Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.2% → 99.2%

Time: 1.5min

Alternatives: 13

Speedup: 1.1×

• could not determine a ground truth

  1. z = -9.221110898697147e+163

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: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\left(z - 7.5\right) + \log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)}\right) \cdot \left(\left(\left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right) - \left(\left(\frac{\frac{\frac{1585431.567088306}{2 - z}}{2 - z} - \frac{\frac{457679.80848377093}{1 - z}}{1 - z}}{\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}} - 0.9999999999998099\right) - \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (/ PI (sin (* PI z)))
   (exp (+ (- z 7.5) (log (* (sqrt (* PI 2.0)) (pow (- 7.5 z) (- 0.5 z)))))))
  (+
   (-
    (+
     (/ 12.507343278686905 (- (- 1.0 z) -4.0))
     (/ -0.13857109526572012 (- (- 1.0 z) -5.0)))
    (-
     (-
      (/
       (-
        (/ (/ 1585431.567088306 (- 2.0 z)) (- 2.0 z))
        (/ (/ 457679.80848377093 (- 1.0 z)) (- 1.0 z)))
       (+ (/ 676.5203681218851 (- 1.0 z)) (/ 1259.1392167224028 (- 2.0 z))))
      0.9999999999998099)
     (+ (/ 771.3234287776531 (- 3.0 z)) (/ -176.6150291621406 (- 4.0 z)))))
   (+
    (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))
    (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0))))))

C

double code(double z) {
	return ((((double) M_PI) / sin((((double) M_PI) * z))) * exp(((z - 7.5) + log((sqrt((((double) M_PI) * 2.0)) * pow((7.5 - z), (0.5 - z))))))) * ((((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0))) - ((((((1585431.567088306 / (2.0 - z)) / (2.0 - z)) - ((457679.80848377093 / (1.0 - z)) / (1.0 - z))) / ((676.5203681218851 / (1.0 - z)) + (1259.1392167224028 / (2.0 - z)))) - 0.9999999999998099) - ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z))))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))));
}

Java

public static double code(double z) {
	return ((Math.PI / Math.sin((Math.PI * z))) * Math.exp(((z - 7.5) + Math.log((Math.sqrt((Math.PI * 2.0)) * Math.pow((7.5 - z), (0.5 - z))))))) * ((((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0))) - ((((((1585431.567088306 / (2.0 - z)) / (2.0 - z)) - ((457679.80848377093 / (1.0 - z)) / (1.0 - z))) / ((676.5203681218851 / (1.0 - z)) + (1259.1392167224028 / (2.0 - z)))) - 0.9999999999998099) - ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z))))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))));
}

Python

def code(z):
	return ((math.pi / math.sin((math.pi * z))) * math.exp(((z - 7.5) + math.log((math.sqrt((math.pi * 2.0)) * math.pow((7.5 - z), (0.5 - z))))))) * ((((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0))) - ((((((1585431.567088306 / (2.0 - z)) / (2.0 - z)) - ((457679.80848377093 / (1.0 - z)) / (1.0 - z))) / ((676.5203681218851 / (1.0 - z)) + (1259.1392167224028 / (2.0 - z)))) - 0.9999999999998099) - ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z))))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))))

Julia

function code(z)
	return Float64(Float64(Float64(pi / sin(Float64(pi * z))) * exp(Float64(Float64(z - 7.5) + log(Float64(sqrt(Float64(pi * 2.0)) * (Float64(7.5 - z) ^ Float64(0.5 - z))))))) * Float64(Float64(Float64(Float64(12.507343278686905 / Float64(Float64(1.0 - z) - -4.0)) + Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) - -5.0))) - Float64(Float64(Float64(Float64(Float64(Float64(1585431.567088306 / Float64(2.0 - z)) / Float64(2.0 - z)) - Float64(Float64(457679.80848377093 / Float64(1.0 - z)) / Float64(1.0 - z))) / Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(1259.1392167224028 / Float64(2.0 - z)))) - 0.9999999999998099) - Float64(Float64(771.3234287776531 / Float64(3.0 - z)) + Float64(-176.6150291621406 / Float64(4.0 - z))))) + Float64(Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0)) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0)))))
end

MATLAB

function tmp = code(z)
	tmp = ((pi / sin((pi * z))) * exp(((z - 7.5) + log((sqrt((pi * 2.0)) * ((7.5 - z) ^ (0.5 - z))))))) * ((((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0))) - ((((((1585431.567088306 / (2.0 - z)) / (2.0 - z)) - ((457679.80848377093 / (1.0 - z)) / (1.0 - z))) / ((676.5203681218851 / (1.0 - z)) + (1259.1392167224028 / (2.0 - z)))) - 0.9999999999998099) - ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z))))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))));
end

Wolfram

code[z_] := N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(z - 7.5), $MachinePrecision] + N[Log[N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] - -4.0), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] - -5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(N[(N[(1585431.567088306 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] - N[(N[(457679.80848377093 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.9999999999998099), $MachinePrecision] - N[(N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

3. Simplified 97.8%

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

5. Applied egg-rr 98.0%

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

   1. frac-add 98.0%

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

   2. sub-neg 98.0%

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

   3. metadata-eval 98.0%

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

   4. --rgt-identity 98.0%

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

   5. --rgt-identity 98.0%

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

   6. sub-neg 98.0%

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

   7. metadata-eval 98.0%

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

   8. frac-add 98.0%

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

   9. flip-+ 99.3%

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

7. Applied egg-rr 99.3%

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

   1. associate-*l/ 99.3%

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

   2. associate-*r/ 99.3%

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

   3. metadata-eval 99.3%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\left(-\left(7.5 + \left(-z\right)\right)\right) + \log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{\frac{\frac{\color{blue}{457679.80848377093}}{1 - z}}{1 - z} - \frac{-1259.1392167224028}{2 - z} \cdot \frac{-1259.1392167224028}{2 - z}}{\frac{676.5203681218851}{1 - z} - \frac{-1259.1392167224028}{2 - z}}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   4. associate-*l/ 99.3%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\left(-\left(7.5 + \left(-z\right)\right)\right) + \log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{\frac{\frac{457679.80848377093}{1 - z}}{1 - z} - \color{blue}{\frac{-1259.1392167224028 \cdot \frac{-1259.1392167224028}{2 - z}}{2 - z}}}{\frac{676.5203681218851}{1 - z} - \frac{-1259.1392167224028}{2 - z}}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   5. associate-*r/ 99.3%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\left(-\left(7.5 + \left(-z\right)\right)\right) + \log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{\frac{\frac{457679.80848377093}{1 - z}}{1 - z} - \frac{\color{blue}{\frac{-1259.1392167224028 \cdot -1259.1392167224028}{2 - z}}}{2 - z}}{\frac{676.5203681218851}{1 - z} - \frac{-1259.1392167224028}{2 - z}}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   6. metadata-eval 99.3%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\left(-\left(7.5 + \left(-z\right)\right)\right) + \log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{\frac{\frac{457679.80848377093}{1 - z}}{1 - z} - \frac{\frac{\color{blue}{1585431.567088306}}{2 - z}}{2 - z}}{\frac{676.5203681218851}{1 - z} - \frac{-1259.1392167224028}{2 - z}}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   7. sub-neg 99.3%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\left(-\left(7.5 + \left(-z\right)\right)\right) + \log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{\frac{\frac{457679.80848377093}{1 - z}}{1 - z} - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{\color{blue}{\frac{676.5203681218851}{1 - z} + \left(-\frac{-1259.1392167224028}{2 - z}\right)}}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   8. distribute-neg-frac 99.3%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\left(-\left(7.5 + \left(-z\right)\right)\right) + \log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{\frac{\frac{457679.80848377093}{1 - z}}{1 - z} - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{\frac{676.5203681218851}{1 - z} + \color{blue}{\frac{--1259.1392167224028}{2 - z}}}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   9. metadata-eval 99.3%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\left(-\left(7.5 + \left(-z\right)\right)\right) + \log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{\frac{\frac{457679.80848377093}{1 - z}}{1 - z} - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{\frac{676.5203681218851}{1 - z} + \frac{\color{blue}{1259.1392167224028}}{2 - z}}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

9. Simplified 99.3%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\left(-\left(7.5 + \left(-z\right)\right)\right) + \log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \color{blue}{\frac{\frac{\frac{457679.80848377093}{1 - z}}{1 - z} - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}}}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
10. Step-by-step derivation

    1. expm1-log1p-u 99.3%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\left(-\left(7.5 + \left(-z\right)\right)\right) + \log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{\frac{\frac{457679.80848377093}{1 - z}}{1 - z} - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}}\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right)}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    2. expm1-udef 99.3%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\left(-\left(7.5 + \left(-z\right)\right)\right) + \log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{\frac{\frac{457679.80848377093}{1 - z}}{1 - z} - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}}\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)} - 1\right)}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    3. sub-neg 99.3%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\left(-\left(7.5 + \left(-z\right)\right)\right) + \log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{\frac{\frac{457679.80848377093}{1 - z}}{1 - z} - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}}\right) + \left(e^{\mathsf{log1p}\left(\frac{771.3234287776531}{\color{blue}{\left(1 - z\right) + \left(--2\right)}} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)} - 1\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    4. metadata-eval 99.3%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\left(-\left(7.5 + \left(-z\right)\right)\right) + \log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{\frac{\frac{457679.80848377093}{1 - z}}{1 - z} - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}}\right) + \left(e^{\mathsf{log1p}\left(\frac{771.3234287776531}{\left(1 - z\right) + \color{blue}{2}} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)} - 1\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    5. sub-neg 99.3%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\left(-\left(7.5 + \left(-z\right)\right)\right) + \log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{\frac{\frac{457679.80848377093}{1 - z}}{1 - z} - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}}\right) + \left(e^{\mathsf{log1p}\left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\color{blue}{\left(1 - z\right) + \left(--3\right)}}\right)} - 1\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    6. metadata-eval 99.3%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\left(-\left(7.5 + \left(-z\right)\right)\right) + \log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{\frac{\frac{457679.80848377093}{1 - z}}{1 - z} - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}}\right) + \left(e^{\mathsf{log1p}\left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + \color{blue}{3}}\right)} - 1\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

11. Applied egg-rr 99.3%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\left(-\left(7.5 + \left(-z\right)\right)\right) + \log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{\frac{\frac{457679.80848377093}{1 - z}}{1 - z} - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}}\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)} - 1\right)}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
12. Step-by-step derivation

    1. expm1-def 99.3%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\left(-\left(7.5 + \left(-z\right)\right)\right) + \log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{\frac{\frac{457679.80848377093}{1 - z}}{1 - z} - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}}\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    2. expm1-log1p 99.3%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\left(-\left(7.5 + \left(-z\right)\right)\right) + \log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{\frac{\frac{457679.80848377093}{1 - z}}{1 - z} - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}}\right) + \color{blue}{\left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    3. +-commutative 99.3%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\left(-\left(7.5 + \left(-z\right)\right)\right) + \log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{\frac{\frac{457679.80848377093}{1 - z}}{1 - z} - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}}\right) + \left(\frac{771.3234287776531}{\color{blue}{2 + \left(1 - z\right)}} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    4. associate-+r- 99.3%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\left(-\left(7.5 + \left(-z\right)\right)\right) + \log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{\frac{\frac{457679.80848377093}{1 - z}}{1 - z} - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}}\right) + \left(\frac{771.3234287776531}{\color{blue}{\left(2 + 1\right) - z}} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    5. metadata-eval 99.3%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\left(-\left(7.5 + \left(-z\right)\right)\right) + \log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{\frac{\frac{457679.80848377093}{1 - z}}{1 - z} - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}}\right) + \left(\frac{771.3234287776531}{\color{blue}{3} - z} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    6. +-commutative 99.3%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\left(-\left(7.5 + \left(-z\right)\right)\right) + \log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{\frac{\frac{457679.80848377093}{1 - z}}{1 - z} - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}}\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{\color{blue}{3 + \left(1 - z\right)}}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    7. associate-+r- 99.3%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\left(-\left(7.5 + \left(-z\right)\right)\right) + \log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{\frac{\frac{457679.80848377093}{1 - z}}{1 - z} - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}}\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{\color{blue}{\left(3 + 1\right) - z}}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    8. metadata-eval 99.3%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\left(-\left(7.5 + \left(-z\right)\right)\right) + \log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{\frac{\frac{457679.80848377093}{1 - z}}{1 - z} - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}}\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{\color{blue}{4} - z}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

13. Simplified 99.3%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\left(-\left(7.5 + \left(-z\right)\right)\right) + \log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{\frac{\frac{457679.80848377093}{1 - z}}{1 - z} - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}}\right) + \color{blue}{\left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

14. Final simplification 99.3%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{\left(z - 7.5\right) + \log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)}\right) \cdot \left(\left(\left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right) - \left(\left(\frac{\frac{\frac{1585431.567088306}{2 - z}}{2 - z} - \frac{\frac{457679.80848377093}{1 - z}}{1 - z}}{\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}} - 0.9999999999998099\right) - \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

Alternative 2: 96.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(z)
	return Float64(Float64(Float64(Float64(Float64(pi * sqrt(Float64(pi * 2.0))) * (Float64(7.5 - z) ^ Float64(0.5 - z))) * exp(Float64(z + -7.5))) / sin(Float64(pi * z))) * Float64(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)))) + Float64(0.9999999999998099 + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(Float64(771.3234287776531 / Float64(3.0 - z)) + Float64(-176.6150291621406 / Float64(4.0 - z))) + Float64(-1259.1392167224028 / Float64(2.0 - z)))))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

3. Simplified 98.0%

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

   1. expm1-log1p-u 96.6%

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

   2. expm1-udef 96.6%

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

   3. associate-+l+ 96.7%

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

   4. +-commutative 96.7%

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

5. Applied egg-rr 96.7%

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

   1. expm1-def 96.7%

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

   2. expm1-log1p 96.6%

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

   3. +-commutative 96.6%

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

7. Simplified 96.6%

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

8. Final simplification 96.6%

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

Alternative 3: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(z)
	return Float64(Float64(Float64(Float64(Float64(pi * sqrt(Float64(pi * 2.0))) * (Float64(7.5 - z) ^ Float64(0.5 - z))) * exp(Float64(z + -7.5))) / sin(Float64(pi * z))) * Float64(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)))) + Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 - z))) + Float64(Float64(Float64(771.3234287776531 / Float64(3.0 - z)) + Float64(-176.6150291621406 / Float64(4.0 - z))) + Float64(-1259.1392167224028 / Float64(2.0 - z))))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

3. Simplified 98.0%

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

4. Final simplification 98.0%

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

Alternative 4: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(z)
	return Float64(Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(0.9999999999998099 + Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(Float64(771.3234287776531 / Float64(3.0 - z)) + Float64(-1259.1392167224028 / Float64(2.0 - z)))))) + 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))))) * Float64(Float64(Float64(Float64(pi * sqrt(Float64(pi * 2.0))) * (Float64(7.5 - z) ^ Float64(0.5 - z))) * exp(Float64(z + -7.5))) / sin(Float64(pi * z))))
end

MATLAB

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

Wolfram

code[z_] := N[(N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $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] * N[(N[(N[(N[(Pi * N[Sqrt[N[(Pi * 2.0), $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] / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

3. Simplified 98.0%

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

   1. expm1-log1p-u 96.6%

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

   2. expm1-udef 96.6%

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

   3. associate-+l+ 96.7%

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

   4. +-commutative 96.7%

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

5. Applied egg-rr 96.7%

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

   1. expm1-def 96.7%

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

   2. expm1-log1p 96.6%

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

   3. +-commutative 96.6%

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

   4. associate-+l+ 96.7%

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

   5. +-commutative 96.7%

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

   6. associate-+r+ 98.0%

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

7. Simplified 98.0%

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

8. Final simplification 98.0%

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

Alternative 5: 97.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(\pi \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(2.4783734731930944 + z \cdot 0.49644453405676175\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/
   (* (* (* PI (sqrt (* PI 2.0))) (pow (- 7.5 z) (- 0.5 z))) (exp (+ z -7.5)))
   (sin (* PI z)))
  (+
   (+
    (/ 676.5203681218851 (- 1.0 z))
    (+
     0.9999999999998099
     (+
      (/ -176.6150291621406 (- 4.0 z))
      (+ (/ 771.3234287776531 (- 3.0 z)) (/ -1259.1392167224028 (- 2.0 z))))))
   (+
    (+ (/ 9.984369578019572e-6 (- 7.0 z)) (/ 1.5056327351493116e-7 (- 8.0 z)))
    (+ 2.4783734731930944 (* z 0.49644453405676175))))))

C

double code(double z) {
	return ((((((double) M_PI) * sqrt((((double) M_PI) * 2.0))) * pow((7.5 - z), (0.5 - z))) * exp((z + -7.5))) / sin((((double) M_PI) * z))) * (((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + ((-176.6150291621406 / (4.0 - z)) + ((771.3234287776531 / (3.0 - z)) + (-1259.1392167224028 / (2.0 - z)))))) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (2.4783734731930944 + (z * 0.49644453405676175))));
}

Java

public static double code(double z) {
	return ((((Math.PI * Math.sqrt((Math.PI * 2.0))) * Math.pow((7.5 - z), (0.5 - z))) * Math.exp((z + -7.5))) / Math.sin((Math.PI * z))) * (((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + ((-176.6150291621406 / (4.0 - z)) + ((771.3234287776531 / (3.0 - z)) + (-1259.1392167224028 / (2.0 - z)))))) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (2.4783734731930944 + (z * 0.49644453405676175))));
}

Python

def code(z):
	return ((((math.pi * math.sqrt((math.pi * 2.0))) * math.pow((7.5 - z), (0.5 - z))) * math.exp((z + -7.5))) / math.sin((math.pi * z))) * (((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + ((-176.6150291621406 / (4.0 - z)) + ((771.3234287776531 / (3.0 - z)) + (-1259.1392167224028 / (2.0 - z)))))) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (2.4783734731930944 + (z * 0.49644453405676175))))

Julia

function code(z)
	return Float64(Float64(Float64(Float64(Float64(pi * sqrt(Float64(pi * 2.0))) * (Float64(7.5 - z) ^ Float64(0.5 - z))) * exp(Float64(z + -7.5))) / sin(Float64(pi * z))) * Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(0.9999999999998099 + Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(Float64(771.3234287776531 / Float64(3.0 - z)) + Float64(-1259.1392167224028 / Float64(2.0 - z)))))) + Float64(Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))) + Float64(2.4783734731930944 + Float64(z * 0.49644453405676175)))))
end

MATLAB

function tmp = code(z)
	tmp = ((((pi * sqrt((pi * 2.0))) * ((7.5 - z) ^ (0.5 - z))) * exp((z + -7.5))) / sin((pi * z))) * (((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + ((-176.6150291621406 / (4.0 - z)) + ((771.3234287776531 / (3.0 - z)) + (-1259.1392167224028 / (2.0 - z)))))) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (2.4783734731930944 + (z * 0.49644453405676175))));
end

Wolfram

code[z_] := N[(N[(N[(N[(N[(Pi * N[Sqrt[N[(Pi * 2.0), $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] / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $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[(2.4783734731930944 + N[(z * 0.49644453405676175), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

3. Simplified 98.0%

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

   1. expm1-log1p-u 96.6%

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

   2. expm1-udef 96.6%

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

   3. associate-+l+ 96.7%

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

   4. +-commutative 96.7%

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

5. Applied egg-rr 96.7%

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

   1. expm1-def 96.7%

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

   2. expm1-log1p 96.6%

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

   3. +-commutative 96.6%

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

   4. associate-+l+ 96.7%

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

   5. +-commutative 96.7%

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

   6. associate-+r+ 98.0%

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

7. Simplified 98.0%

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

8. Taylor expanded in z around 0 96.5%

   \[\leadsto \left(\left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + 0.9999999999998099\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \color{blue}{\left(2.4783734731930944 + 0.49644453405676175 \cdot z\right)}\right)\right) \cdot \frac{\left(\left(\pi \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}}{\sin \left(\pi \cdot z\right)} \]
9. Step-by-step derivation

   1. *-commutative 96.2%

      \[\leadsto \left(\left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(-229.08220098308368 \cdot z - 372.46179876865034\right) + \frac{-176.6150291621406}{4 - z}\right) + 0.9999999999998099\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(2.4783734731930944 + \color{blue}{z \cdot 0.49644453405676175}\right)\right)\right) \cdot \frac{\left(\left(\pi \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}}{\sin \left(\pi \cdot z\right)} \]

10. Simplified 96.5%

    \[\leadsto \left(\left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + 0.9999999999998099\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \color{blue}{\left(2.4783734731930944 + z \cdot 0.49644453405676175\right)}\right)\right) \cdot \frac{\left(\left(\pi \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}}{\sin \left(\pi \cdot z\right)} \]

11. Final simplification 96.5%

    \[\leadsto \frac{\left(\left(\pi \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(2.4783734731930944 + z \cdot 0.49644453405676175\right)\right)\right) \]

Alternative 6: 97.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(\pi \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(2.4783734731930944 + z \cdot 0.49644453405676175\right)\right) + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \left(\frac{-176.6150291621406}{4 - z} + \left(z \cdot -229.08220098308368 - 372.46179876865034\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/
   (* (* (* PI (sqrt (* PI 2.0))) (pow (- 7.5 z) (- 0.5 z))) (exp (+ z -7.5)))
   (sin (* PI z)))
  (+
   (+
    (+ (/ 9.984369578019572e-6 (- 7.0 z)) (/ 1.5056327351493116e-7 (- 8.0 z)))
    (+ 2.4783734731930944 (* z 0.49644453405676175)))
   (+
    (/ 676.5203681218851 (- 1.0 z))
    (+
     0.9999999999998099
     (+
      (/ -176.6150291621406 (- 4.0 z))
      (- (* z -229.08220098308368) 372.46179876865034)))))))

C

double code(double z) {
	return ((((((double) M_PI) * sqrt((((double) M_PI) * 2.0))) * pow((7.5 - z), (0.5 - z))) * exp((z + -7.5))) / sin((((double) M_PI) * z))) * ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (2.4783734731930944 + (z * 0.49644453405676175))) + ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + ((-176.6150291621406 / (4.0 - z)) + ((z * -229.08220098308368) - 372.46179876865034)))));
}

Java

public static double code(double z) {
	return ((((Math.PI * Math.sqrt((Math.PI * 2.0))) * Math.pow((7.5 - z), (0.5 - z))) * Math.exp((z + -7.5))) / Math.sin((Math.PI * z))) * ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (2.4783734731930944 + (z * 0.49644453405676175))) + ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + ((-176.6150291621406 / (4.0 - z)) + ((z * -229.08220098308368) - 372.46179876865034)))));
}

Python

def code(z):
	return ((((math.pi * math.sqrt((math.pi * 2.0))) * math.pow((7.5 - z), (0.5 - z))) * math.exp((z + -7.5))) / math.sin((math.pi * z))) * ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (2.4783734731930944 + (z * 0.49644453405676175))) + ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + ((-176.6150291621406 / (4.0 - z)) + ((z * -229.08220098308368) - 372.46179876865034)))))

Julia

function code(z)
	return Float64(Float64(Float64(Float64(Float64(pi * sqrt(Float64(pi * 2.0))) * (Float64(7.5 - z) ^ Float64(0.5 - z))) * exp(Float64(z + -7.5))) / sin(Float64(pi * z))) * Float64(Float64(Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))) + Float64(2.4783734731930944 + Float64(z * 0.49644453405676175))) + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(0.9999999999998099 + Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(Float64(z * -229.08220098308368) - 372.46179876865034))))))
end

MATLAB

function tmp = code(z)
	tmp = ((((pi * sqrt((pi * 2.0))) * ((7.5 - z) ^ (0.5 - z))) * exp((z + -7.5))) / sin((pi * z))) * ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (2.4783734731930944 + (z * 0.49644453405676175))) + ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + ((-176.6150291621406 / (4.0 - z)) + ((z * -229.08220098308368) - 372.46179876865034)))));
end

Wolfram

code[z_] := N[(N[(N[(N[(N[(Pi * N[Sqrt[N[(Pi * 2.0), $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] / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.4783734731930944 + N[(z * 0.49644453405676175), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(z * -229.08220098308368), $MachinePrecision] - 372.46179876865034), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

3. Simplified 98.0%

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

   1. expm1-log1p-u 96.6%

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

   2. expm1-udef 96.6%

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

   3. associate-+l+ 96.7%

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

   4. +-commutative 96.7%

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

5. Applied egg-rr 96.7%

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

   1. expm1-def 96.7%

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

   2. expm1-log1p 96.6%

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

   3. +-commutative 96.6%

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

   4. associate-+l+ 96.7%

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

   5. +-commutative 96.7%

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

   6. associate-+r+ 98.0%

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

7. Simplified 98.0%

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

8. Taylor expanded in z around 0 96.2%

   \[\leadsto \left(\left(\frac{676.5203681218851}{1 - z} + \left(\left(\color{blue}{\left(-229.08220098308368 \cdot z - 372.46179876865034\right)} + \frac{-176.6150291621406}{4 - z}\right) + 0.9999999999998099\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right)\right) \cdot \frac{\left(\left(\pi \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}}{\sin \left(\pi \cdot z\right)} \]

9. Taylor expanded in z around 0 96.2%

   \[\leadsto \left(\left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(-229.08220098308368 \cdot z - 372.46179876865034\right) + \frac{-176.6150291621406}{4 - z}\right) + 0.9999999999998099\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \color{blue}{\left(2.4783734731930944 + 0.49644453405676175 \cdot z\right)}\right)\right) \cdot \frac{\left(\left(\pi \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}}{\sin \left(\pi \cdot z\right)} \]
10. Step-by-step derivation

    1. *-commutative 96.2%

       \[\leadsto \left(\left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(-229.08220098308368 \cdot z - 372.46179876865034\right) + \frac{-176.6150291621406}{4 - z}\right) + 0.9999999999998099\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(2.4783734731930944 + \color{blue}{z \cdot 0.49644453405676175}\right)\right)\right) \cdot \frac{\left(\left(\pi \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}}{\sin \left(\pi \cdot z\right)} \]

11. Simplified 96.2%

    \[\leadsto \left(\left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(-229.08220098308368 \cdot z - 372.46179876865034\right) + \frac{-176.6150291621406}{4 - z}\right) + 0.9999999999998099\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \color{blue}{\left(2.4783734731930944 + z \cdot 0.49644453405676175\right)}\right)\right) \cdot \frac{\left(\left(\pi \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}}{\sin \left(\pi \cdot z\right)} \]

12. Final simplification 96.2%

    \[\leadsto \frac{\left(\left(\pi \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(2.4783734731930944 + z \cdot 0.49644453405676175\right)\right) + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \left(\frac{-176.6150291621406}{4 - z} + \left(z \cdot -229.08220098308368 - 372.46179876865034\right)\right)\right)\right)\right) \]

Alternative 7: 96.7% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/
   (* (* (* PI (sqrt (* PI 2.0))) (pow (- 7.5 z) (- 0.5 z))) (exp (+ z -7.5)))
   (sin (* PI z)))
  (+
   (+
    (+ (/ 9.984369578019572e-6 (- 7.0 z)) (/ 1.5056327351493116e-7 (- 8.0 z)))
    (+ (/ 12.507343278686905 (- 5.0 z)) (/ -0.13857109526572012 (- 6.0 z))))
   (+ 260.9048120626994 (* z 436.3997278161676)))))

C

double code(double z) {
	return ((((((double) M_PI) * sqrt((((double) M_PI) * 2.0))) * pow((7.5 - z), (0.5 - z))) * exp((z + -7.5))) / sin((((double) M_PI) * z))) * ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z)))) + (260.9048120626994 + (z * 436.3997278161676)));
}

Java

public static double code(double z) {
	return ((((Math.PI * Math.sqrt((Math.PI * 2.0))) * Math.pow((7.5 - z), (0.5 - z))) * Math.exp((z + -7.5))) / Math.sin((Math.PI * z))) * ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z)))) + (260.9048120626994 + (z * 436.3997278161676)));
}

Python

def code(z):
	return ((((math.pi * math.sqrt((math.pi * 2.0))) * math.pow((7.5 - z), (0.5 - z))) * math.exp((z + -7.5))) / math.sin((math.pi * z))) * ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z)))) + (260.9048120626994 + (z * 436.3997278161676)))

Julia

function code(z)
	return Float64(Float64(Float64(Float64(Float64(pi * sqrt(Float64(pi * 2.0))) * (Float64(7.5 - z) ^ Float64(0.5 - z))) * exp(Float64(z + -7.5))) / sin(Float64(pi * z))) * Float64(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)))) + Float64(260.9048120626994 + Float64(z * 436.3997278161676))))
end

MATLAB

function tmp = code(z)
	tmp = ((((pi * sqrt((pi * 2.0))) * ((7.5 - z) ^ (0.5 - z))) * exp((z + -7.5))) / sin((pi * z))) * ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z)))) + (260.9048120626994 + (z * 436.3997278161676)));
end

Wolfram

code[z_] := N[(N[(N[(N[(N[(Pi * N[Sqrt[N[(Pi * 2.0), $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] / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(260.9048120626994 + N[(z * 436.3997278161676), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

3. Simplified 98.0%

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

4. Taylor expanded in z around 0 96.0%

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

   1. *-commutative 96.0%

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

6. Simplified 96.0%

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

7. Final simplification 96.0%

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

Alternative 8: 96.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(\pi \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + 2.4783734731930944\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/
   (* (* (* PI (sqrt (* PI 2.0))) (pow (- 7.5 z) (- 0.5 z))) (exp (+ z -7.5)))
   (sin (* PI z)))
  (+
   (+ 260.9048120626994 (* z 436.3997278161676))
   (+
    (+ (/ 9.984369578019572e-6 (- 7.0 z)) (/ 1.5056327351493116e-7 (- 8.0 z)))
    2.4783734731930944))))

C

double code(double z) {
	return ((((((double) M_PI) * sqrt((((double) M_PI) * 2.0))) * pow((7.5 - z), (0.5 - z))) * exp((z + -7.5))) / sin((((double) M_PI) * z))) * ((260.9048120626994 + (z * 436.3997278161676)) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + 2.4783734731930944));
}

Java

public static double code(double z) {
	return ((((Math.PI * Math.sqrt((Math.PI * 2.0))) * Math.pow((7.5 - z), (0.5 - z))) * Math.exp((z + -7.5))) / Math.sin((Math.PI * z))) * ((260.9048120626994 + (z * 436.3997278161676)) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + 2.4783734731930944));
}

Python

def code(z):
	return ((((math.pi * math.sqrt((math.pi * 2.0))) * math.pow((7.5 - z), (0.5 - z))) * math.exp((z + -7.5))) / math.sin((math.pi * z))) * ((260.9048120626994 + (z * 436.3997278161676)) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + 2.4783734731930944))

Julia

function code(z)
	return Float64(Float64(Float64(Float64(Float64(pi * sqrt(Float64(pi * 2.0))) * (Float64(7.5 - z) ^ Float64(0.5 - z))) * exp(Float64(z + -7.5))) / sin(Float64(pi * z))) * Float64(Float64(260.9048120626994 + Float64(z * 436.3997278161676)) + Float64(Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))) + 2.4783734731930944)))
end

MATLAB

function tmp = code(z)
	tmp = ((((pi * sqrt((pi * 2.0))) * ((7.5 - z) ^ (0.5 - z))) * exp((z + -7.5))) / sin((pi * z))) * ((260.9048120626994 + (z * 436.3997278161676)) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + 2.4783734731930944));
end

Wolfram

code[z_] := N[(N[(N[(N[(N[(Pi * N[Sqrt[N[(Pi * 2.0), $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] / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(260.9048120626994 + N[(z * 436.3997278161676), $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] + 2.4783734731930944), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

3. Simplified 98.0%

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

4. Taylor expanded in z around 0 96.0%

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

   1. *-commutative 96.0%

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

6. Simplified 96.0%

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

7. Taylor expanded in z around 0 95.9%

   \[\leadsto \left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \color{blue}{2.4783734731930944}\right)\right) \cdot \frac{\left(\left(\pi \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}}{\sin \left(\pi \cdot z\right)} \]

8. Final simplification 95.9%

   \[\leadsto \frac{\left(\left(\pi \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + 2.4783734731930944\right)\right) \]

Alternative 9: 96.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

3. Simplified 96.5%

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

4. Taylor expanded in z around 0 95.0%

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

   1. *-commutative 95.0%

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

6. Simplified 95.0%

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

7. Taylor expanded in z around 0 96.0%

   \[\leadsto \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \color{blue}{\left(212.9540523020159 + 74.66416387488323 \cdot z\right)}\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right) \]
8. Step-by-step derivation

   1. *-commutative 96.0%

      \[\leadsto \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \left(212.9540523020159 + \color{blue}{z \cdot 74.66416387488323}\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right) \]

9. Simplified 96.0%

   \[\leadsto \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \color{blue}{\left(212.9540523020159 + z \cdot 74.66416387488323\right)}\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right) \]

10. Taylor expanded in z around 0 95.5%

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

    1. *-commutative 95.5%

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

12. Simplified 95.5%

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

13. Final simplification 95.5%

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

Alternative 10: 95.5% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (+
   (+
    (+ (/ 9.984369578019572e-6 (- 7.0 z)) (/ 1.5056327351493116e-7 (- 8.0 z)))
    (+ (/ 12.507343278686905 (- 5.0 z)) (/ -0.13857109526572012 (- 6.0 z))))
   (+ 260.9048120626994 (* z 436.3997278161676)))
  (* (sqrt PI) (/ (* (exp -7.5) (* (sqrt 2.0) (sqrt 7.5))) z))))

C

double code(double z) {
	return ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z)))) + (260.9048120626994 + (z * 436.3997278161676))) * (sqrt(((double) M_PI)) * ((exp(-7.5) * (sqrt(2.0) * sqrt(7.5))) / z));
}

Java

public static double code(double z) {
	return ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z)))) + (260.9048120626994 + (z * 436.3997278161676))) * (Math.sqrt(Math.PI) * ((Math.exp(-7.5) * (Math.sqrt(2.0) * Math.sqrt(7.5))) / z));
}

Python

def code(z):
	return ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z)))) + (260.9048120626994 + (z * 436.3997278161676))) * (math.sqrt(math.pi) * ((math.exp(-7.5) * (math.sqrt(2.0) * math.sqrt(7.5))) / z))

Julia

function code(z)
	return Float64(Float64(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)))) + Float64(260.9048120626994 + Float64(z * 436.3997278161676))) * Float64(sqrt(pi) * Float64(Float64(exp(-7.5) * Float64(sqrt(2.0) * sqrt(7.5))) / z)))
end

MATLAB

function tmp = code(z)
	tmp = ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z)))) + (260.9048120626994 + (z * 436.3997278161676))) * (sqrt(pi) * ((exp(-7.5) * (sqrt(2.0) * sqrt(7.5))) / z));
end

Wolfram

code[z_] := N[(N[(N[(N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(260.9048120626994 + N[(z * 436.3997278161676), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[(N[Exp[-7.5], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

3. Simplified 98.0%

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

4. Taylor expanded in z around 0 96.0%

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

   1. *-commutative 96.0%

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

6. Simplified 96.0%

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

7. Taylor expanded in z around 0 95.4%

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

8. Final simplification 95.4%

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

Alternative 11: 95.5% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (+
   (+
    (+ (/ 9.984369578019572e-6 (- 7.0 z)) (/ 1.5056327351493116e-7 (- 8.0 z)))
    (+ (/ 12.507343278686905 (- 5.0 z)) (/ -0.13857109526572012 (- 6.0 z))))
   (+ 260.9048120626994 (* z 436.3997278161676)))
  (/ (* (exp -7.5) (* (sqrt 2.0) (sqrt 7.5))) (/ z (sqrt PI)))))

C

double code(double z) {
	return ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z)))) + (260.9048120626994 + (z * 436.3997278161676))) * ((exp(-7.5) * (sqrt(2.0) * sqrt(7.5))) / (z / sqrt(((double) M_PI))));
}

Java

public static double code(double z) {
	return ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z)))) + (260.9048120626994 + (z * 436.3997278161676))) * ((Math.exp(-7.5) * (Math.sqrt(2.0) * Math.sqrt(7.5))) / (z / Math.sqrt(Math.PI)));
}

Python

def code(z):
	return ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z)))) + (260.9048120626994 + (z * 436.3997278161676))) * ((math.exp(-7.5) * (math.sqrt(2.0) * math.sqrt(7.5))) / (z / math.sqrt(math.pi)))

Julia

function code(z)
	return Float64(Float64(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)))) + Float64(260.9048120626994 + Float64(z * 436.3997278161676))) * Float64(Float64(exp(-7.5) * Float64(sqrt(2.0) * sqrt(7.5))) / Float64(z / sqrt(pi))))
end

MATLAB

function tmp = code(z)
	tmp = ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z)))) + (260.9048120626994 + (z * 436.3997278161676))) * ((exp(-7.5) * (sqrt(2.0) * sqrt(7.5))) / (z / sqrt(pi)));
end

Wolfram

code[z_] := N[(N[(N[(N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(260.9048120626994 + N[(z * 436.3997278161676), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Exp[-7.5], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

3. Simplified 98.0%

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

4. Taylor expanded in z around 0 96.0%

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

   1. *-commutative 96.0%

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

6. Simplified 96.0%

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

7. Taylor expanded in z around 0 95.4%

   \[\leadsto \left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right)\right) \cdot \color{blue}{\left(\frac{e^{-7.5} \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)}{z} \cdot \sqrt{\pi}\right)} \]
8. Step-by-step derivation

   1. associate-*l/ 95.5%

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

   2. associate-/l* 95.4%

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

9. Simplified 95.4%

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

10. Final simplification 95.4%

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

Alternative 12: 95.6% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (+
   (+
    (+ (/ 9.984369578019572e-6 (- 7.0 z)) (/ 1.5056327351493116e-7 (- 8.0 z)))
    (+ (/ 12.507343278686905 (- 5.0 z)) (/ -0.13857109526572012 (- 6.0 z))))
   (+ 260.9048120626994 (* z 436.3997278161676)))
  (/ (* (* (exp -7.5) (* (sqrt 2.0) (sqrt 7.5))) (sqrt PI)) z)))

C

double code(double z) {
	return ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z)))) + (260.9048120626994 + (z * 436.3997278161676))) * (((exp(-7.5) * (sqrt(2.0) * sqrt(7.5))) * sqrt(((double) M_PI))) / z);
}

Java

public static double code(double z) {
	return ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z)))) + (260.9048120626994 + (z * 436.3997278161676))) * (((Math.exp(-7.5) * (Math.sqrt(2.0) * Math.sqrt(7.5))) * Math.sqrt(Math.PI)) / z);
}

Python

def code(z):
	return ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z)))) + (260.9048120626994 + (z * 436.3997278161676))) * (((math.exp(-7.5) * (math.sqrt(2.0) * math.sqrt(7.5))) * math.sqrt(math.pi)) / z)

Julia

function code(z)
	return Float64(Float64(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)))) + Float64(260.9048120626994 + Float64(z * 436.3997278161676))) * Float64(Float64(Float64(exp(-7.5) * Float64(sqrt(2.0) * sqrt(7.5))) * sqrt(pi)) / z))
end

MATLAB

function tmp = code(z)
	tmp = ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z)))) + (260.9048120626994 + (z * 436.3997278161676))) * (((exp(-7.5) * (sqrt(2.0) * sqrt(7.5))) * sqrt(pi)) / z);
end

Wolfram

code[z_] := N[(N[(N[(N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(260.9048120626994 + N[(z * 436.3997278161676), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Exp[-7.5], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

3. Simplified 98.0%

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

4. Taylor expanded in z around 0 96.0%

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

   1. *-commutative 96.0%

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

6. Simplified 96.0%

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

7. Taylor expanded in z around 0 95.4%

   \[\leadsto \left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right)\right) \cdot \color{blue}{\left(\frac{e^{-7.5} \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)}{z} \cdot \sqrt{\pi}\right)} \]
8. Step-by-step derivation

   1. associate-*l/ 95.5%

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

9. Simplified 95.5%

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

10. Final simplification 95.5%

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

Alternative 13: 14.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{\pi} \cdot \frac{e^{-7.5} \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)}{z}\right) \cdot 3.4783749183518244 \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (* (sqrt PI) (/ (* (exp -7.5) (* (sqrt 2.0) (sqrt 7.5))) z))
  3.4783749183518244))

C

double code(double z) {
	return (sqrt(((double) M_PI)) * ((exp(-7.5) * (sqrt(2.0) * sqrt(7.5))) / z)) * 3.4783749183518244;
}

Java

public static double code(double z) {
	return (Math.sqrt(Math.PI) * ((Math.exp(-7.5) * (Math.sqrt(2.0) * Math.sqrt(7.5))) / z)) * 3.4783749183518244;
}

Python

def code(z):
	return (math.sqrt(math.pi) * ((math.exp(-7.5) * (math.sqrt(2.0) * math.sqrt(7.5))) / z)) * 3.4783749183518244

Julia

function code(z)
	return Float64(Float64(sqrt(pi) * Float64(Float64(exp(-7.5) * Float64(sqrt(2.0) * sqrt(7.5))) / z)) * 3.4783749183518244)
end

MATLAB

function tmp = code(z)
	tmp = (sqrt(pi) * ((exp(-7.5) * (sqrt(2.0) * sqrt(7.5))) / z)) * 3.4783749183518244;
end

Wolfram

code[z_] := N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[(N[Exp[-7.5], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] * 3.4783749183518244), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

3. Simplified 98.0%

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

4. Taylor expanded in z around inf 14.5%

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

5. Taylor expanded in z around 0 14.5%

   \[\leadsto \left(0.9999999999998099 + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \color{blue}{2.4783734731930944}\right)\right) \cdot \frac{\left(\left(\pi \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}}{\sin \left(\pi \cdot z\right)} \]

6. Taylor expanded in z around 0 14.5%

   \[\leadsto \left(0.9999999999998099 + \color{blue}{2.4783749183520145}\right) \cdot \frac{\left(\left(\pi \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}}{\sin \left(\pi \cdot z\right)} \]

7. Taylor expanded in z around 0 14.4%

   \[\leadsto \left(0.9999999999998099 + 2.4783749183520145\right) \cdot \color{blue}{\left(\frac{e^{-7.5} \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)}{z} \cdot \sqrt{\pi}\right)} \]

8. Final simplification 14.4%

   \[\leadsto \left(\sqrt{\pi} \cdot \frac{e^{-7.5} \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)}{z}\right) \cdot 3.4783749183518244 \]

Reproduce

herbie shell --seed 2023305 
(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))))))