Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.2% → 98.1%

Time: 1.4min

Alternatives: 9

Speedup: 1.6×

• could not determine a ground truth

  1. z = -4.2307865106641256e+40

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: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(1 - z\right) + 6.5\right)}^{\left(0.5 - z\right)} \cdot e^{\left(z + -1\right) + -6.5}\\ \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{1 + \left(1 - z\right)}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right) \cdot \left(\left(\sqrt[3]{t_0 \cdot \left(t_0 \cdot t_0\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Julia

function code(z)
	t_0 = Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(0.5 - z)) * exp(Float64(Float64(z + -1.0) + -6.5)))
	return Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(-1259.1392167224028 / Float64(1.0 + Float64(1.0 - z))))) + Float64(Float64(771.3234287776531 / Float64(Float64(1.0 - z) + 2.0)) + Float64(Float64(-176.6150291621406 / Float64(Float64(1.0 - z) + 3.0)) + Float64(12.507343278686905 / Float64(Float64(1.0 - z) + 4.0))))) + Float64(Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) + 5.0)) + Float64(Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) + 6.0)) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) + 7.0))))) * Float64(Float64(cbrt(Float64(t_0 * Float64(t_0 * t_0))) * sqrt(Float64(2.0 * pi))) * Float64(pi / sin(Float64(z * pi)))))
end

Wolfram

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

Derivation

1. Initial program 97.3%

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

3. Simplified 99.0%

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

   1. add-cbrt-cube 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 2: 97.7% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (sqrt (* 2.0 PI))
  (*
   (+
    (/ 9.984369578019572e-6 (- 7.0 z))
    (+
     (/ 1.5056327351493116e-7 (- 8.0 z))
     (+
      (+
       0.9999999999998099
       (+ (/ 676.5203681218851 (- 1.0 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))))))))
   (* (/ PI (sin (* z PI))) (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5)))))))

C

double code(double z) {
	return sqrt((2.0 * ((double) M_PI))) * (((9.984369578019572e-6 / (7.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + ((0.9999999999998099 + ((676.5203681218851 / (1.0 - 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)))))))) * ((((double) M_PI) / sin((z * ((double) M_PI)))) * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5)))));
}

Java

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

Python

def code(z):
	return math.sqrt((2.0 * math.pi)) * (((9.984369578019572e-6 / (7.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + ((0.9999999999998099 + ((676.5203681218851 / (1.0 - 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)))))))) * ((math.pi / math.sin((z * math.pi))) * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5)))))

Julia

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

MATLAB

function tmp = code(z)
	tmp = sqrt((2.0 * pi)) * (((9.984369578019572e-6 / (7.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + ((0.9999999999998099 + ((676.5203681218851 / (1.0 - 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)))))))) * ((pi / sin((z * pi))) * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5)))));
end

Wolfram

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

Derivation

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 96.4%

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

5. Applied egg-rr 44.8%

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

7. Simplified 97.6%

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

9. Applied egg-rr 44.7%

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

11. Simplified 98.9%

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

12. Final simplification 98.9%

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

Alternative 3: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 96.4%

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

5. Applied egg-rr 44.8%

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

7. Simplified 99.2%

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-udef 98.2%

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

9. Applied egg-rr 98.2%

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

11. Simplified 99.2%

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

12. Final simplification 99.2%

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

Alternative 4: 96.7% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (sqrt (* 2.0 PI))
  (*
   (/ PI (sin (* z PI)))
   (*
    (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5)))
    (+
     (+
      (+
       (+ (/ -0.13857109526572012 (- 6.0 z)) (/ -176.6150291621406 (- 4.0 z)))
       (/ 12.507343278686905 (- 5.0 z)))
      (+
       (/ 9.984369578019572e-6 (- 7.0 z))
       (/ 1.5056327351493116e-7 (- 8.0 z))))
     (+
      0.9999999999998099
      (+
       (/ 771.3234287776531 (- 3.0 z))
       (+ 46.9507597606837 (* z 361.7355639412844)))))))))

C

double code(double z) {
	return sqrt((2.0 * ((double) M_PI))) * ((((double) M_PI) / sin((z * ((double) M_PI)))) * ((pow((7.5 - z), (0.5 - z)) * exp((z + -7.5))) * (((((-0.13857109526572012 / (6.0 - z)) + (-176.6150291621406 / (4.0 - z))) + (12.507343278686905 / (5.0 - z))) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + (0.9999999999998099 + ((771.3234287776531 / (3.0 - z)) + (46.9507597606837 + (z * 361.7355639412844)))))));
}

Java

public static double code(double z) {
	return Math.sqrt((2.0 * Math.PI)) * ((Math.PI / Math.sin((z * Math.PI))) * ((Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5))) * (((((-0.13857109526572012 / (6.0 - z)) + (-176.6150291621406 / (4.0 - z))) + (12.507343278686905 / (5.0 - z))) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + (0.9999999999998099 + ((771.3234287776531 / (3.0 - z)) + (46.9507597606837 + (z * 361.7355639412844)))))));
}

Python

def code(z):
	return math.sqrt((2.0 * math.pi)) * ((math.pi / math.sin((z * math.pi))) * ((math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5))) * (((((-0.13857109526572012 / (6.0 - z)) + (-176.6150291621406 / (4.0 - z))) + (12.507343278686905 / (5.0 - z))) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + (0.9999999999998099 + ((771.3234287776531 / (3.0 - z)) + (46.9507597606837 + (z * 361.7355639412844)))))))

Julia

function code(z)
	return Float64(sqrt(Float64(2.0 * pi)) * Float64(Float64(pi / sin(Float64(z * pi))) * Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5))) * Float64(Float64(Float64(Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(-176.6150291621406 / Float64(4.0 - z))) + 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(Float64(771.3234287776531 / Float64(3.0 - z)) + Float64(46.9507597606837 + Float64(z * 361.7355639412844))))))))
end

MATLAB

function tmp = code(z)
	tmp = sqrt((2.0 * pi)) * ((pi / sin((z * pi))) * ((((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5))) * (((((-0.13857109526572012 / (6.0 - z)) + (-176.6150291621406 / (4.0 - z))) + (12.507343278686905 / (5.0 - z))) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + (0.9999999999998099 + ((771.3234287776531 / (3.0 - z)) + (46.9507597606837 + (z * 361.7355639412844)))))));
end

Wolfram

code[z_] := N[(N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(Pi / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision] + N[(46.9507597606837 + N[(z * 361.7355639412844), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 96.4%

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

5. Applied egg-rr 44.8%

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

7. Simplified 99.2%

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-udef 98.2%

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

9. Applied egg-rr 98.2%

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

11. Simplified 99.2%

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

12. Taylor expanded in z around 0 98.5%

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

    1. *-commutative 98.5%

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

14. Simplified 98.5%

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

15. Final simplification 98.5%

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

Alternative 5: 96.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot \pi} \cdot \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{-176.6150291621406}{4 - z}\right) + \frac{12.507343278686905}{5 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) + \left(305.05856935323453 + z \cdot 447.4381671388014\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (sqrt (* 2.0 PI))
  (*
   (/ PI (sin (* z PI)))
   (*
    (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5)))
    (+
     (+
      (+
       (+ (/ -0.13857109526572012 (- 6.0 z)) (/ -176.6150291621406 (- 4.0 z)))
       (/ 12.507343278686905 (- 5.0 z)))
      (+
       (/ 9.984369578019572e-6 (- 7.0 z))
       (/ 1.5056327351493116e-7 (- 8.0 z))))
     (+ 305.05856935323453 (* z 447.4381671388014)))))))

C

double code(double z) {
	return sqrt((2.0 * ((double) M_PI))) * ((((double) M_PI) / sin((z * ((double) M_PI)))) * ((pow((7.5 - z), (0.5 - z)) * exp((z + -7.5))) * (((((-0.13857109526572012 / (6.0 - z)) + (-176.6150291621406 / (4.0 - z))) + (12.507343278686905 / (5.0 - z))) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + (305.05856935323453 + (z * 447.4381671388014)))));
}

Java

public static double code(double z) {
	return Math.sqrt((2.0 * Math.PI)) * ((Math.PI / Math.sin((z * Math.PI))) * ((Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5))) * (((((-0.13857109526572012 / (6.0 - z)) + (-176.6150291621406 / (4.0 - z))) + (12.507343278686905 / (5.0 - z))) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + (305.05856935323453 + (z * 447.4381671388014)))));
}

Python

def code(z):
	return math.sqrt((2.0 * math.pi)) * ((math.pi / math.sin((z * math.pi))) * ((math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5))) * (((((-0.13857109526572012 / (6.0 - z)) + (-176.6150291621406 / (4.0 - z))) + (12.507343278686905 / (5.0 - z))) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + (305.05856935323453 + (z * 447.4381671388014)))))

Julia

function code(z)
	return Float64(sqrt(Float64(2.0 * pi)) * Float64(Float64(pi / sin(Float64(z * pi))) * Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5))) * Float64(Float64(Float64(Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(-176.6150291621406 / Float64(4.0 - z))) + 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(305.05856935323453 + Float64(z * 447.4381671388014))))))
end

MATLAB

function tmp = code(z)
	tmp = sqrt((2.0 * pi)) * ((pi / sin((z * pi))) * ((((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5))) * (((((-0.13857109526572012 / (6.0 - z)) + (-176.6150291621406 / (4.0 - z))) + (12.507343278686905 / (5.0 - z))) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + (305.05856935323453 + (z * 447.4381671388014)))));
end

Wolfram

code[z_] := N[(N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(Pi / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(305.05856935323453 + N[(z * 447.4381671388014), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 96.4%

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

5. Applied egg-rr 44.8%

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

7. Simplified 99.2%

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-udef 98.2%

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

9. Applied egg-rr 98.2%

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

11. Simplified 99.2%

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

12. Taylor expanded in z around 0 98.3%

    \[\leadsto \sqrt{2 \cdot \pi} \cdot \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(e^{-7.5 + z} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\color{blue}{\left(305.05856935323453 + 447.4381671388014 \cdot z\right)} + \left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{-176.6150291621406}{4 - z}\right) + \frac{12.507343278686905}{5 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]

13. Final simplification 98.3%

    \[\leadsto \sqrt{2 \cdot \pi} \cdot \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{-176.6150291621406}{4 - z}\right) + \frac{12.507343278686905}{5 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) + \left(305.05856935323453 + z \cdot 447.4381671388014\right)\right)\right)\right) \]

Alternative 6: 96.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 96.4%

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

4. Taylor expanded in z around 0 96.9%

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

   1. associate-/l* 97.0%

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

6. Simplified 97.0%

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

7. Taylor expanded in z around 0 96.9%

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

   1. associate-*l/ 96.7%

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

   2. *-commutative 96.7%

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

   3. associate-*l* 97.5%

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

   4. *-commutative 97.5%

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

9. Simplified 97.5%

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

10. Final simplification 97.5%

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

Alternative 7: 95.8% accurate, 2.0× 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 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 96.4%

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

4. Taylor expanded in z around 0 96.9%

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

   1. associate-/l* 97.0%

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

6. Simplified 97.0%

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

   1. associate-/r/ 97.0%

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

   2. sqrt-unprod 97.0%

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

   3. metadata-eval 97.0%

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

8. Applied egg-rr 97.0%

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

9. Final simplification 97.0%

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

Alternative 8: 96.0% accurate, 2.0× speedup

Math

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

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(263.3831869810514 * Float64(Float64(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[(263.3831869810514 * N[(N[(N[Exp[-7.5], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] / N[(z / N[Sqrt[15.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 96.4%

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

4. Taylor expanded in z around 0 96.9%

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

   1. associate-/l* 97.0%

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

6. Simplified 97.0%

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

   1. associate-*l/ 97.3%

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

   2. sqrt-unprod 97.3%

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

   3. metadata-eval 97.3%

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

8. Applied egg-rr 97.3%

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

9. Final simplification 97.3%

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

Alternative 9: 95.4% accurate, 2.6× 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 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 96.4%

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

4. Taylor expanded in z around 0 96.9%

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

   1. associate-/l* 97.0%

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

6. Simplified 97.0%

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

   1. associate-/r/ 97.0%

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

   2. sqrt-unprod 97.0%

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

   3. metadata-eval 97.0%

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

8. Applied egg-rr 97.0%

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

   1. associate-/r/ 97.0%

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

   2. *-commutative 97.0%

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

   3. expm1-log1p-u 45.1%

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

   4. expm1-udef 45.1%

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

10. Applied egg-rr 45.1%

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

    1. expm1-def 45.1%

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

    2. expm1-log1p 96.7%

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

    3. associate-*r* 96.7%

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

    4. unpow1/2 96.7%

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

12. Simplified 96.7%

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

    1. expm1-log1p-u 45.1%

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

    2. expm1-udef 45.1%

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

    3. associate-*l* 45.1%

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

14. Applied egg-rr 45.1%

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

    1. expm1-def 45.1%

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

    2. expm1-log1p 96.7%

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

    3. *-commutative 96.7%

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

16. Simplified 96.7%

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

17. Final simplification 96.7%

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

Reproduce

herbie shell --seed 2023326 
(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))))))