left(z + -2\right)}\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right)\right)} + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   2. expm1-log1p 98.3%

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

   3. associate--r+ 98.3%

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

   4. metadata-eval 98.3%

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

   5. +-commutative 98.3%

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

7. Simplified 98.3%

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

   1. metadata-eval 98.3%

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

   2. sub-neg 98.3%

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

   3. associate--r+ 98.3%

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

   4. expm1-log1p-u 98.3%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{1 - \left(z + -2\right)}\right) + \frac{-176.6150291621406}{3 + \left(1 - z\right)}\right)\right)\right) + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   5. expm1-udef 98.3%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{1 - \left(z + -2\right)}\right) + \frac{-176.6150291621406}{3 + \left(1 - z\right)}\right)\right)\right) + \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)} - 1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   6. sub-neg 98.3%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{1 - \left(z + -2\right)}\right) + \frac{-176.6150291621406}{3 + \left(1 - z\right)}\right)\right)\right) + \left(\left(e^{\mathsf{log1p}\left(\frac{12.507343278686905}{\color{blue}{\left(1 - z\right) + \left(--4\right)}} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)} - 1\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   7. metadata-eval 98.3%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{1 - \left(z + -2\right)}\right) + \frac{-176.6150291621406}{3 + \left(1 - z\right)}\right)\right)\right) + \left(\left(e^{\mathsf{log1p}\left(\frac{12.507343278686905}{\left(1 - z\right) + \color{blue}{4}} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)} - 1\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. sub-neg 98.3%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{1 - \left(z + -2\right)}\right) + \frac{-176.6150291621406}{3 + \left(1 - z\right)}\right)\right)\right) + \left(\left(e^{\mathsf{log1p}\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\color{blue}{\left(1 - z\right) + \left(--5\right)}}\right)} - 1\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   9. metadata-eval 98.3%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{1 - \left(z + -2\right)}\right) + \frac{-176.6150291621406}{3 + \left(1 - z\right)}\right)\right)\right) + \left(\left(e^{\mathsf{log1p}\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + \color{blue}{5}}\right)} - 1\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

9. Applied egg-rr 98.3%

   \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{1 - \left(z + -2\right)}\right) + \frac{-176.6150291621406}{3 + \left(1 - z\right)}\right)\right)\right) + \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right)} - 1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
10. Step-by-step derivation

    1. expm1-def 98.3%

       \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{1 - \left(z + -2\right)}\right) + \frac{-176.6150291621406}{3 + \left(1 - z\right)}\right)\right)\right) + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

    2. expm1-log1p 98.3%

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

    3. +-commutative 98.3%

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

    4. associate-+r- 98.3%

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

    5. metadata-eval 98.3%

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

    6. +-commutative 98.3%

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

    7. associate-+r- 98.3%

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

    8. metadata-eval 98.3%

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

11. Simplified 98.3%

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

12. Final simplification 98.3%

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

Alternative 6: 97.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

3. Simplified 98.3%

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

   1. expm1-log1p-u 97.0%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right)\right)} + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   2. expm1-udef 97.0%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right)} - 1\right)} + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

5. Applied egg-rr 97.0%

   \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(e^{\mathsf{log1p}\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-1259.1392167224028}{3 - \left(1 + z\right)} + \frac{771.3234287776531}{1 - \left(z + -2\right)}\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right)} - 1\right)} + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
6. Step-by-step derivation

   1. expm1-def 97.0%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-1259.1392167224028}{3 - \left(1 + z\right)} + \frac{771.3234287776531}{1 - \left(z + -2\right)}\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right)\right)} + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   2. expm1-log1p 98.3%

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

   3. associate--r+ 98.3%

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

   4. metadata-eval 98.3%

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

   5. +-commutative 98.3%

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

7. Simplified 98.3%

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

   1. expm1-log1p-u 96.3%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{1 - \left(z + -2\right)}\right) + \frac{-176.6150291621406}{3 + \left(1 - z\right)}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right)\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   2. expm1-udef 96.3%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{1 - \left(z + -2\right)}\right) + \frac{-176.6150291621406}{3 + \left(1 - z\right)}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right)} - 1\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

9. Applied egg-rr 96.2%

   \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(0.9999999999998099 + \left(\left(\left(\frac{676.5203681218851}{1 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \left(\frac{-0.13857109526572012}{1 - \left(z + -5\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right)\right)\right)} - 1\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

11. Simplified 98.3%

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

12. Final simplification 98.3%

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

Alternative 7: 98.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-15} \lor \neg \left(z \leq 5 \cdot 10^{-17}\right):\\ \;\;\;\;\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right)\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)}{\sin \left(\pi \cdot z\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(263.3831869810514 \cdot e^{-7.5}\right) \cdot \sqrt{\pi}}{\frac{z}{\sqrt{15}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (if (or (<= z -1e-15) (not (<= z 5e-17)))
   (*
    (+
     (+ (/ 9.984369578019572e-6 (- 7.0 z)) (/ 1.5056327351493116e-7 (- 8.0 z)))
     (+
      (+ (/ -176.6150291621406 (- 4.0 z)) (/ 12.507343278686905 (- 5.0 z)))
      (+
       (+
        (/ -1259.1392167224028 (- 2.0 z))
        (+ 0.9999999999998099 (/ 771.3234287776531 (- 3.0 z))))
       (+
        (/ -0.13857109526572012 (- 6.0 z))
        (/ 676.5203681218851 (- 1.0 z))))))
    (*
     (exp (+ z -7.5))
     (/
      (* PI (* (sqrt (* PI 2.0)) (pow (- 7.5 z) (- 0.5 z))))
      (sin (* PI z)))))
   (/ (* (* 263.3831869810514 (exp -7.5)) (sqrt PI)) (/ z (sqrt 15.0)))))

C

double code(double z) {
	double tmp;
	if ((z <= -1e-15) || !(z <= 5e-17)) {
		tmp = (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + (((-1259.1392167224028 / (2.0 - z)) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z)))) + ((-0.13857109526572012 / (6.0 - z)) + (676.5203681218851 / (1.0 - z)))))) * (exp((z + -7.5)) * ((((double) M_PI) * (sqrt((((double) M_PI) * 2.0)) * pow((7.5 - z), (0.5 - z)))) / sin((((double) M_PI) * z))));
	} else {
		tmp = ((263.3831869810514 * exp(-7.5)) * sqrt(((double) M_PI))) / (z / sqrt(15.0));
	}
	return tmp;
}

Java

public static double code(double z) {
	double tmp;
	if ((z <= -1e-15) || !(z <= 5e-17)) {
		tmp = (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + (((-1259.1392167224028 / (2.0 - z)) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z)))) + ((-0.13857109526572012 / (6.0 - z)) + (676.5203681218851 / (1.0 - z)))))) * (Math.exp((z + -7.5)) * ((Math.PI * (Math.sqrt((Math.PI * 2.0)) * Math.pow((7.5 - z), (0.5 - z)))) / Math.sin((Math.PI * z))));
	} else {
		tmp = ((263.3831869810514 * Math.exp(-7.5)) * Math.sqrt(Math.PI)) / (z / Math.sqrt(15.0));
	}
	return tmp;
}

Python

def code(z):
	tmp = 0
	if (z <= -1e-15) or not (z <= 5e-17):
		tmp = (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + (((-1259.1392167224028 / (2.0 - z)) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z)))) + ((-0.13857109526572012 / (6.0 - z)) + (676.5203681218851 / (1.0 - z)))))) * (math.exp((z + -7.5)) * ((math.pi * (math.sqrt((math.pi * 2.0)) * math.pow((7.5 - z), (0.5 - z)))) / math.sin((math.pi * z))))
	else:
		tmp = ((263.3831869810514 * math.exp(-7.5)) * math.sqrt(math.pi)) / (z / math.sqrt(15.0))
	return tmp

Julia

function code(z)
	tmp = 0.0
	if ((z <= -1e-15) || !(z <= 5e-17))
		tmp = Float64(Float64(Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))) + Float64(Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(12.507343278686905 / Float64(5.0 - z))) + Float64(Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) + Float64(0.9999999999998099 + Float64(771.3234287776531 / Float64(3.0 - z)))) + Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(676.5203681218851 / Float64(1.0 - z)))))) * Float64(exp(Float64(z + -7.5)) * Float64(Float64(pi * Float64(sqrt(Float64(pi * 2.0)) * (Float64(7.5 - z) ^ Float64(0.5 - z)))) / sin(Float64(pi * z)))));
	else
		tmp = Float64(Float64(Float64(263.3831869810514 * exp(-7.5)) * sqrt(pi)) / Float64(z / sqrt(15.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	tmp = 0.0;
	if ((z <= -1e-15) || ~((z <= 5e-17)))
		tmp = (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + (((-1259.1392167224028 / (2.0 - z)) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z)))) + ((-0.13857109526572012 / (6.0 - z)) + (676.5203681218851 / (1.0 - z)))))) * (exp((z + -7.5)) * ((pi * (sqrt((pi * 2.0)) * ((7.5 - z) ^ (0.5 - z)))) / sin((pi * z))));
	else
		tmp = ((263.3831869810514 * exp(-7.5)) * sqrt(pi)) / (z / sqrt(15.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := If[Or[LessEqual[z, -1e-15], N[Not[LessEqual[z, 5e-17]], $MachinePrecision]], N[(N[(N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(N[(Pi * 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[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(263.3831869810514 * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] / N[(z / N[Sqrt[15.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.0000000000000001e-15 or 4.9999999999999999e-17 < z

   1. Initial program 97.8%

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

   3. Simplified 97.9%

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

   5. Applied egg-rr 97.9%

      \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right) + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)}\right)\right)} - 1}}{\sin \left(\pi \cdot z\right)}\right) \]
   6. Step-by-step derivation

      1. expm1-def 97.8%

         \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right) + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)}\right)\right)\right)}}{\sin \left(\pi \cdot z\right)}\right) \]

      2. expm1-log1p 97.9%

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

      3. *-commutative 97.9%

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

      4. fma-udef 97.9%

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

      5. neg-mul-1 97.9%

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

      6. +-commutative 97.9%

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

      7. sub-neg 97.9%

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

      8. *-commutative 97.9%

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

   7. Simplified 97.9%

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

   if -1.0000000000000001e-15 < z < 4.9999999999999999e-17

   1. Initial program 97.4%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in z around 0 98.0%

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

   5. Taylor expanded in z around 0 98.1%

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

   6. Taylor expanded in z around 0 99.0%

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

      1. associate-*r* 98.9%

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

      2. associate-/l* 99.1%

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

      3. associate-*r/ 99.1%

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

   8. Simplified 99.1%

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

      1. associate-*l/ 99.6%

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

      2. sqrt-unprod 99.6%

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

      3. metadata-eval 99.6%

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

   10. Applied egg-rr 99.6%

       \[\leadsto \color{blue}{\frac{\left(263.3831869810514 \cdot e^{-7.5}\right) \cdot \sqrt{\pi}}{\frac{z}{\sqrt{15}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-15} \lor \neg \left(z \leq 5 \cdot 10^{-17}\right):\\ \;\;\;\;\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right)\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)}{\sin \left(\pi \cdot z\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(263.3831869810514 \cdot e^{-7.5}\right) \cdot \sqrt{\pi}}{\frac{z}{\sqrt{15}}}\\ \end{array} \]

Alternative 8: 98.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\\ t_1 := \sin \left(\pi \cdot z\right)\\ t_2 := e^{z + -7.5}\\ t_3 := \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right)\right)\right)\\ t_4 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-15}:\\ \;\;\;\;\left(t_2 \cdot \frac{t_4 \cdot \left(\pi \cdot t_0\right)}{t_1}\right) \cdot t_3\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(263.3831869810514 \cdot e^{-7.5}\right) \cdot \sqrt{\pi}}{\frac{z}{\sqrt{15}}}\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(t_2 \cdot \frac{\pi \cdot \left(t_4 \cdot t_0\right)}{t_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (pow (- 7.5 z) (- 0.5 z)))
        (t_1 (sin (* PI z)))
        (t_2 (exp (+ z -7.5)))
        (t_3
         (+
          (+
           (/ 9.984369578019572e-6 (- 7.0 z))
           (/ 1.5056327351493116e-7 (- 8.0 z)))
          (+
           (+
            (/ -176.6150291621406 (- 4.0 z))
            (/ 12.507343278686905 (- 5.0 z)))
           (+
            (+
             (/ -1259.1392167224028 (- 2.0 z))
             (+ 0.9999999999998099 (/ 771.3234287776531 (- 3.0 z))))
            (+
             (/ -0.13857109526572012 (- 6.0 z))
             (/ 676.5203681218851 (- 1.0 z)))))))
        (t_4 (sqrt (* PI 2.0))))
   (if (<= z -1e-15)
     (* (* t_2 (/ (* t_4 (* PI t_0)) t_1)) t_3)
     (if (<= z 5e-17)
       (/ (* (* 263.3831869810514 (exp -7.5)) (sqrt PI)) (/ z (sqrt 15.0)))
       (* t_3 (* t_2 (/ (* PI (* t_4 t_0)) t_1)))))))

C

double code(double z) {
	double t_0 = pow((7.5 - z), (0.5 - z));
	double t_1 = sin((((double) M_PI) * z));
	double t_2 = exp((z + -7.5));
	double t_3 = ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + (((-1259.1392167224028 / (2.0 - z)) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z)))) + ((-0.13857109526572012 / (6.0 - z)) + (676.5203681218851 / (1.0 - z)))));
	double t_4 = sqrt((((double) M_PI) * 2.0));
	double tmp;
	if (z <= -1e-15) {
		tmp = (t_2 * ((t_4 * (((double) M_PI) * t_0)) / t_1)) * t_3;
	} else if (z <= 5e-17) {
		tmp = ((263.3831869810514 * exp(-7.5)) * sqrt(((double) M_PI))) / (z / sqrt(15.0));
	} else {
		tmp = t_3 * (t_2 * ((((double) M_PI) * (t_4 * t_0)) / t_1));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.pow((7.5 - z), (0.5 - z));
	double t_1 = Math.sin((Math.PI * z));
	double t_2 = Math.exp((z + -7.5));
	double t_3 = ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + (((-1259.1392167224028 / (2.0 - z)) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z)))) + ((-0.13857109526572012 / (6.0 - z)) + (676.5203681218851 / (1.0 - z)))));
	double t_4 = Math.sqrt((Math.PI * 2.0));
	double tmp;
	if (z <= -1e-15) {
		tmp = (t_2 * ((t_4 * (Math.PI * t_0)) / t_1)) * t_3;
	} else if (z <= 5e-17) {
		tmp = ((263.3831869810514 * Math.exp(-7.5)) * Math.sqrt(Math.PI)) / (z / Math.sqrt(15.0));
	} else {
		tmp = t_3 * (t_2 * ((Math.PI * (t_4 * t_0)) / t_1));
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.pow((7.5 - z), (0.5 - z))
	t_1 = math.sin((math.pi * z))
	t_2 = math.exp((z + -7.5))
	t_3 = ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + (((-1259.1392167224028 / (2.0 - z)) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z)))) + ((-0.13857109526572012 / (6.0 - z)) + (676.5203681218851 / (1.0 - z)))))
	t_4 = math.sqrt((math.pi * 2.0))
	tmp = 0
	if z <= -1e-15:
		tmp = (t_2 * ((t_4 * (math.pi * t_0)) / t_1)) * t_3
	elif z <= 5e-17:
		tmp = ((263.3831869810514 * math.exp(-7.5)) * math.sqrt(math.pi)) / (z / math.sqrt(15.0))
	else:
		tmp = t_3 * (t_2 * ((math.pi * (t_4 * t_0)) / t_1))
	return tmp

Julia

function code(z)
	t_0 = Float64(7.5 - z) ^ Float64(0.5 - z)
	t_1 = sin(Float64(pi * z))
	t_2 = exp(Float64(z + -7.5))
	t_3 = Float64(Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))) + Float64(Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(12.507343278686905 / Float64(5.0 - z))) + Float64(Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) + Float64(0.9999999999998099 + Float64(771.3234287776531 / Float64(3.0 - z)))) + Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(676.5203681218851 / Float64(1.0 - z))))))
	t_4 = sqrt(Float64(pi * 2.0))
	tmp = 0.0
	if (z <= -1e-15)
		tmp = Float64(Float64(t_2 * Float64(Float64(t_4 * Float64(pi * t_0)) / t_1)) * t_3);
	elseif (z <= 5e-17)
		tmp = Float64(Float64(Float64(263.3831869810514 * exp(-7.5)) * sqrt(pi)) / Float64(z / sqrt(15.0)));
	else
		tmp = Float64(t_3 * Float64(t_2 * Float64(Float64(pi * Float64(t_4 * t_0)) / t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = (7.5 - z) ^ (0.5 - z);
	t_1 = sin((pi * z));
	t_2 = exp((z + -7.5));
	t_3 = ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + (((-1259.1392167224028 / (2.0 - z)) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z)))) + ((-0.13857109526572012 / (6.0 - z)) + (676.5203681218851 / (1.0 - z)))));
	t_4 = sqrt((pi * 2.0));
	tmp = 0.0;
	if (z <= -1e-15)
		tmp = (t_2 * ((t_4 * (pi * t_0)) / t_1)) * t_3;
	elseif (z <= 5e-17)
		tmp = ((263.3831869810514 * exp(-7.5)) * sqrt(pi)) / (z / sqrt(15.0));
	else
		tmp = t_3 * (t_2 * ((pi * (t_4 * t_0)) / t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1e-15], N[(N[(t$95$2 * N[(N[(t$95$4 * N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[z, 5e-17], N[(N[(N[(263.3831869810514 * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] / N[(z / N[Sqrt[15.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * N[(t$95$2 * N[(N[(Pi * N[(t$95$4 * t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.0000000000000001e-15

   1. Initial program 97.9%

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

   3. Simplified 97.8%

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

   5. Applied egg-rr 97.9%

      \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right) + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)}\right)\right)} - 1}}{\sin \left(\pi \cdot z\right)}\right) \]
   6. Step-by-step derivation

      1. expm1-def 97.7%

         \[\leadsto \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{771.3234287776531}{3 - z} + 0.9999999999998099\right)\right) + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)}\right)\right)\right)}}{\sin \left(\pi \cdot z\right)}\right) \]

      2. expm1-log1p 97.5%

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

      3. *-commutative 97.5%

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

      4. fma-udef 97.5%

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

      5. neg-mul-1 97.5%

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

      6. +-commutative 97.5%

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

      7. sub-neg 97.5%

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

      8. exp-to-pow 97.2%

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

      9. associate-*l* 97.1%

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

      10. *-commutative 97.1%

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

      11. exp-to-pow 97.5%

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

   7. Simplified 97.8%

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

   if -1.0000000000000001e-15 < z < 4.9999999999999999e-17

   1. Initial program 97.4%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in z around 0 98.0%

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

   5. Taylor expanded in z around 0 98.1%

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

   6. Taylor expanded in z around 0 99.0%

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

      1. associate-*r* 98.9%

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

      2. associate-/l* 99.1%

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

      3. associate-*r/ 99.1%

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

   8. Simplified 99.1%

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

      1. associate-*l/ 99.6%

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

      2. sqrt-unprod 99.6%

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

      3. metadata-eval 99.6%

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

   10. Applied egg-rr 99.6%

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

   if 4.9999999999999999e-17 < z

   1. Initial program 97.6%

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

   3. Simplified 98.2%

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

   5. Applied egg-rr 97.9%

      \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right) + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)}\right)\right)} - 1}}{\sin \left(\pi \cdot z\right)}\right) \]
   6. Step-by-step derivation

      1. expm1-def 97.9%

         \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right) + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)}\right)\right)\right)}}{\sin \left(\pi \cdot z\right)}\right) \]

      2. expm1-log1p 98.4%

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

      3. *-commutative 98.4%

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

      4. fma-udef 98.4%

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

      5. neg-mul-1 98.4%

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

      6. +-commutative 98.4%

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

      7. sub-neg 98.4%

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

      8. *-commutative 98.4%

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

   7. Simplified 98.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-15}:\\ \;\;\;\;\left(e^{z + -7.5} \cdot \frac{\sqrt{\pi \cdot 2} \cdot \left(\pi \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right)\right)\right)\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(263.3831869810514 \cdot e^{-7.5}\right) \cdot \sqrt{\pi}}{\frac{z}{\sqrt{15}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right)\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)}{\sin \left(\pi \cdot z\right)}\right)\\ \end{array} \]

Alternative 9: 98.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\\ t_1 := e^{z + -7.5} \cdot \frac{\sqrt{\pi \cdot 2} \cdot \left(\pi \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)}{\sin \left(\pi \cdot z\right)}\\ t_2 := \frac{-0.13857109526572012}{6 - z}\\ t_3 := \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\\ t_4 := \frac{676.5203681218851}{1 - z}\\ t_5 := \frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\\ t_6 := \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-15}:\\ \;\;\;\;t_1 \cdot \left(\left(t_3 + t_6\right) + \left(t_0 + \left(t_5 + \left(t_2 + t_4\right)\right)\right)\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(263.3831869810514 \cdot e^{-7.5}\right) \cdot \sqrt{\pi}}{\frac{z}{\sqrt{15}}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\left(t_3 + \left(t_2 + t_6\right)\right) + \left(t_4 + \left(t_0 + t_5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0
         (+ (/ -176.6150291621406 (- 4.0 z)) (/ 12.507343278686905 (- 5.0 z))))
        (t_1
         (*
          (exp (+ z -7.5))
          (/
           (* (sqrt (* PI 2.0)) (* PI (pow (- 7.5 z) (- 0.5 z))))
           (sin (* PI z)))))
        (t_2 (/ -0.13857109526572012 (- 6.0 z)))
        (t_3 (/ 9.984369578019572e-6 (- 7.0 z)))
        (t_4 (/ 676.5203681218851 (- 1.0 z)))
        (t_5
         (+
          (/ -1259.1392167224028 (- 2.0 z))
          (+ 0.9999999999998099 (/ 771.3234287776531 (- 3.0 z)))))
        (t_6 (/ 1.5056327351493116e-7 (- 8.0 z))))
   (if (<= z -1e-15)
     (* t_1 (+ (+ t_3 t_6) (+ t_0 (+ t_5 (+ t_2 t_4)))))
     (if (<= z 4e-18)
       (/ (* (* 263.3831869810514 (exp -7.5)) (sqrt PI)) (/ z (sqrt 15.0)))
       (* t_1 (+ (+ t_3 (+ t_2 t_6)) (+ t_4 (+ t_0 t_5))))))))

C

double code(double z) {
	double t_0 = (-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z));
	double t_1 = exp((z + -7.5)) * ((sqrt((((double) M_PI) * 2.0)) * (((double) M_PI) * pow((7.5 - z), (0.5 - z)))) / sin((((double) M_PI) * z)));
	double t_2 = -0.13857109526572012 / (6.0 - z);
	double t_3 = 9.984369578019572e-6 / (7.0 - z);
	double t_4 = 676.5203681218851 / (1.0 - z);
	double t_5 = (-1259.1392167224028 / (2.0 - z)) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z)));
	double t_6 = 1.5056327351493116e-7 / (8.0 - z);
	double tmp;
	if (z <= -1e-15) {
		tmp = t_1 * ((t_3 + t_6) + (t_0 + (t_5 + (t_2 + t_4))));
	} else if (z <= 4e-18) {
		tmp = ((263.3831869810514 * exp(-7.5)) * sqrt(((double) M_PI))) / (z / sqrt(15.0));
	} else {
		tmp = t_1 * ((t_3 + (t_2 + t_6)) + (t_4 + (t_0 + t_5)));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = (-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z));
	double t_1 = Math.exp((z + -7.5)) * ((Math.sqrt((Math.PI * 2.0)) * (Math.PI * Math.pow((7.5 - z), (0.5 - z)))) / Math.sin((Math.PI * z)));
	double t_2 = -0.13857109526572012 / (6.0 - z);
	double t_3 = 9.984369578019572e-6 / (7.0 - z);
	double t_4 = 676.5203681218851 / (1.0 - z);
	double t_5 = (-1259.1392167224028 / (2.0 - z)) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z)));
	double t_6 = 1.5056327351493116e-7 / (8.0 - z);
	double tmp;
	if (z <= -1e-15) {
		tmp = t_1 * ((t_3 + t_6) + (t_0 + (t_5 + (t_2 + t_4))));
	} else if (z <= 4e-18) {
		tmp = ((263.3831869810514 * Math.exp(-7.5)) * Math.sqrt(Math.PI)) / (z / Math.sqrt(15.0));
	} else {
		tmp = t_1 * ((t_3 + (t_2 + t_6)) + (t_4 + (t_0 + t_5)));
	}
	return tmp;
}

Python

def code(z):
	t_0 = (-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))
	t_1 = math.exp((z + -7.5)) * ((math.sqrt((math.pi * 2.0)) * (math.pi * math.pow((7.5 - z), (0.5 - z)))) / math.sin((math.pi * z)))
	t_2 = -0.13857109526572012 / (6.0 - z)
	t_3 = 9.984369578019572e-6 / (7.0 - z)
	t_4 = 676.5203681218851 / (1.0 - z)
	t_5 = (-1259.1392167224028 / (2.0 - z)) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z)))
	t_6 = 1.5056327351493116e-7 / (8.0 - z)
	tmp = 0
	if z <= -1e-15:
		tmp = t_1 * ((t_3 + t_6) + (t_0 + (t_5 + (t_2 + t_4))))
	elif z <= 4e-18:
		tmp = ((263.3831869810514 * math.exp(-7.5)) * math.sqrt(math.pi)) / (z / math.sqrt(15.0))
	else:
		tmp = t_1 * ((t_3 + (t_2 + t_6)) + (t_4 + (t_0 + t_5)))
	return tmp

Julia

function code(z)
	t_0 = Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(12.507343278686905 / Float64(5.0 - z)))
	t_1 = Float64(exp(Float64(z + -7.5)) * Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64(pi * (Float64(7.5 - z) ^ Float64(0.5 - z)))) / sin(Float64(pi * z))))
	t_2 = Float64(-0.13857109526572012 / Float64(6.0 - z))
	t_3 = Float64(9.984369578019572e-6 / Float64(7.0 - z))
	t_4 = Float64(676.5203681218851 / Float64(1.0 - z))
	t_5 = Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) + Float64(0.9999999999998099 + Float64(771.3234287776531 / Float64(3.0 - z))))
	t_6 = Float64(1.5056327351493116e-7 / Float64(8.0 - z))
	tmp = 0.0
	if (z <= -1e-15)
		tmp = Float64(t_1 * Float64(Float64(t_3 + t_6) + Float64(t_0 + Float64(t_5 + Float64(t_2 + t_4)))));
	elseif (z <= 4e-18)
		tmp = Float64(Float64(Float64(263.3831869810514 * exp(-7.5)) * sqrt(pi)) / Float64(z / sqrt(15.0)));
	else
		tmp = Float64(t_1 * Float64(Float64(t_3 + Float64(t_2 + t_6)) + Float64(t_4 + Float64(t_0 + t_5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = (-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z));
	t_1 = exp((z + -7.5)) * ((sqrt((pi * 2.0)) * (pi * ((7.5 - z) ^ (0.5 - z)))) / sin((pi * z)));
	t_2 = -0.13857109526572012 / (6.0 - z);
	t_3 = 9.984369578019572e-6 / (7.0 - z);
	t_4 = 676.5203681218851 / (1.0 - z);
	t_5 = (-1259.1392167224028 / (2.0 - z)) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z)));
	t_6 = 1.5056327351493116e-7 / (8.0 - z);
	tmp = 0.0;
	if (z <= -1e-15)
		tmp = t_1 * ((t_3 + t_6) + (t_0 + (t_5 + (t_2 + t_4))));
	elseif (z <= 4e-18)
		tmp = ((263.3831869810514 * exp(-7.5)) * sqrt(pi)) / (z / sqrt(15.0));
	else
		tmp = t_1 * ((t_3 + (t_2 + t_6)) + (t_4 + (t_0 + t_5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(Pi * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e-15], N[(t$95$1 * N[(N[(t$95$3 + t$95$6), $MachinePrecision] + N[(t$95$0 + N[(t$95$5 + N[(t$95$2 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4e-18], N[(N[(N[(263.3831869810514 * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] / N[(z / N[Sqrt[15.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(t$95$3 + N[(t$95$2 + t$95$6), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 + N[(t$95$0 + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.0000000000000001e-15

   1. Initial program 97.9%

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

   3. Simplified 97.8%

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

   5. Applied egg-rr 97.9%

      \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right) + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)}\right)\right)} - 1}}{\sin \left(\pi \cdot z\right)}\right) \]
   6. Step-by-step derivation

      1. expm1-def 97.7%

         \[\leadsto \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{771.3234287776531}{3 - z} + 0.9999999999998099\right)\right) + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)}\right)\right)\right)}}{\sin \left(\pi \cdot z\right)}\right) \]

      2. expm1-log1p 97.5%

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

      3. *-commutative 97.5%

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

      4. fma-udef 97.5%

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

      5. neg-mul-1 97.5%

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

      6. +-commutative 97.5%

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

      7. sub-neg 97.5%

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

      8. exp-to-pow 97.2%

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

      9. associate-*l* 97.1%

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

      10. *-commutative 97.1%

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

      11. exp-to-pow 97.5%

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

   7. Simplified 97.8%

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

   if -1.0000000000000001e-15 < z < 4.0000000000000003e-18

   1. Initial program 97.4%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in z around 0 98.0%

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

   5. Taylor expanded in z around 0 98.1%

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

   6. Taylor expanded in z around 0 99.0%

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

      1. associate-*r* 98.9%

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

      2. associate-/l* 99.1%

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

      3. associate-*r/ 99.1%

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

   8. Simplified 99.1%

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

      1. associate-*l/ 99.6%

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

      2. sqrt-unprod 99.6%

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

      3. metadata-eval 99.6%

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

   10. Applied egg-rr 99.6%

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

   if 4.0000000000000003e-18 < z

   1. Initial program 97.6%

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

   3. Simplified 98.2%

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

      1. expm1-log1p-u 97.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right) + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)} \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

      2. expm1-udef 97.9%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right) + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)} - 1\right)} \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

   5. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)} - 1\right)} \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

   7. Simplified 98.8%

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

   9. Applied egg-rr 98.4%

      \[\leadsto \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{771.3234287776531}{3 - z} + 0.9999999999998099\right)\right) + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)}\right)\right)} - 1}}{\sin \left(\pi \cdot z\right)}\right) \]
   10. Step-by-step derivation

       1. expm1-def 98.4%

          \[\leadsto \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{771.3234287776531}{3 - z} + 0.9999999999998099\right)\right) + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)}\right)\right)\right)}}{\sin \left(\pi \cdot z\right)}\right) \]

       2. expm1-log1p 98.5%

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

       3. *-commutative 98.5%

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

       4. fma-udef 98.5%

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

       5. neg-mul-1 98.5%

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

       6. +-commutative 98.5%

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

       7. sub-neg 98.5%

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

       8. exp-to-pow 98.4%

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

       9. associate-*l* 98.9%

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

       10. *-commutative 98.9%

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

       11. exp-to-pow 98.8%

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

   11. Simplified 98.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-15}:\\ \;\;\;\;\left(e^{z + -7.5} \cdot \frac{\sqrt{\pi \cdot 2} \cdot \left(\pi \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right)\right)\right)\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(263.3831869810514 \cdot e^{-7.5}\right) \cdot \sqrt{\pi}}{\frac{z}{\sqrt{15}}}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{z + -7.5} \cdot \frac{\sqrt{\pi \cdot 2} \cdot \left(\pi \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{-0.13857109526572012}{6 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) + \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 10: 97.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

3. Simplified 96.6%

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

   1. expm1-log1p-u 96.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right) + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)} \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

   2. expm1-udef 96.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right) + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)} - 1\right)} \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

5. Applied egg-rr 96.5%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)} - 1\right)} \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

7. Simplified 97.2%

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

   1. expm1-log1p-u 0.4%

      \[\leadsto \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{676.5203681218851}{1 - z} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{771.3234287776531}{3 - z} + 0.9999999999998099\right)\right) + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

   2. expm1-udef 0.4%

      \[\leadsto \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{676.5203681218851}{1 - z} + \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{771.3234287776531}{3 - z} + 0.9999999999998099\right)\right) + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)} - 1\right)}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

   3. associate-+l+ 0.4%

      \[\leadsto \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{676.5203681218851}{1 - z} + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{-1259.1392167224028}{2 - z} + \left(\left(\frac{771.3234287776531}{3 - z} + 0.9999999999998099\right) + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)}\right)} - 1\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

   4. +-commutative 0.4%

      \[\leadsto \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{676.5203681218851}{1 - z} + \left(e^{\mathsf{log1p}\left(\frac{-1259.1392167224028}{2 - z} + \left(\color{blue}{\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)} - 1\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

9. Applied egg-rr 0.4%

   \[\leadsto \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{676.5203681218851}{1 - z} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-1259.1392167224028}{2 - z} + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)} - 1\right)}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]
10. Step-by-step derivation

    1. expm1-def 0.4%

       \[\leadsto \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{676.5203681218851}{1 - z} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1259.1392167224028}{2 - z} + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

    2. expm1-log1p 97.2%

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

    3. associate-+r+ 97.2%

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

    4. associate-+r+ 98.2%

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

11. Simplified 98.2%

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

12. Final simplification 98.2%

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

Alternative 11: 97.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

3. Simplified 96.6%

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

   1. expm1-log1p-u 96.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right) + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)} \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

   2. expm1-udef 96.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right) + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)} - 1\right)} \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

5. Applied egg-rr 96.5%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)} - 1\right)} \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]
6. Step-by-step derivation

   1. expm1-def 96.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)} \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

   2. expm1-log1p 97.2%

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

   3. associate-+l+ 98.2%

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

   4. associate-+l+ 98.2%

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

7. Simplified 98.2%

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

8. Final simplification 98.2%

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

Alternative 12: 96.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \left(-1 + \left(1 - z\right)\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{\left(\left(z + -1\right) - -1\right) - 7.5}\right) \cdot \frac{1}{z}\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right) + \left(\left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \left(47.95075976068351 + z \cdot 361.7355639412844\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (*
    (* (sqrt (* PI 2.0)) (pow (+ 7.5 (+ -1.0 (- 1.0 z))) (- (- 1.0 z) 0.5)))
    (exp (- (- (+ z -1.0) -1.0) 7.5)))
   (/ 1.0 z))
  (+
   (+
    (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))
    (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0)))
   (+
    (+
     (/ 12.507343278686905 (- (- 1.0 z) -4.0))
     (/ -0.13857109526572012 (- (- 1.0 z) -5.0)))
    (+
     (+
      (/ 771.3234287776531 (- (- 1.0 z) -2.0))
      (/ -176.6150291621406 (- (- 1.0 z) -3.0)))
     (+ 47.95075976068351 (* z 361.7355639412844)))))))

C

double code(double z) {
	return (((sqrt((((double) M_PI) * 2.0)) * pow((7.5 + (-1.0 + (1.0 - z))), ((1.0 - z) - 0.5))) * exp((((z + -1.0) - -1.0) - 7.5))) * (1.0 / z)) * (((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))) + (((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0))) + (((771.3234287776531 / ((1.0 - z) - -2.0)) + (-176.6150291621406 / ((1.0 - z) - -3.0))) + (47.95075976068351 + (z * 361.7355639412844)))));
}

Java

public static double code(double z) {
	return (((Math.sqrt((Math.PI * 2.0)) * Math.pow((7.5 + (-1.0 + (1.0 - z))), ((1.0 - z) - 0.5))) * Math.exp((((z + -1.0) - -1.0) - 7.5))) * (1.0 / z)) * (((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))) + (((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0))) + (((771.3234287776531 / ((1.0 - z) - -2.0)) + (-176.6150291621406 / ((1.0 - z) - -3.0))) + (47.95075976068351 + (z * 361.7355639412844)))));
}

Python

def code(z):
	return (((math.sqrt((math.pi * 2.0)) * math.pow((7.5 + (-1.0 + (1.0 - z))), ((1.0 - z) - 0.5))) * math.exp((((z + -1.0) - -1.0) - 7.5))) * (1.0 / z)) * (((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))) + (((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0))) + (((771.3234287776531 / ((1.0 - z) - -2.0)) + (-176.6150291621406 / ((1.0 - z) - -3.0))) + (47.95075976068351 + (z * 361.7355639412844)))))

Julia

function code(z)
	return Float64(Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (Float64(7.5 + Float64(-1.0 + Float64(1.0 - z))) ^ Float64(Float64(1.0 - z) - 0.5))) * exp(Float64(Float64(Float64(z + -1.0) - -1.0) - 7.5))) * Float64(1.0 / z)) * 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(Float64(Float64(12.507343278686905 / Float64(Float64(1.0 - z) - -4.0)) + Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) - -5.0))) + Float64(Float64(Float64(771.3234287776531 / Float64(Float64(1.0 - z) - -2.0)) + Float64(-176.6150291621406 / Float64(Float64(1.0 - z) - -3.0))) + Float64(47.95075976068351 + Float64(z * 361.7355639412844))))))
end

MATLAB

function tmp = code(z)
	tmp = (((sqrt((pi * 2.0)) * ((7.5 + (-1.0 + (1.0 - z))) ^ ((1.0 - z) - 0.5))) * exp((((z + -1.0) - -1.0) - 7.5))) * (1.0 / z)) * (((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))) + (((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0))) + (((771.3234287776531 / ((1.0 - z) - -2.0)) + (-176.6150291621406 / ((1.0 - z) - -3.0))) + (47.95075976068351 + (z * 361.7355639412844)))));
end

Wolfram

code[z_] := N[(N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 + N[(-1.0 + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(N[(z + -1.0), $MachinePrecision] - -1.0), $MachinePrecision] - 7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $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[(N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] - -4.0), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] - -5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(771.3234287776531 / N[(N[(1.0 - z), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(47.95075976068351 + N[(z * 361.7355639412844), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.4%

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

3. Simplified 98.9%

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

4. Taylor expanded in z around 0 96.6%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\color{blue}{\left(47.95075976068351 + 361.7355639412844 \cdot z\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
5. Step-by-step derivation

   1. *-commutative 96.6%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\left(47.95075976068351 + \color{blue}{z \cdot 361.7355639412844}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

6. Simplified 96.6%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\color{blue}{\left(47.95075976068351 + z \cdot 361.7355639412844\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

7. Taylor expanded in z around 0 97.1%

   \[\leadsto \left(\color{blue}{\frac{1}{z}} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\left(47.95075976068351 + z \cdot 361.7355639412844\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

8. Final simplification 97.1%

   \[\leadsto \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \left(-1 + \left(1 - z\right)\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{\left(\left(z + -1\right) - -1\right) - 7.5}\right) \cdot \frac{1}{z}\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right) + \left(\left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \left(47.95075976068351 + z \cdot 361.7355639412844\right)\right)\right)\right) \]

Alternative 13: 96.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{1}{z} \cdot \left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{8 + \left(-1 - z\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 + \left(-1 - z\right)}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (pow (+ (- 1.0 z) 6.5) (- 1.0 (+ z 0.5)))
   (* (sqrt (* PI 2.0)) (exp (+ z -7.5))))
  (*
   (/ 1.0 z)
   (+
    (+ 260.9048120626994 (* z 436.3997278161676))
    (+
     (+
      (/ 9.984369578019572e-6 (+ 8.0 (- -1.0 z)))
      (/ 1.5056327351493116e-7 (+ 9.0 (- -1.0 z))))
     (+
      (/ 12.507343278686905 (+ (- 1.0 z) 4.0))
      (/ -0.13857109526572012 (- 1.0 (+ z -5.0)))))))))

C

double code(double z) {
	return (pow(((1.0 - z) + 6.5), (1.0 - (z + 0.5))) * (sqrt((((double) M_PI) * 2.0)) * exp((z + -7.5)))) * ((1.0 / z) * ((260.9048120626994 + (z * 436.3997278161676)) + (((9.984369578019572e-6 / (8.0 + (-1.0 - z))) + (1.5056327351493116e-7 / (9.0 + (-1.0 - z)))) + ((12.507343278686905 / ((1.0 - z) + 4.0)) + (-0.13857109526572012 / (1.0 - (z + -5.0)))))));
}

Java

public static double code(double z) {
	return (Math.pow(((1.0 - z) + 6.5), (1.0 - (z + 0.5))) * (Math.sqrt((Math.PI * 2.0)) * Math.exp((z + -7.5)))) * ((1.0 / z) * ((260.9048120626994 + (z * 436.3997278161676)) + (((9.984369578019572e-6 / (8.0 + (-1.0 - z))) + (1.5056327351493116e-7 / (9.0 + (-1.0 - z)))) + ((12.507343278686905 / ((1.0 - z) + 4.0)) + (-0.13857109526572012 / (1.0 - (z + -5.0)))))));
}

Python

def code(z):
	return (math.pow(((1.0 - z) + 6.5), (1.0 - (z + 0.5))) * (math.sqrt((math.pi * 2.0)) * math.exp((z + -7.5)))) * ((1.0 / z) * ((260.9048120626994 + (z * 436.3997278161676)) + (((9.984369578019572e-6 / (8.0 + (-1.0 - z))) + (1.5056327351493116e-7 / (9.0 + (-1.0 - z)))) + ((12.507343278686905 / ((1.0 - z) + 4.0)) + (-0.13857109526572012 / (1.0 - (z + -5.0)))))))

Julia

function code(z)
	return Float64(Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(1.0 - Float64(z + 0.5))) * Float64(sqrt(Float64(pi * 2.0)) * exp(Float64(z + -7.5)))) * Float64(Float64(1.0 / z) * Float64(Float64(260.9048120626994 + Float64(z * 436.3997278161676)) + Float64(Float64(Float64(9.984369578019572e-6 / Float64(8.0 + Float64(-1.0 - z))) + Float64(1.5056327351493116e-7 / Float64(9.0 + Float64(-1.0 - z)))) + Float64(Float64(12.507343278686905 / Float64(Float64(1.0 - z) + 4.0)) + Float64(-0.13857109526572012 / Float64(1.0 - Float64(z + -5.0))))))))
end

MATLAB

function tmp = code(z)
	tmp = ((((1.0 - z) + 6.5) ^ (1.0 - (z + 0.5))) * (sqrt((pi * 2.0)) * exp((z + -7.5)))) * ((1.0 / z) * ((260.9048120626994 + (z * 436.3997278161676)) + (((9.984369578019572e-6 / (8.0 + (-1.0 - z))) + (1.5056327351493116e-7 / (9.0 + (-1.0 - z)))) + ((12.507343278686905 / ((1.0 - z) + 4.0)) + (-0.13857109526572012 / (1.0 - (z + -5.0)))))));
end

Wolfram

code[z_] := N[(N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(1.0 - N[(z + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / z), $MachinePrecision] * N[(N[(260.9048120626994 + N[(z * 436.3997278161676), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(9.984369578019572e-6 / N[(8.0 + N[(-1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(9.0 + N[(-1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(1.0 - N[(z + -5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.4%

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

3. Simplified 98.3%

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

4. Taylor expanded in z around 0 96.0%

   \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(260.9048120626994 + 436.3997278161676 \cdot z\right)} + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
5. Step-by-step derivation

   1. *-commutative 96.0%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + \color{blue}{z \cdot 436.3997278161676}\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

6. Simplified 96.0%

   \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(260.9048120626994 + z \cdot 436.3997278161676\right)} + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
7. Step-by-step derivation

   1. expm1-log1p-u 96.0%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)}\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   2. expm1-udef 84.3%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)} - 1\right)}\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

8. Applied egg-rr 84.3%

   \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\pi \cdot 2} \cdot e^{-\mathsf{fma}\left(-1, z, 7.5\right)}\right)} - 1\right)}\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
9. Step-by-step derivation

   1. expm1-def 96.0%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi \cdot 2} \cdot e^{-\mathsf{fma}\left(-1, z, 7.5\right)}\right)\right)}\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   2. expm1-log1p 96.0%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot e^{-\mathsf{fma}\left(-1, z, 7.5\right)}\right)}\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   3. *-commutative 96.0%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\color{blue}{2 \cdot \pi}} \cdot e^{-\mathsf{fma}\left(-1, z, 7.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   4. neg-mul-1 96.0%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\color{blue}{-1 \cdot \mathsf{fma}\left(-1, z, 7.5\right)}}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   5. fma-udef 96.0%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{-1 \cdot \color{blue}{\left(-1 \cdot z + 7.5\right)}}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   6. neg-mul-1 96.0%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{-1 \cdot \left(\color{blue}{\left(-z\right)} + 7.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   7. distribute-lft-in 96.0%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\color{blue}{-1 \cdot \left(-z\right) + -1 \cdot 7.5}}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. neg-mul-1 96.0%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\color{blue}{\left(-\left(-z\right)\right)} + -1 \cdot 7.5}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   9. remove-double-neg 96.0%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\color{blue}{z} + -1 \cdot 7.5}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   10. metadata-eval 96.0%

       \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{z + \color{blue}{-7.5}}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

10. Simplified 96.0%

    \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \color{blue}{\left(\sqrt{2 \cdot \pi} \cdot e^{z + -7.5}\right)}\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

11. Taylor expanded in z around 0 96.4%

    \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{z + -7.5}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \color{blue}{\frac{1}{z}}\right) \]

12. Final simplification 96.4%

    \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{1}{z} \cdot \left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{8 + \left(-1 - z\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 + \left(-1 - z\right)}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right)\right)\right)\right) \]

Alternative 14: 95.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

3. Simplified 98.3%

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

4. Taylor expanded in z around 0 94.6%

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

5. Taylor expanded in z around 0 94.2%

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

6. Taylor expanded in z around 0 95.2%

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

   1. associate-*r* 95.2%

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

   2. associate-/l* 95.3%

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

   3. associate-*r/ 95.3%

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

8. Simplified 95.3%

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

   1. expm1-log1p-u 40.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{263.3831869810514 \cdot e^{-7.5}}{\frac{z}{\sqrt{2} \cdot \sqrt{7.5}}}\right)\right)} \cdot \sqrt{\pi} \]

   2. expm1-udef 40.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{263.3831869810514 \cdot e^{-7.5}}{\frac{z}{\sqrt{2} \cdot \sqrt{7.5}}}\right)} - 1\right)} \cdot \sqrt{\pi} \]

   3. associate-/r/ 40.2%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{263.3831869810514 \cdot e^{-7.5}}{z} \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)}\right)} - 1\right) \cdot \sqrt{\pi} \]

   4. sqrt-unprod 40.2%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{263.3831869810514 \cdot e^{-7.5}}{z} \cdot \color{blue}{\sqrt{2 \cdot 7.5}}\right)} - 1\right) \cdot \sqrt{\pi} \]

   5. metadata-eval 40.2%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{263.3831869810514 \cdot e^{-7.5}}{z} \cdot \sqrt{\color{blue}{15}}\right)} - 1\right) \cdot \sqrt{\pi} \]

10. Applied egg-rr 40.2%

    \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{263.3831869810514 \cdot e^{-7.5}}{z} \cdot \sqrt{15}\right)} - 1\right)} \cdot \sqrt{\pi} \]
11. Step-by-step derivation

    1. expm1-def 40.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{263.3831869810514 \cdot e^{-7.5}}{z} \cdot \sqrt{15}\right)\right)} \cdot \sqrt{\pi} \]

    2. expm1-log1p 95.3%

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

    3. associate-/l* 95.4%

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

12. Simplified 95.4%

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

13. Final simplification 95.4%

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

Alternative 15: 96.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

3. Simplified 98.3%

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

4. Taylor expanded in z around 0 94.6%

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

5. Taylor expanded in z around 0 94.2%

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

6. Taylor expanded in z around 0 95.2%

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

   1. associate-*r* 95.2%

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

   2. associate-/l* 95.3%

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

   3. associate-*r/ 95.3%

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

8. Simplified 95.3%

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

   1. associate-*l/ 95.7%

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

   2. sqrt-unprod 95.7%

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

   3. metadata-eval 95.7%

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

10. Applied egg-rr 95.7%

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

11. Final simplification 95.7%

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

Reproduce

herbie shell --seed 2023271 
(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))))))