Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.4% → 99.0%

Time: 1.7min

Alternatives: 12

Speedup: 1.1×

• could not determine a ground truth

  1. z = -7.482858345119027e+165

Specification

\[z \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (pow (- 7.5 z) (- 0.5 z)))
        (t_1 (sqrt (* PI 2.0)))
        (t_2 (/ PI (sin (* PI z))))
        (t_3 (+ (- 1.0 z) -1.0))
        (t_4 (+ 7.0 t_3)))
   (if (<=
        (*
         t_2
         (*
          (*
           (* t_1 (pow (+ 0.5 t_4) (+ 0.5 t_3)))
           (exp (- (- (+ 1.0 (+ z -1.0)) 7.0) 0.5)))
          (+
           (+
            (+
             (+
              (+
               (+
                (+
                 (+ 0.9999999999998099 (/ 676.5203681218851 (+ 1.0 t_3)))
                 (/ -1259.1392167224028 (+ 2.0 t_3)))
                (/ 771.3234287776531 (+ 3.0 t_3)))
               (/ -176.6150291621406 (+ 4.0 t_3)))
              (/ 12.507343278686905 (+ 5.0 t_3)))
             (/ -0.13857109526572012 (+ 6.0 t_3)))
            (/ 9.984369578019572e-6 t_4))
           (/ 1.5056327351493116e-7 (+ 8.0 t_3)))))
        1e+307)
     (*
      t_1
      (*
       t_2
       (*
        (exp (+ z -7.5))
        (*
         t_0
         (+
          (+
           (+
            (/ -1259.1392167224028 (- 2.0 z))
            (+
             (/ 771.3234287776531 (- 3.0 z))
             (/ -176.6150291621406 (- 4.0 z))))
           (+
            (/ -0.13857109526572012 (- 6.0 z))
            (+
             (/ 12.507343278686905 (- 5.0 z))
             (+
              (/ 9.984369578019572e-6 (- 7.0 z))
              (/ 1.5056327351493116e-7 (- 8.0 z))))))
          (+ 0.9999999999998099 (/ 676.5203681218851 (- 1.0 z))))))))
     (* t_1 (* t_2 (exp (+ (+ z -7.5) (log (* 0.9999999999998099 t_0)))))))))

C

double code(double z) {
	double t_0 = pow((7.5 - z), (0.5 - z));
	double t_1 = sqrt((((double) M_PI) * 2.0));
	double t_2 = ((double) M_PI) / sin((((double) M_PI) * z));
	double t_3 = (1.0 - z) + -1.0;
	double t_4 = 7.0 + t_3;
	double tmp;
	if ((t_2 * (((t_1 * pow((0.5 + t_4), (0.5 + t_3))) * exp((((1.0 + (z + -1.0)) - 7.0) - 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / (1.0 + t_3))) + (-1259.1392167224028 / (2.0 + t_3))) + (771.3234287776531 / (3.0 + t_3))) + (-176.6150291621406 / (4.0 + t_3))) + (12.507343278686905 / (5.0 + t_3))) + (-0.13857109526572012 / (6.0 + t_3))) + (9.984369578019572e-6 / t_4)) + (1.5056327351493116e-7 / (8.0 + t_3))))) <= 1e+307) {
		tmp = t_1 * (t_2 * (exp((z + -7.5)) * (t_0 * ((((-1259.1392167224028 / (2.0 - z)) + ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z)))) + ((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))))) + (0.9999999999998099 + (676.5203681218851 / (1.0 - z)))))));
	} else {
		tmp = t_1 * (t_2 * exp(((z + -7.5) + log((0.9999999999998099 * t_0)))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.pow((7.5 - z), (0.5 - z));
	double t_1 = Math.sqrt((Math.PI * 2.0));
	double t_2 = Math.PI / Math.sin((Math.PI * z));
	double t_3 = (1.0 - z) + -1.0;
	double t_4 = 7.0 + t_3;
	double tmp;
	if ((t_2 * (((t_1 * Math.pow((0.5 + t_4), (0.5 + t_3))) * Math.exp((((1.0 + (z + -1.0)) - 7.0) - 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / (1.0 + t_3))) + (-1259.1392167224028 / (2.0 + t_3))) + (771.3234287776531 / (3.0 + t_3))) + (-176.6150291621406 / (4.0 + t_3))) + (12.507343278686905 / (5.0 + t_3))) + (-0.13857109526572012 / (6.0 + t_3))) + (9.984369578019572e-6 / t_4)) + (1.5056327351493116e-7 / (8.0 + t_3))))) <= 1e+307) {
		tmp = t_1 * (t_2 * (Math.exp((z + -7.5)) * (t_0 * ((((-1259.1392167224028 / (2.0 - z)) + ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z)))) + ((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))))) + (0.9999999999998099 + (676.5203681218851 / (1.0 - z)))))));
	} else {
		tmp = t_1 * (t_2 * Math.exp(((z + -7.5) + Math.log((0.9999999999998099 * t_0)))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.pow((7.5 - z), (0.5 - z))
	t_1 = math.sqrt((math.pi * 2.0))
	t_2 = math.pi / math.sin((math.pi * z))
	t_3 = (1.0 - z) + -1.0
	t_4 = 7.0 + t_3
	tmp = 0
	if (t_2 * (((t_1 * math.pow((0.5 + t_4), (0.5 + t_3))) * math.exp((((1.0 + (z + -1.0)) - 7.0) - 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / (1.0 + t_3))) + (-1259.1392167224028 / (2.0 + t_3))) + (771.3234287776531 / (3.0 + t_3))) + (-176.6150291621406 / (4.0 + t_3))) + (12.507343278686905 / (5.0 + t_3))) + (-0.13857109526572012 / (6.0 + t_3))) + (9.984369578019572e-6 / t_4)) + (1.5056327351493116e-7 / (8.0 + t_3))))) <= 1e+307:
		tmp = t_1 * (t_2 * (math.exp((z + -7.5)) * (t_0 * ((((-1259.1392167224028 / (2.0 - z)) + ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z)))) + ((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))))) + (0.9999999999998099 + (676.5203681218851 / (1.0 - z)))))))
	else:
		tmp = t_1 * (t_2 * math.exp(((z + -7.5) + math.log((0.9999999999998099 * t_0)))))
	return tmp

Julia

function code(z)
	t_0 = Float64(7.5 - z) ^ Float64(0.5 - z)
	t_1 = sqrt(Float64(pi * 2.0))
	t_2 = Float64(pi / sin(Float64(pi * z)))
	t_3 = Float64(Float64(1.0 - z) + -1.0)
	t_4 = Float64(7.0 + t_3)
	tmp = 0.0
	if (Float64(t_2 * Float64(Float64(Float64(t_1 * (Float64(0.5 + t_4) ^ Float64(0.5 + t_3))) * exp(Float64(Float64(Float64(1.0 + Float64(z + -1.0)) - 7.0) - 0.5))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 + t_3))) + Float64(-1259.1392167224028 / Float64(2.0 + t_3))) + Float64(771.3234287776531 / Float64(3.0 + t_3))) + Float64(-176.6150291621406 / Float64(4.0 + t_3))) + Float64(12.507343278686905 / Float64(5.0 + t_3))) + Float64(-0.13857109526572012 / Float64(6.0 + t_3))) + Float64(9.984369578019572e-6 / t_4)) + Float64(1.5056327351493116e-7 / Float64(8.0 + t_3))))) <= 1e+307)
		tmp = Float64(t_1 * Float64(t_2 * Float64(exp(Float64(z + -7.5)) * Float64(t_0 * Float64(Float64(Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) + Float64(Float64(771.3234287776531 / Float64(3.0 - z)) + Float64(-176.6150291621406 / Float64(4.0 - z)))) + Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z)))))) + Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 - z))))))));
	else
		tmp = Float64(t_1 * Float64(t_2 * exp(Float64(Float64(z + -7.5) + log(Float64(0.9999999999998099 * t_0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = (7.5 - z) ^ (0.5 - z);
	t_1 = sqrt((pi * 2.0));
	t_2 = pi / sin((pi * z));
	t_3 = (1.0 - z) + -1.0;
	t_4 = 7.0 + t_3;
	tmp = 0.0;
	if ((t_2 * (((t_1 * ((0.5 + t_4) ^ (0.5 + t_3))) * exp((((1.0 + (z + -1.0)) - 7.0) - 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / (1.0 + t_3))) + (-1259.1392167224028 / (2.0 + t_3))) + (771.3234287776531 / (3.0 + t_3))) + (-176.6150291621406 / (4.0 + t_3))) + (12.507343278686905 / (5.0 + t_3))) + (-0.13857109526572012 / (6.0 + t_3))) + (9.984369578019572e-6 / t_4)) + (1.5056327351493116e-7 / (8.0 + t_3))))) <= 1e+307)
		tmp = t_1 * (t_2 * (exp((z + -7.5)) * (t_0 * ((((-1259.1392167224028 / (2.0 - z)) + ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z)))) + ((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))))) + (0.9999999999998099 + (676.5203681218851 / (1.0 - z)))))));
	else
		tmp = t_1 * (t_2 * exp(((z + -7.5) + log((0.9999999999998099 * t_0)))));
	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[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(1.0 - z), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$4 = N[(7.0 + t$95$3), $MachinePrecision]}, If[LessEqual[N[(t$95$2 * N[(N[(N[(t$95$1 * N[Power[N[(0.5 + t$95$4), $MachinePrecision], N[(0.5 + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(N[(1.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision] - 7.0), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(3.0 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(4.0 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(5.0 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(6.0 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$4), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+307], N[(t$95$1 * N[(t$95$2 * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(N[(N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(t$95$2 * N[Exp[N[(N[(z + -7.5), $MachinePrecision] + N[Log[N[(0.9999999999998099 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (PI.f64) (sin.f64 (*.f64 (PI.f64) z))) (*.f64 (*.f64 (*.f64 (sqrt.f64 (*.f64 (PI.f64) 2)) (pow.f64 (+.f64 (+.f64 (-.f64 (-.f64 1 z) 1) 7) 1/2) (+.f64 (-.f64 (-.f64 1 z) 1) 1/2))) (exp.f64 (neg.f64 (+.f64 (+.f64 (-.f64 (-.f64 1 z) 1) 7) 1/2)))) (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 9999999999998099/10000000000000000 (/.f64 6765203681218851/10000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 1))) (/.f64 -3147848041806007/2500000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 2))) (/.f64 7713234287776531/10000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 3))) (/.f64 -883075145810703/5000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 4))) (/.f64 2501468655737381/200000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 5))) (/.f64 -3464277381643003/25000000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 6))) (/.f64 2496092394504893/250000000000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 7))) (/.f64 3764081837873279/25000000000000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 8))))) < 9.99999999999999986e306

   1. Initial program 97.3%

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

   3. Simplified 98.2%

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

   6. Applied egg-rr 98.2%

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

      1. expm1-def 98.2%

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

      2. expm1-log1p 98.2%

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

      3. associate-*l* 97.5%

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

   8. Simplified 97.5%

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

      1. expm1-log1p-u 97.0%

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

      2. expm1-udef 98.2%

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

   10. Applied egg-rr 98.2%

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

   12. Simplified 99.2%

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

   if 9.99999999999999986e306 < (*.f64 (/.f64 (PI.f64) (sin.f64 (*.f64 (PI.f64) z))) (*.f64 (*.f64 (*.f64 (sqrt.f64 (*.f64 (PI.f64) 2)) (pow.f64 (+.f64 (+.f64 (-.f64 (-.f64 1 z) 1) 7) 1/2) (+.f64 (-.f64 (-.f64 1 z) 1) 1/2))) (exp.f64 (neg.f64 (+.f64 (+.f64 (-.f64 (-.f64 1 z) 1) 7) 1/2)))) (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 9999999999998099/10000000000000000 (/.f64 6765203681218851/10000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 1))) (/.f64 -3147848041806007/2500000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 2))) (/.f64 7713234287776531/10000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 3))) (/.f64 -883075145810703/5000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 4))) (/.f64 2501468655737381/200000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 5))) (/.f64 -3464277381643003/25000000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 6))) (/.f64 2496092394504893/250000000000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 7))) (/.f64 3764081837873279/25000000000000000000000 (+.f64 (-.f64 (-.f64 1 z) 1) 8)))))

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. expm1-def 0.0%

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

      2. expm1-log1p 0.0%

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

      3. associate-*l* 0.0%

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

   8. Simplified 0.0%

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

   10. Applied egg-rr 100.0%

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

   11. Taylor expanded in z around inf 100.0%

       \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(e^{\left(z + -7.5\right) + \color{blue}{\log \left(0.9999999999998099 \cdot e^{\log \left(7.5 - z\right) \cdot \left(0.5 - z\right)}\right)}} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   12. Step-by-step derivation

       1. exp-to-pow 100.0%

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

   13. Simplified 100.0%

       \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(e^{\left(z + -7.5\right) + \color{blue}{\log \left(0.9999999999998099 \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)}} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

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

Alternative 2: 98.1% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (sqrt (* PI 2.0))
  (*
   (exp
    (+
     (+ z -7.5)
     (log
      (*
       (+
        0.9999999999998099
        (+
         (/ 676.5203681218851 (- 1.0 z))
         (+
          (/ -1259.1392167224028 (- 2.0 z))
          (+
           (+ (/ 771.3234287776531 (- 3.0 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)))))))))
       (pow (- 7.5 z) (- 0.5 z))))))
   (/ PI (sin (* PI z))))))

C

double code(double z) {
	return sqrt((((double) M_PI) * 2.0)) * (exp(((z + -7.5) + log(((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (((771.3234287776531 / (3.0 - 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))))))))) * pow((7.5 - z), (0.5 - z)))))) * (((double) M_PI) / sin((((double) M_PI) * z))));
}

Java

public static double code(double z) {
	return Math.sqrt((Math.PI * 2.0)) * (Math.exp(((z + -7.5) + Math.log(((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (((771.3234287776531 / (3.0 - 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))))))))) * Math.pow((7.5 - z), (0.5 - z)))))) * (Math.PI / Math.sin((Math.PI * z))));
}

Python

def code(z):
	return math.sqrt((math.pi * 2.0)) * (math.exp(((z + -7.5) + math.log(((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (((771.3234287776531 / (3.0 - 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))))))))) * math.pow((7.5 - z), (0.5 - z)))))) * (math.pi / math.sin((math.pi * z))))

Julia

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

MATLAB

function tmp = code(z)
	tmp = sqrt((pi * 2.0)) * (exp(((z + -7.5) + log(((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (((771.3234287776531 / (3.0 - 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))))))))) * ((7.5 - z) ^ (0.5 - z)))))) * (pi / sin((pi * z))));
end

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 96.3%

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

6. Applied egg-rr 96.3%

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

   1. expm1-def 96.3%

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

   2. expm1-log1p 96.3%

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

   3. associate-*l* 95.6%

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

8. Simplified 95.6%

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

10. Applied egg-rr 98.3%

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

11. Final simplification 98.3%

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

Alternative 3: 98.2% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (sqrt (* PI 2.0))
  (*
   (/ PI (sin (* PI z)))
   (*
    (*
     (+
      0.9999999999998099
      (+
       (/ 676.5203681218851 (- 1.0 z))
       (+
        (/ -1259.1392167224028 (- 2.0 z))
        (+
         (+ (/ 771.3234287776531 (- 3.0 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)))))))))
     (pow (- 7.5 z) (- 0.5 z)))
    (exp (+ z -7.5))))))

C

double code(double z) {
	return sqrt((((double) M_PI) * 2.0)) * ((((double) M_PI) / sin((((double) M_PI) * z))) * (((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (((771.3234287776531 / (3.0 - 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))))))))) * pow((7.5 - z), (0.5 - z))) * exp((z + -7.5))));
}

Java

public static double code(double z) {
	return Math.sqrt((Math.PI * 2.0)) * ((Math.PI / Math.sin((Math.PI * z))) * (((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (((771.3234287776531 / (3.0 - 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))))))))) * Math.pow((7.5 - z), (0.5 - z))) * Math.exp((z + -7.5))));
}

Python

def code(z):
	return math.sqrt((math.pi * 2.0)) * ((math.pi / math.sin((math.pi * z))) * (((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (((771.3234287776531 / (3.0 - 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))))))))) * math.pow((7.5 - z), (0.5 - z))) * math.exp((z + -7.5))))

Julia

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

MATLAB

function tmp = code(z)
	tmp = sqrt((pi * 2.0)) * ((pi / sin((pi * z))) * (((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (((771.3234287776531 / (3.0 - 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))))))))) * ((7.5 - z) ^ (0.5 - z))) * exp((z + -7.5))));
end

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 96.3%

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

6. Applied egg-rr 96.3%

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

   1. expm1-def 96.3%

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

   2. expm1-log1p 96.3%

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

   3. associate-*l* 95.6%

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

8. Simplified 95.6%

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

   1. expm1-log1p-u 95.1%

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

   2. expm1-udef 96.3%

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

10. Applied egg-rr 96.3%

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

    1. expm1-def 97.3%

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

    2. expm1-log1p 97.3%

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

    3. associate-*l* 97.2%

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

12. Simplified 97.2%

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

13. Final simplification 97.2%

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

Alternative 4: 98.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 96.3%

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

6. Applied egg-rr 96.3%

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

   1. expm1-def 96.3%

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

   2. expm1-log1p 96.3%

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

   3. associate-*l* 95.6%

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

8. Simplified 95.6%

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

   1. expm1-log1p-u 95.1%

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

   2. expm1-udef 96.3%

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

10. Applied egg-rr 96.3%

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

12. Simplified 97.2%

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

13. Final simplification 97.2%

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

Alternative 5: 96.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\pi \cdot 2} \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(46.9507597606837 + z \cdot 361.7355639412844\right) + \left(212.9540523020159 + z \cdot 74.66416387488323\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (sqrt (* PI 2.0))
  (*
   (/ PI (sin (* PI z)))
   (*
    (pow (- 7.5 z) (- 0.5 z))
    (*
     (exp (+ z -7.5))
     (+
      (+
       0.9999999999998099
       (+
        (+ 46.9507597606837 (* z 361.7355639412844))
        (+ 212.9540523020159 (* z 74.66416387488323))))
      (+
       (+
        (/ 9.984369578019572e-6 (- 7.0 z))
        (/ 1.5056327351493116e-7 (- 8.0 z)))
       (+
        (/ 12.507343278686905 (- 5.0 z))
        (/ -0.13857109526572012 (- 6.0 z))))))))))

C

double code(double z) {
	return sqrt((((double) M_PI) * 2.0)) * ((((double) M_PI) / sin((((double) M_PI) * z))) * (pow((7.5 - z), (0.5 - z)) * (exp((z + -7.5)) * ((0.9999999999998099 + ((46.9507597606837 + (z * 361.7355639412844)) + (212.9540523020159 + (z * 74.66416387488323)))) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z))))))));
}

Java

public static double code(double z) {
	return Math.sqrt((Math.PI * 2.0)) * ((Math.PI / Math.sin((Math.PI * z))) * (Math.pow((7.5 - z), (0.5 - z)) * (Math.exp((z + -7.5)) * ((0.9999999999998099 + ((46.9507597606837 + (z * 361.7355639412844)) + (212.9540523020159 + (z * 74.66416387488323)))) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z))))))));
}

Python

def code(z):
	return math.sqrt((math.pi * 2.0)) * ((math.pi / math.sin((math.pi * z))) * (math.pow((7.5 - z), (0.5 - z)) * (math.exp((z + -7.5)) * ((0.9999999999998099 + ((46.9507597606837 + (z * 361.7355639412844)) + (212.9540523020159 + (z * 74.66416387488323)))) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z))))))))

Julia

function code(z)
	return Float64(sqrt(Float64(pi * 2.0)) * Float64(Float64(pi / sin(Float64(pi * z))) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * Float64(exp(Float64(z + -7.5)) * Float64(Float64(0.9999999999998099 + Float64(Float64(46.9507597606837 + Float64(z * 361.7355639412844)) + Float64(212.9540523020159 + Float64(z * 74.66416387488323)))) + Float64(Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))) + Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(-0.13857109526572012 / Float64(6.0 - z)))))))))
end

MATLAB

function tmp = code(z)
	tmp = sqrt((pi * 2.0)) * ((pi / sin((pi * z))) * (((7.5 - z) ^ (0.5 - z)) * (exp((z + -7.5)) * ((0.9999999999998099 + ((46.9507597606837 + (z * 361.7355639412844)) + (212.9540523020159 + (z * 74.66416387488323)))) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z))))))));
end

Wolfram

code[z_] := N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(N[(0.9999999999998099 + N[(N[(46.9507597606837 + N[(z * 361.7355639412844), $MachinePrecision]), $MachinePrecision] + N[(212.9540523020159 + N[(z * 74.66416387488323), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.4%

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

3. Simplified 96.3%

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

5. Taylor expanded in z around 0 95.0%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\color{blue}{\left(46.9507597606837 + 361.7355639412844 \cdot z\right)} + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

6. Taylor expanded in z around 0 95.9%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(46.9507597606837 + 361.7355639412844 \cdot z\right) + \color{blue}{\left(212.9540523020159 + 74.66416387488323 \cdot z\right)}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
7. Step-by-step derivation

   1. *-commutative 95.9%

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(46.9507597606837 + 361.7355639412844 \cdot z\right) + \left(212.9540523020159 + \color{blue}{z \cdot 74.66416387488323}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

8. Simplified 95.9%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(46.9507597606837 + 361.7355639412844 \cdot z\right) + \color{blue}{\left(212.9540523020159 + z \cdot 74.66416387488323\right)}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

9. Final simplification 95.9%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(46.9507597606837 + z \cdot 361.7355639412844\right) + \left(212.9540523020159 + z \cdot 74.66416387488323\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 6: 95.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 96.3%

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

5. Taylor expanded in z around 0 94.8%

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

   1. expm1-log1p-u 48.2%

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

   2. expm1-udef 48.2%

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

   3. associate-*l* 48.2%

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

   4. *-commutative 48.2%

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

7. Applied egg-rr 48.2%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\pi \cdot 2} \cdot \left(263.3831869810514 \cdot \left(\left(e^{-7.5} \cdot \sqrt{7.5}\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right)\right)\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 48.2%

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

   2. expm1-log1p 94.5%

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

   3. *-commutative 94.5%

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

   4. associate-*r* 94.8%

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

   5. associate-*r/ 95.1%

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

9. Simplified 95.1%

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

10. Final simplification 95.1%

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

Alternative 7: 95.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 96.3%

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

5. Taylor expanded in z around 0 94.8%

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

6. Taylor expanded in z around 0 94.7%

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

   1. expm1-log1p-u 48.2%

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

   2. expm1-udef 48.2%

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

   3. associate-/l* 48.2%

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

   4. sqrt-unprod 48.2%

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

   5. metadata-eval 48.2%

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

8. Applied egg-rr 48.2%

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

   1. expm1-def 48.2%

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

   2. expm1-log1p 94.8%

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

   3. associate-/r/ 94.8%

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

10. Simplified 94.8%

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

11. Final simplification 94.8%

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

Alternative 8: 95.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 96.3%

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

5. Taylor expanded in z around 0 94.8%

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

6. Taylor expanded in z around 0 94.7%

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

   1. expm1-log1p-u 48.2%

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

   2. expm1-udef 48.2%

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

   3. associate-/l* 48.2%

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

   4. sqrt-unprod 48.2%

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

   5. metadata-eval 48.2%

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

8. Applied egg-rr 48.2%

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

   1. expm1-def 48.2%

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

   2. expm1-log1p 94.8%

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

   3. associate-/r/ 94.8%

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

10. Simplified 94.8%

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

11. Taylor expanded in z around 0 94.7%

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

    1. associate-*r* 94.7%

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

    2. associate-/l* 94.9%

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

13. Simplified 94.9%

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

14. Final simplification 94.9%

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

Alternative 9: 95.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 96.3%

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

5. Taylor expanded in z around 0 94.8%

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

6. Taylor expanded in z around 0 94.7%

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

   1. expm1-log1p-u 48.2%

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

   2. expm1-udef 48.2%

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

   3. associate-/l* 48.2%

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

   4. sqrt-unprod 48.2%

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

   5. metadata-eval 48.2%

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

8. Applied egg-rr 48.2%

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

   1. expm1-def 48.2%

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

   2. expm1-log1p 94.8%

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

   3. associate-/r/ 94.8%

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

10. Simplified 94.8%

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

11. Taylor expanded in z around 0 94.7%

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

    1. *-commutative 94.7%

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

    2. associate-*l* 94.7%

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

    3. associate-/l* 95.0%

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

    4. *-commutative 95.0%

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

13. Simplified 95.0%

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

14. Final simplification 95.0%

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

Alternative 10: 96.0% accurate, 1.7× 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 95.4%

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

3. Simplified 96.3%

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

5. Taylor expanded in z around 0 94.8%

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

6. Taylor expanded in z around 0 94.7%

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

   1. associate-*r* 94.7%

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

   2. associate-/l* 94.9%

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

   3. associate-*r/ 94.8%

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

   4. *-commutative 94.8%

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

   5. associate-/r* 94.9%

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

8. Simplified 94.9%

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

   1. associate-*l/ 95.0%

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

   2. associate-/l/ 95.1%

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

   3. pow1/2 95.1%

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

   4. pow1/2 95.1%

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

   5. pow-prod-down 95.1%

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

   6. metadata-eval 95.1%

      \[\leadsto \frac{\left(263.3831869810514 \cdot e^{-7.5}\right) \cdot \sqrt{\pi}}{\frac{z}{{\color{blue}{15}}^{0.5}}} \]

   7. pow1/2 95.1%

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

10. Applied egg-rr 95.1%

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

11. Final simplification 95.1%

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

Alternative 11: 95.3% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 96.3%

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

5. Taylor expanded in z around 0 94.8%

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

6. Taylor expanded in z around 0 94.7%

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

   1. expm1-log1p-u 48.2%

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

   2. expm1-udef 48.2%

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

   3. associate-/l* 48.2%

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

   4. sqrt-unprod 48.2%

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

   5. metadata-eval 48.2%

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

8. Applied egg-rr 48.2%

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

   1. expm1-def 48.2%

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

   2. expm1-log1p 94.8%

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

   3. associate-/r/ 94.8%

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

10. Simplified 94.8%

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

    1. expm1-log1p-u 48.2%

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

    2. expm1-udef 48.2%

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

    3. associate-*l* 48.2%

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

    4. pow1/2 48.2%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \left(\frac{e^{-7.5}}{z} \cdot \left(\color{blue}{{15}^{0.5}} \cdot \sqrt{\pi}\right)\right)\right)} - 1 \]

    5. pow1/2 48.2%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \left(\frac{e^{-7.5}}{z} \cdot \left({15}^{0.5} \cdot \color{blue}{{\pi}^{0.5}}\right)\right)\right)} - 1 \]

    6. pow-prod-down 48.2%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \left(\frac{e^{-7.5}}{z} \cdot \color{blue}{{\left(15 \cdot \pi\right)}^{0.5}}\right)\right)} - 1 \]

12. Applied egg-rr 48.2%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(263.3831869810514 \cdot \left(\frac{e^{-7.5}}{z} \cdot {\left(15 \cdot \pi\right)}^{0.5}\right)\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 48.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(263.3831869810514 \cdot \left(\frac{e^{-7.5}}{z} \cdot {\left(15 \cdot \pi\right)}^{0.5}\right)\right)\right)} \]

    2. expm1-log1p 94.4%

       \[\leadsto \color{blue}{263.3831869810514 \cdot \left(\frac{e^{-7.5}}{z} \cdot {\left(15 \cdot \pi\right)}^{0.5}\right)} \]

    3. associate-*l/ 94.4%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{\frac{e^{-7.5} \cdot {\left(15 \cdot \pi\right)}^{0.5}}{z}} \]

    4. rem-exp-log 48.1%

       \[\leadsto \color{blue}{e^{\log \left(263.3831869810514 \cdot \frac{e^{-7.5} \cdot {\left(15 \cdot \pi\right)}^{0.5}}{z}\right)}} \]

    5. log-prod 47.9%

       \[\leadsto e^{\color{blue}{\log 263.3831869810514 + \log \left(\frac{e^{-7.5} \cdot {\left(15 \cdot \pi\right)}^{0.5}}{z}\right)}} \]

    6. exp-sum 48.3%

       \[\leadsto \color{blue}{e^{\log 263.3831869810514} \cdot e^{\log \left(\frac{e^{-7.5} \cdot {\left(15 \cdot \pi\right)}^{0.5}}{z}\right)}} \]

    7. rem-exp-log 48.3%

       \[\leadsto \color{blue}{263.3831869810514} \cdot e^{\log \left(\frac{e^{-7.5} \cdot {\left(15 \cdot \pi\right)}^{0.5}}{z}\right)} \]

    8. rem-exp-log 94.4%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{\frac{e^{-7.5} \cdot {\left(15 \cdot \pi\right)}^{0.5}}{z}} \]

    9. associate-*l/ 94.4%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{\left(\frac{e^{-7.5}}{z} \cdot {\left(15 \cdot \pi\right)}^{0.5}\right)} \]

    10. unpow1/2 94.4%

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

14. Simplified 94.4%

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

15. Final simplification 94.4%

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

Alternative 12: 95.3% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 96.3%

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

5. Taylor expanded in z around 0 94.8%

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

6. Taylor expanded in z around 0 94.7%

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

   1. expm1-log1p-u 48.2%

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

   2. expm1-udef 48.2%

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

   3. associate-/l* 48.2%

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

   4. sqrt-unprod 48.2%

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

   5. metadata-eval 48.2%

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

8. Applied egg-rr 48.2%

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

   1. expm1-def 48.2%

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

   2. expm1-log1p 94.8%

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

   3. associate-/r/ 94.8%

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

10. Simplified 94.8%

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

    1. expm1-log1p-u 48.2%

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

    2. expm1-udef 48.2%

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

    3. associate-*l* 48.2%

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

    4. pow1/2 48.2%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \left(\frac{e^{-7.5}}{z} \cdot \left(\color{blue}{{15}^{0.5}} \cdot \sqrt{\pi}\right)\right)\right)} - 1 \]

    5. pow1/2 48.2%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \left(\frac{e^{-7.5}}{z} \cdot \left({15}^{0.5} \cdot \color{blue}{{\pi}^{0.5}}\right)\right)\right)} - 1 \]

    6. pow-prod-down 48.2%

       \[\leadsto e^{\mathsf{log1p}\left(263.3831869810514 \cdot \left(\frac{e^{-7.5}}{z} \cdot \color{blue}{{\left(15 \cdot \pi\right)}^{0.5}}\right)\right)} - 1 \]

12. Applied egg-rr 48.2%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(263.3831869810514 \cdot \left(\frac{e^{-7.5}}{z} \cdot {\left(15 \cdot \pi\right)}^{0.5}\right)\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 48.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(263.3831869810514 \cdot \left(\frac{e^{-7.5}}{z} \cdot {\left(15 \cdot \pi\right)}^{0.5}\right)\right)\right)} \]

    2. expm1-log1p 94.4%

       \[\leadsto \color{blue}{263.3831869810514 \cdot \left(\frac{e^{-7.5}}{z} \cdot {\left(15 \cdot \pi\right)}^{0.5}\right)} \]

    3. associate-*l/ 94.4%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{\frac{e^{-7.5} \cdot {\left(15 \cdot \pi\right)}^{0.5}}{z}} \]

    4. *-commutative 94.4%

       \[\leadsto 263.3831869810514 \cdot \frac{\color{blue}{{\left(15 \cdot \pi\right)}^{0.5} \cdot e^{-7.5}}}{z} \]

    5. associate-/l* 94.5%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{\frac{{\left(15 \cdot \pi\right)}^{0.5}}{\frac{z}{e^{-7.5}}}} \]

    6. unpow1/2 94.5%

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

14. Simplified 94.5%

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

15. Final simplification 94.5%

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

Reproduce

herbie shell --seed 2024027 
(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))))))