Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.4% → 98.0%

Time: 1.0min

Alternatives: 14

Speedup: 1.3×

• could not determine a ground truth

  1. z = -7.191511103999593e+239

Specification

\[z \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[z_] := N[(N[(Pi * N[(N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-0.5 + N[(z - 7.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(N[(1.0 - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(N[(1.0 - z), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] - -4.0), $MachinePrecision]), $MachinePrecision] - N[(-0.13857109526572012 / N[(-5.0 + N[(z + -1.0), $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 96.5%

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

3. Simplified 98.3%

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

   1. associate-*l/ 98.5%

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

6. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\frac{\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-0.5 + \left(-\left(\left(1 - z\right) + 6\right)\right)}\right)\right)}{\sin \left(\pi \cdot z\right)}} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
7. Step-by-step derivation

   1. associate-/l* 98.3%

      \[\leadsto \color{blue}{\left(\pi \cdot \frac{\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-0.5 + \left(-\left(\left(1 - z\right) + 6\right)\right)}\right)}{\sin \left(\pi \cdot z\right)}\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

8. Simplified 98.3%

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

9. Final simplification 98.3%

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

Alternative 2: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

3. Simplified 98.3%

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

   1. associate-*l/ 98.5%

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

6. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\frac{\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-0.5 + \left(-\left(\left(1 - z\right) + 6\right)\right)}\right)\right)}{\sin \left(\pi \cdot z\right)}} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
7. Step-by-step derivation

   1. associate-/l* 98.3%

      \[\leadsto \color{blue}{\left(\pi \cdot \frac{\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-0.5 + \left(-\left(\left(1 - z\right) + 6\right)\right)}\right)}{\sin \left(\pi \cdot z\right)}\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

8. Simplified 98.3%

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

10. Applied egg-rr 97.3%

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

12. Simplified 98.2%

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

13. Final simplification 98.2%

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

Alternative 3: 97.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 + \left(z - 7\right)}\right) \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* PI z)))
  (*
   (* (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ -0.5 (- z 7.0)))) (sqrt (* PI 2.0)))
   (+
    (+
     263.3831855358925
     (*
      z
      (+
       436.8961723502244
       (* z (+ 545.0353078134797 (* z 606.6766809125655))))))
    (+
     (/ 1.5056327351493116e-7 (- 8.0 z))
     (/ 9.984369578019572e-6 (- 7.0 z)))))))

C

double code(double z) {
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (((pow((7.5 - z), (0.5 - z)) * exp((-0.5 + (z - 7.0)))) * sqrt((((double) M_PI) * 2.0))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))) + ((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z)))));
}

Java

public static double code(double z) {
	return (Math.PI / Math.sin((Math.PI * z))) * (((Math.pow((7.5 - z), (0.5 - z)) * Math.exp((-0.5 + (z - 7.0)))) * Math.sqrt((Math.PI * 2.0))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))) + ((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z)))));
}

Python

def code(z):
	return (math.pi / math.sin((math.pi * z))) * (((math.pow((7.5 - z), (0.5 - z)) * math.exp((-0.5 + (z - 7.0)))) * math.sqrt((math.pi * 2.0))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))) + ((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z)))))

Julia

function code(z)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(-0.5 + Float64(z - 7.0)))) * sqrt(Float64(pi * 2.0))) * Float64(Float64(263.3831855358925 + Float64(z * Float64(436.8961723502244 + Float64(z * Float64(545.0353078134797 + Float64(z * 606.6766809125655)))))) + Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) + Float64(9.984369578019572e-6 / Float64(7.0 - z))))))
end

MATLAB

function tmp = code(z)
	tmp = (pi / sin((pi * z))) * (((((7.5 - z) ^ (0.5 - z)) * exp((-0.5 + (z - 7.0)))) * sqrt((pi * 2.0))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))) + ((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z)))));
end

Wolfram

code[z_] := N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-0.5 + N[(z - 7.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(263.3831855358925 + N[(z * N[(436.8961723502244 + N[(z * N[(545.0353078134797 + N[(z * 606.6766809125655), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.5%

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

3. Simplified 96.4%

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

6. Applied egg-rr 96.4%

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

8. Simplified 97.6%

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

9. Taylor expanded in z around 0 98.2%

   \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 - \left(7 - z\right)}\right)\right)\right) \]
10. Step-by-step derivation

    1. *-commutative 98.2%

       \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + \color{blue}{z \cdot 606.6766809125655}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 - \left(7 - z\right)}\right)\right)\right) \]

11. Simplified 98.2%

    \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 - \left(7 - z\right)}\right)\right)\right) \]

12. Final simplification 98.2%

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 + \left(z - 7\right)}\right) \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right)\right) \]
13. Add Preprocessing

Alternative 4: 97.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 + \left(z - 7\right)}\right) \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* PI z)))
  (*
   (* (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ -0.5 (- z 7.0)))) (sqrt (* PI 2.0)))
   (+
    (+ (/ 1.5056327351493116e-7 (- 8.0 z)) (/ 9.984369578019572e-6 (- 7.0 z)))
    (+
     263.3831855358925
     (* z (+ 436.8961723502244 (* z 545.0353078134797))))))))

C

double code(double z) {
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (((pow((7.5 - z), (0.5 - z)) * exp((-0.5 + (z - 7.0)))) * sqrt((((double) M_PI) * 2.0))) * (((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (263.3831855358925 + (z * (436.8961723502244 + (z * 545.0353078134797))))));
}

Java

public static double code(double z) {
	return (Math.PI / Math.sin((Math.PI * z))) * (((Math.pow((7.5 - z), (0.5 - z)) * Math.exp((-0.5 + (z - 7.0)))) * Math.sqrt((Math.PI * 2.0))) * (((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (263.3831855358925 + (z * (436.8961723502244 + (z * 545.0353078134797))))));
}

Python

def code(z):
	return (math.pi / math.sin((math.pi * z))) * (((math.pow((7.5 - z), (0.5 - z)) * math.exp((-0.5 + (z - 7.0)))) * math.sqrt((math.pi * 2.0))) * (((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (263.3831855358925 + (z * (436.8961723502244 + (z * 545.0353078134797))))))

Julia

function code(z)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(-0.5 + Float64(z - 7.0)))) * sqrt(Float64(pi * 2.0))) * Float64(Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) + Float64(9.984369578019572e-6 / Float64(7.0 - z))) + Float64(263.3831855358925 + Float64(z * Float64(436.8961723502244 + Float64(z * 545.0353078134797)))))))
end

MATLAB

function tmp = code(z)
	tmp = (pi / sin((pi * z))) * (((((7.5 - z) ^ (0.5 - z)) * exp((-0.5 + (z - 7.0)))) * sqrt((pi * 2.0))) * (((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (263.3831855358925 + (z * (436.8961723502244 + (z * 545.0353078134797))))));
end

Wolfram

code[z_] := N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-0.5 + N[(z - 7.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(263.3831855358925 + N[(z * N[(436.8961723502244 + N[(z * 545.0353078134797), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.5%

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

3. Simplified 96.4%

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

6. Applied egg-rr 96.4%

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

8. Simplified 97.6%

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

9. Taylor expanded in z around 0 98.0%

   \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + 545.0353078134797 \cdot z\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 - \left(7 - z\right)}\right)\right)\right) \]
10. Step-by-step derivation

    1. *-commutative 98.0%

       \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + \color{blue}{z \cdot 545.0353078134797}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 - \left(7 - z\right)}\right)\right)\right) \]

11. Simplified 98.0%

    \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 - \left(7 - z\right)}\right)\right)\right) \]

12. Final simplification 98.0%

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 + \left(z - 7\right)}\right) \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 5: 97.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\pi \cdot \frac{\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 + \left(z - 7\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right) + \left(1.4451589203350195 \cdot 10^{-6} + z \cdot 2.0611519559804982 \cdot 10^{-7}\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   PI
   (/
    (*
     (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ -0.5 (- z 7.0))))
     (sqrt (* PI 2.0)))
    (sin (* PI z))))
  (+
   (+
    263.3831855358925
    (*
     z
     (+
      436.8961723502244
      (* z (+ 545.0353078134797 (* z 606.6766809125655))))))
   (+ 1.4451589203350195e-6 (* z 2.0611519559804982e-7)))))

C

double code(double z) {
	return (((double) M_PI) * (((pow((7.5 - z), (0.5 - z)) * exp((-0.5 + (z - 7.0)))) * sqrt((((double) M_PI) * 2.0))) / sin((((double) M_PI) * z)))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))) + (1.4451589203350195e-6 + (z * 2.0611519559804982e-7)));
}

Java

public static double code(double z) {
	return (Math.PI * (((Math.pow((7.5 - z), (0.5 - z)) * Math.exp((-0.5 + (z - 7.0)))) * Math.sqrt((Math.PI * 2.0))) / Math.sin((Math.PI * z)))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))) + (1.4451589203350195e-6 + (z * 2.0611519559804982e-7)));
}

Python

def code(z):
	return (math.pi * (((math.pow((7.5 - z), (0.5 - z)) * math.exp((-0.5 + (z - 7.0)))) * math.sqrt((math.pi * 2.0))) / math.sin((math.pi * z)))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))) + (1.4451589203350195e-6 + (z * 2.0611519559804982e-7)))

Julia

function code(z)
	return Float64(Float64(pi * Float64(Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(-0.5 + Float64(z - 7.0)))) * sqrt(Float64(pi * 2.0))) / sin(Float64(pi * z)))) * Float64(Float64(263.3831855358925 + Float64(z * Float64(436.8961723502244 + Float64(z * Float64(545.0353078134797 + Float64(z * 606.6766809125655)))))) + Float64(1.4451589203350195e-6 + Float64(z * 2.0611519559804982e-7))))
end

MATLAB

function tmp = code(z)
	tmp = (pi * (((((7.5 - z) ^ (0.5 - z)) * exp((-0.5 + (z - 7.0)))) * sqrt((pi * 2.0))) / sin((pi * z)))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))) + (1.4451589203350195e-6 + (z * 2.0611519559804982e-7)));
end

Wolfram

code[z_] := N[(N[(Pi * N[(N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-0.5 + N[(z - 7.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(263.3831855358925 + N[(z * N[(436.8961723502244 + N[(z * N[(545.0353078134797 + N[(z * 606.6766809125655), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.4451589203350195e-6 + N[(z * 2.0611519559804982e-7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.5%

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

3. Simplified 98.3%

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

   1. associate-*l/ 98.5%

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

6. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\frac{\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-0.5 + \left(-\left(\left(1 - z\right) + 6\right)\right)}\right)\right)}{\sin \left(\pi \cdot z\right)}} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
7. Step-by-step derivation

   1. associate-/l* 98.3%

      \[\leadsto \color{blue}{\left(\pi \cdot \frac{\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-0.5 + \left(-\left(\left(1 - z\right) + 6\right)\right)}\right)}{\sin \left(\pi \cdot z\right)}\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

8. Simplified 98.3%

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

9. Taylor expanded in z around 0 98.0%

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

    1. *-commutative 98.0%

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

11. Simplified 98.0%

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

12. Taylor expanded in z around 0 98.0%

    \[\leadsto \left(\pi \cdot \frac{\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 - \left(7 - z\right)}\right)}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right) + \color{blue}{\left(1.4451589203350195 \cdot 10^{-6} + 2.0611519559804982 \cdot 10^{-7} \cdot z\right)}\right) \]
13. Step-by-step derivation

    1. *-commutative 98.0%

       \[\leadsto \left(\pi \cdot \frac{\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 - \left(7 - z\right)}\right)}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right) + \left(1.4451589203350195 \cdot 10^{-6} + \color{blue}{z \cdot 2.0611519559804982 \cdot 10^{-7}}\right)\right) \]

14. Simplified 98.0%

    \[\leadsto \left(\pi \cdot \frac{\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 - \left(7 - z\right)}\right)}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right) + \color{blue}{\left(1.4451589203350195 \cdot 10^{-6} + z \cdot 2.0611519559804982 \cdot 10^{-7}\right)}\right) \]

15. Final simplification 98.0%

    \[\leadsto \left(\pi \cdot \frac{\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 + \left(z - 7\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right) + \left(1.4451589203350195 \cdot 10^{-6} + z \cdot 2.0611519559804982 \cdot 10^{-7}\right)\right) \]
16. Add Preprocessing

Alternative 6: 97.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\pi \cdot \frac{\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 + \left(z - 7\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + z \cdot 606.6766809167608\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   PI
   (/
    (*
     (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ -0.5 (- z 7.0))))
     (sqrt (* PI 2.0)))
    (sin (* PI z))))
  (+
   263.3831869810514
   (*
    z
    (+
     436.8961725563396
     (* z (+ 545.0353078428827 (* z 606.6766809167608))))))))

C

double code(double z) {
	return (((double) M_PI) * (((pow((7.5 - z), (0.5 - z)) * exp((-0.5 + (z - 7.0)))) * sqrt((((double) M_PI) * 2.0))) / sin((((double) M_PI) * z)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (z * 606.6766809167608))))));
}

Java

public static double code(double z) {
	return (Math.PI * (((Math.pow((7.5 - z), (0.5 - z)) * Math.exp((-0.5 + (z - 7.0)))) * Math.sqrt((Math.PI * 2.0))) / Math.sin((Math.PI * z)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (z * 606.6766809167608))))));
}

Python

def code(z):
	return (math.pi * (((math.pow((7.5 - z), (0.5 - z)) * math.exp((-0.5 + (z - 7.0)))) * math.sqrt((math.pi * 2.0))) / math.sin((math.pi * z)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (z * 606.6766809167608))))))

Julia

function code(z)
	return Float64(Float64(pi * Float64(Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(-0.5 + Float64(z - 7.0)))) * sqrt(Float64(pi * 2.0))) / sin(Float64(pi * z)))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * Float64(545.0353078428827 + Float64(z * 606.6766809167608)))))))
end

MATLAB

function tmp = code(z)
	tmp = (pi * (((((7.5 - z) ^ (0.5 - z)) * exp((-0.5 + (z - 7.0)))) * sqrt((pi * 2.0))) / sin((pi * z)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (z * 606.6766809167608))))));
end

Wolfram

code[z_] := N[(N[(Pi * N[(N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-0.5 + N[(z - 7.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * N[(545.0353078428827 + N[(z * 606.6766809167608), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.5%

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

3. Simplified 98.3%

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

   1. associate-*l/ 98.5%

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

6. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\frac{\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-0.5 + \left(-\left(\left(1 - z\right) + 6\right)\right)}\right)\right)}{\sin \left(\pi \cdot z\right)}} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
7. Step-by-step derivation

   1. associate-/l* 98.3%

      \[\leadsto \color{blue}{\left(\pi \cdot \frac{\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-0.5 + \left(-\left(\left(1 - z\right) + 6\right)\right)}\right)}{\sin \left(\pi \cdot z\right)}\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

8. Simplified 98.3%

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

9. Taylor expanded in z around 0 97.7%

   \[\leadsto \left(\pi \cdot \frac{\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 - \left(7 - z\right)}\right)}{\sin \left(z \cdot \pi\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)} \]
10. Step-by-step derivation

    1. *-commutative 97.7%

       \[\leadsto \left(\pi \cdot \frac{\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 - \left(7 - z\right)}\right)}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + \color{blue}{z \cdot 606.6766809167608}\right)\right)\right) \]

11. Simplified 97.7%

    \[\leadsto \left(\pi \cdot \frac{\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 - \left(7 - z\right)}\right)}{\sin \left(z \cdot \pi\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + z \cdot 606.6766809167608\right)\right)\right)} \]

12. Final simplification 97.7%

    \[\leadsto \left(\pi \cdot \frac{\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 + \left(z - 7\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + z \cdot 606.6766809167608\right)\right)\right) \]
13. Add Preprocessing

Alternative 7: 97.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\pi \cdot \frac{1}{\frac{\sin \left(\pi \cdot z\right)}{\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \sqrt{\pi \cdot 2}}}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   PI
   (/
    1.0
    (/
     (sin (* PI z))
     (* (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5))) (sqrt (* PI 2.0))))))
  (+ 263.3831869810514 (* z (+ 436.8961725563396 (* z 545.0353078428827))))))

C

double code(double z) {
	return (((double) M_PI) * (1.0 / (sin((((double) M_PI) * z)) / ((pow((7.5 - z), (0.5 - z)) * exp((z + -7.5))) * sqrt((((double) M_PI) * 2.0)))))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827))));
}

Java

public static double code(double z) {
	return (Math.PI * (1.0 / (Math.sin((Math.PI * z)) / ((Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5))) * Math.sqrt((Math.PI * 2.0)))))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827))));
}

Python

def code(z):
	return (math.pi * (1.0 / (math.sin((math.pi * z)) / ((math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5))) * math.sqrt((math.pi * 2.0)))))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827))))

Julia

function code(z)
	return Float64(Float64(pi * Float64(1.0 / Float64(sin(Float64(pi * z)) / Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5))) * sqrt(Float64(pi * 2.0)))))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * 545.0353078428827)))))
end

MATLAB

function tmp = code(z)
	tmp = (pi * (1.0 / (sin((pi * z)) / ((((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5))) * sqrt((pi * 2.0)))))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827))));
end

Wolfram

code[z_] := N[(N[(Pi * N[(1.0 / N[(N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision] / N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * 545.0353078428827), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.5%

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

3. Simplified 98.3%

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

   1. associate-*l/ 98.5%

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

6. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\frac{\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-0.5 + \left(-\left(\left(1 - z\right) + 6\right)\right)}\right)\right)}{\sin \left(\pi \cdot z\right)}} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
7. Step-by-step derivation

   1. associate-/l* 98.3%

      \[\leadsto \color{blue}{\left(\pi \cdot \frac{\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-0.5 + \left(-\left(\left(1 - z\right) + 6\right)\right)}\right)}{\sin \left(\pi \cdot z\right)}\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

8. Simplified 98.3%

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

9. Taylor expanded in z around 0 97.5%

   \[\leadsto \left(\pi \cdot \frac{\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 - \left(7 - z\right)}\right)}{\sin \left(z \cdot \pi\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right)} \]
10. Step-by-step derivation

    1. *-commutative 97.5%

       \[\leadsto \left(\pi \cdot \frac{\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 - \left(7 - z\right)}\right)}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + \color{blue}{z \cdot 545.0353078428827}\right)\right) \]

11. Simplified 97.5%

    \[\leadsto \left(\pi \cdot \frac{\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 - \left(7 - z\right)}\right)}{\sin \left(z \cdot \pi\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)} \]
12. Step-by-step derivation

    1. clear-num 97.6%

       \[\leadsto \left(\pi \cdot \color{blue}{\frac{1}{\frac{\sin \left(z \cdot \pi\right)}{\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 - \left(7 - z\right)}\right)}}}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right) \]

    2. inv-pow 97.6%

       \[\leadsto \left(\pi \cdot \color{blue}{{\left(\frac{\sin \left(z \cdot \pi\right)}{\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 - \left(7 - z\right)}\right)}\right)}^{-1}}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right) \]

    3. *-commutative 97.6%

       \[\leadsto \left(\pi \cdot {\left(\frac{\sin \color{blue}{\left(\pi \cdot z\right)}}{\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 - \left(7 - z\right)}\right)}\right)}^{-1}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right) \]

    4. associate-*r* 97.6%

       \[\leadsto \left(\pi \cdot {\left(\frac{\sin \left(\pi \cdot z\right)}{\color{blue}{\left(\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{-0.5 - \left(7 - z\right)}}}\right)}^{-1}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right) \]

    5. associate--r- 97.6%

       \[\leadsto \left(\pi \cdot {\left(\frac{\sin \left(\pi \cdot z\right)}{\left(\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{\color{blue}{\left(-0.5 - 7\right) + z}}}\right)}^{-1}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right) \]

    6. metadata-eval 97.6%

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

13. Applied egg-rr 97.6%

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

    1. unpow-1 97.6%

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

    2. associate-*l* 97.6%

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

    3. *-commutative 97.6%

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

15. Simplified 97.6%

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

16. Final simplification 97.6%

    \[\leadsto \left(\pi \cdot \frac{1}{\frac{\sin \left(\pi \cdot z\right)}{\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \sqrt{\pi \cdot 2}}}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right) \]
17. Add Preprocessing

Alternative 8: 97.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right) \cdot \left(\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \frac{e^{z + -7.5}}{\sin \left(\pi \cdot z\right)}\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (+ 263.3831869810514 (* z (+ 436.8961725563396 (* z 545.0353078428827))))
  (*
   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 (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))) * (((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 (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))) * (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 (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))) * (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(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * 545.0353078428827)))) * Float64(pi * Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * Float64(exp(Float64(z + -7.5)) / sin(Float64(pi * z)))))))
end

MATLAB

function tmp = code(z)
	tmp = (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))) * (pi * (sqrt((pi * 2.0)) * (((7.5 - z) ^ (0.5 - z)) * (exp((z + -7.5)) / sin((pi * z))))));
end

Wolfram

code[z_] := N[(N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * 545.0353078428827), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.5%

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

3. Simplified 98.3%

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

   1. associate-*l/ 98.5%

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

6. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\frac{\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-0.5 + \left(-\left(\left(1 - z\right) + 6\right)\right)}\right)\right)}{\sin \left(\pi \cdot z\right)}} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
7. Step-by-step derivation

   1. associate-/l* 98.3%

      \[\leadsto \color{blue}{\left(\pi \cdot \frac{\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-0.5 + \left(-\left(\left(1 - z\right) + 6\right)\right)}\right)}{\sin \left(\pi \cdot z\right)}\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

8. Simplified 98.3%

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

9. Taylor expanded in z around 0 97.5%

   \[\leadsto \left(\pi \cdot \frac{\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 - \left(7 - z\right)}\right)}{\sin \left(z \cdot \pi\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right)} \]
10. Step-by-step derivation

    1. *-commutative 97.5%

       \[\leadsto \left(\pi \cdot \frac{\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 - \left(7 - z\right)}\right)}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + \color{blue}{z \cdot 545.0353078428827}\right)\right) \]

11. Simplified 97.5%

    \[\leadsto \left(\pi \cdot \frac{\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 - \left(7 - z\right)}\right)}{\sin \left(z \cdot \pi\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)} \]
12. Step-by-step derivation

    1. associate-*r/ 97.3%

       \[\leadsto \color{blue}{\frac{\pi \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 - \left(7 - z\right)}\right)\right)}{\sin \left(z \cdot \pi\right)}} \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right) \]

    2. associate-*r* 97.3%

       \[\leadsto \frac{\pi \cdot \color{blue}{\left(\left(\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{-0.5 - \left(7 - z\right)}\right)}}{\sin \left(z \cdot \pi\right)} \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right) \]

    3. associate--r- 97.3%

       \[\leadsto \frac{\pi \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{\color{blue}{\left(-0.5 - 7\right) + z}}\right)}{\sin \left(z \cdot \pi\right)} \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right) \]

    4. metadata-eval 97.3%

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

    5. *-commutative 97.3%

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

13. Applied egg-rr 97.3%

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

    1. associate-*r/ 97.5%

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

    2. associate-*l* 97.5%

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

    3. associate-*r/ 97.6%

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

    4. *-commutative 97.6%

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

    5. associate-/l* 97.5%

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

15. Simplified 97.5%

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

16. Final simplification 97.5%

    \[\leadsto \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right) \cdot \left(\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \frac{e^{z + -7.5}}{\sin \left(\pi \cdot z\right)}\right)\right)\right) \]
17. Add Preprocessing

Alternative 9: 96.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\pi \cdot \frac{\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 + \left(z - 7\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   PI
   (/
    (*
     (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ -0.5 (- z 7.0))))
     (sqrt (* PI 2.0)))
    (sin (* PI z))))
  (+ 263.3831869810514 (* z 436.8961725563396))))

C

double code(double z) {
	return (((double) M_PI) * (((pow((7.5 - z), (0.5 - z)) * exp((-0.5 + (z - 7.0)))) * sqrt((((double) M_PI) * 2.0))) / sin((((double) M_PI) * z)))) * (263.3831869810514 + (z * 436.8961725563396));
}

Java

public static double code(double z) {
	return (Math.PI * (((Math.pow((7.5 - z), (0.5 - z)) * Math.exp((-0.5 + (z - 7.0)))) * Math.sqrt((Math.PI * 2.0))) / Math.sin((Math.PI * z)))) * (263.3831869810514 + (z * 436.8961725563396));
}

Python

def code(z):
	return (math.pi * (((math.pow((7.5 - z), (0.5 - z)) * math.exp((-0.5 + (z - 7.0)))) * math.sqrt((math.pi * 2.0))) / math.sin((math.pi * z)))) * (263.3831869810514 + (z * 436.8961725563396))

Julia

function code(z)
	return Float64(Float64(pi * Float64(Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(-0.5 + Float64(z - 7.0)))) * sqrt(Float64(pi * 2.0))) / sin(Float64(pi * z)))) * Float64(263.3831869810514 + Float64(z * 436.8961725563396)))
end

MATLAB

function tmp = code(z)
	tmp = (pi * (((((7.5 - z) ^ (0.5 - z)) * exp((-0.5 + (z - 7.0)))) * sqrt((pi * 2.0))) / sin((pi * z)))) * (263.3831869810514 + (z * 436.8961725563396));
end

Wolfram

code[z_] := N[(N[(Pi * N[(N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-0.5 + N[(z - 7.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * 436.8961725563396), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.5%

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

3. Simplified 98.3%

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

   1. associate-*l/ 98.5%

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

6. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\frac{\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-0.5 + \left(-\left(\left(1 - z\right) + 6\right)\right)}\right)\right)}{\sin \left(\pi \cdot z\right)}} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
7. Step-by-step derivation

   1. associate-/l* 98.3%

      \[\leadsto \color{blue}{\left(\pi \cdot \frac{\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-0.5 + \left(-\left(\left(1 - z\right) + 6\right)\right)}\right)}{\sin \left(\pi \cdot z\right)}\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

8. Simplified 98.3%

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

9. Taylor expanded in z around 0 97.0%

   \[\leadsto \left(\pi \cdot \frac{\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 - \left(7 - z\right)}\right)}{\sin \left(z \cdot \pi\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + 436.8961725563396 \cdot z\right)} \]
10. Step-by-step derivation

    1. *-commutative 97.0%

       \[\leadsto \left(\pi \cdot \frac{\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 - \left(7 - z\right)}\right)}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(263.3831869810514 + \color{blue}{z \cdot 436.8961725563396}\right) \]

11. Simplified 97.0%

    \[\leadsto \left(\pi \cdot \frac{\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 - \left(7 - z\right)}\right)}{\sin \left(z \cdot \pi\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot 436.8961725563396\right)} \]

12. Final simplification 97.0%

    \[\leadsto \left(\pi \cdot \frac{\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-0.5 + \left(z - 7\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right) \]
13. Add Preprocessing

Alternative 10: 96.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

3. Simplified 98.3%

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

   1. associate-*l/ 98.5%

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

6. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\frac{\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-0.5 + \left(-\left(\left(1 - z\right) + 6\right)\right)}\right)\right)}{\sin \left(\pi \cdot z\right)}} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
7. Step-by-step derivation

   1. associate-/l* 98.3%

      \[\leadsto \color{blue}{\left(\pi \cdot \frac{\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-0.5 + \left(-\left(\left(1 - z\right) + 6\right)\right)}\right)}{\sin \left(\pi \cdot z\right)}\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

8. Simplified 98.3%

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

10. Applied egg-rr 97.3%

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

12. Simplified 98.2%

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

13. Taylor expanded in z around 0 96.3%

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

    1. associate-*l/ 96.1%

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

    2. *-commutative 96.1%

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

    3. associate-*r* 96.7%

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

15. Simplified 96.7%

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

Alternative 11: 96.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

3. Simplified 96.4%

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

5. Taylor expanded in z around 0 96.6%

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

   1. *-commutative 96.6%

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

7. Simplified 96.6%

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

8. Final simplification 96.6%

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

Alternative 12: 96.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

3. Simplified 98.3%

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

   1. associate-*l/ 98.5%

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

6. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\frac{\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-0.5 + \left(-\left(\left(1 - z\right) + 6\right)\right)}\right)\right)}{\sin \left(\pi \cdot z\right)}} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
7. Step-by-step derivation

   1. associate-/l* 98.3%

      \[\leadsto \color{blue}{\left(\pi \cdot \frac{\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-0.5 + \left(-\left(\left(1 - z\right) + 6\right)\right)}\right)}{\sin \left(\pi \cdot z\right)}\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

8. Simplified 98.3%

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

10. Applied egg-rr 97.3%

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

12. Simplified 98.2%

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

13. Taylor expanded in z around 0 96.3%

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

    1. associate-*l/ 96.1%

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

    2. *-commutative 96.1%

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

    3. associate-*r* 96.7%

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

15. Simplified 96.7%

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

    1. associate-*r/ 96.7%

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

    2. associate-*l* 95.7%

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

    3. sqrt-unprod 95.7%

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

    4. metadata-eval 95.7%

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

17. Applied egg-rr 95.7%

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

    1. associate-/l* 96.1%

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

    2. associate-*r* 96.7%

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

    3. *-commutative 96.7%

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

    4. associate-*r/ 96.6%

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

    5. *-commutative 96.6%

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

19. Simplified 96.6%

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

Alternative 13: 95.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

3. Simplified 96.4%

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

5. Taylor expanded in z around 0 95.6%

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

6. Taylor expanded in z around 0 95.8%

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

7. Taylor expanded in z around 0 95.8%

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

   1. pow1 95.8%

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

   2. associate-*r* 95.8%

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

   3. pow1/2 95.8%

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

   4. *-commutative 95.8%

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

   5. pow1/2 95.8%

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

   6. pow-prod-down 95.8%

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

9. Applied egg-rr 95.8%

   \[\leadsto \color{blue}{{\left({\left(\left(2 \cdot \pi\right) \cdot 7.5\right)}^{0.5} \cdot e^{-7.5}\right)}^{1}} \cdot \frac{263.3831869810514}{z} \]
10. Step-by-step derivation

    1. unpow1 95.8%

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

    2. *-commutative 95.8%

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

    3. unpow1/2 95.8%

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

    4. associate-*l* 95.8%

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

11. Simplified 95.8%

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

Alternative 14: 95.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

3. Simplified 96.4%

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

5. Taylor expanded in z around 0 95.6%

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

6. Taylor expanded in z around 0 95.8%

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

7. Taylor expanded in z around 0 95.8%

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

   1. associate-*r/ 95.7%

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

   2. associate-*r* 95.7%

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

   3. pow1/2 95.7%

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

   4. *-commutative 95.7%

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

   5. pow1/2 95.7%

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

   6. pow-prod-down 95.7%

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

9. Applied egg-rr 95.7%

   \[\leadsto \color{blue}{\frac{\left({\left(\left(2 \cdot \pi\right) \cdot 7.5\right)}^{0.5} \cdot e^{-7.5}\right) \cdot 263.3831869810514}{z}} \]
10. Step-by-step derivation

    1. associate-*r/ 95.8%

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

    2. associate-*l* 95.6%

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

    3. unpow1/2 95.6%

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

    4. associate-*l* 95.6%

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

11. Simplified 95.6%

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

Reproduce

herbie shell --seed 2024150 
(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))))))