Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.4% → 99.2%

Time: 1.8min

Alternatives: 10

Speedup: 1.1×

• could not determine a ground truth

  1. z = -4.667196875147603e+82

Specification

\[z \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.9%

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

3. Simplified 95.1%

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

   1. expm1-log1p-u 95.0%

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

   2. expm1-udef 95.0%

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

   3. associate-+r+ 95.0%

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

   4. +-commutative 95.0%

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

5. Applied egg-rr 95.0%

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

   1. expm1-def 95.0%

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

   2. expm1-log1p 95.0%

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

   3. +-commutative 95.0%

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

   4. associate-+l+ 96.8%

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

7. Simplified 96.8%

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

   1. expm1-log1p-u 45.3%

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

   2. expm1-udef 45.3%

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

9. Applied egg-rr 45.3%

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

    1. expm1-def 45.3%

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

    2. expm1-log1p 97.2%

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

    3. associate-*r/ 97.1%

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

    4. associate-*r* 97.1%

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

    5. fma-udef 97.1%

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

    6. neg-mul-1 97.1%

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

    7. +-commutative 97.1%

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

    8. sub-neg 97.1%

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

    9. *-commutative 97.1%

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

11. Simplified 97.1%

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

    1. add-exp-log 97.7%

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

    2. associate-*l* 97.7%

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

    3. sub-neg 97.7%

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

    4. +-commutative 97.7%

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

    5. log-prod 97.7%

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

    6. add-log-exp 99.2%

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

    7. +-commutative 99.2%

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

    8. sub-neg 99.2%

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

13. Applied egg-rr 99.2%

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

14. Final simplification 99.2%

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

Alternative 2: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.9%

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

3. Simplified 95.1%

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

   1. expm1-log1p-u 95.0%

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

   2. expm1-udef 95.0%

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

   3. associate-+r+ 95.0%

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

   4. +-commutative 95.0%

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

5. Applied egg-rr 95.0%

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

   1. expm1-def 95.0%

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

   2. expm1-log1p 95.0%

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

   3. +-commutative 95.0%

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

   4. associate-+l+ 96.8%

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

7. Simplified 96.8%

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

9. Applied egg-rr 97.7%

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

    1. associate-/l* 97.8%

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

    2. associate-*r* 97.8%

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

    3. *-commutative 97.8%

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

    4. fma-udef 97.8%

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

    5. neg-mul-1 97.8%

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

    6. +-commutative 97.8%

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

    7. sub-neg 97.8%

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

11. Simplified 97.8%

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

12. Final simplification 97.8%

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

Alternative 3: 98.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.9%

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

3. Simplified 95.1%

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

   1. expm1-log1p-u 95.0%

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

   2. expm1-udef 95.0%

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

5. Applied egg-rr 95.1%

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

   1. expm1-def 95.1%

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

   2. expm1-log1p 95.1%

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

   3. associate-+l+ 95.8%

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

   4. associate-+l+ 97.8%

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

7. Simplified 97.8%

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

8. Final simplification 97.8%

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

Alternative 4: 97.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(e^{z + -7.5} \cdot \frac{\sqrt{2 \cdot \pi} \cdot \left(\pi \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\left(\frac{676.5203681218851}{1 - z} + \left(\frac{-0.13857109526572012}{6 - z} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) + \left(z \cdot -10.53814559148631 - 41.65228863479777\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (exp (+ z -7.5))
   (/ (* (sqrt (* 2.0 PI)) (* PI (pow (- 7.5 z) (- 0.5 z)))) (sin (* z PI))))
  (+
   (+
    (+
     (/ 676.5203681218851 (- 1.0 z))
     (+
      (/ -0.13857109526572012 (- 6.0 z))
      (+ 0.9999999999998099 (/ 771.3234287776531 (- 3.0 z)))))
    (+
     (/ -1259.1392167224028 (- 2.0 z))
     (+
      (/ 9.984369578019572e-6 (- 7.0 z))
      (/ 1.5056327351493116e-7 (- 8.0 z)))))
   (- (* z -10.53814559148631) 41.65228863479777))))

C

double code(double z) {
	return (exp((z + -7.5)) * ((sqrt((2.0 * ((double) M_PI))) * (((double) M_PI) * pow((7.5 - z), (0.5 - z)))) / sin((z * ((double) M_PI))))) * ((((676.5203681218851 / (1.0 - z)) + ((-0.13857109526572012 / (6.0 - z)) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z))))) + ((-1259.1392167224028 / (2.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))) + ((z * -10.53814559148631) - 41.65228863479777));
}

Java

public static double code(double z) {
	return (Math.exp((z + -7.5)) * ((Math.sqrt((2.0 * Math.PI)) * (Math.PI * Math.pow((7.5 - z), (0.5 - z)))) / Math.sin((z * Math.PI)))) * ((((676.5203681218851 / (1.0 - z)) + ((-0.13857109526572012 / (6.0 - z)) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z))))) + ((-1259.1392167224028 / (2.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))) + ((z * -10.53814559148631) - 41.65228863479777));
}

Python

def code(z):
	return (math.exp((z + -7.5)) * ((math.sqrt((2.0 * math.pi)) * (math.pi * math.pow((7.5 - z), (0.5 - z)))) / math.sin((z * math.pi)))) * ((((676.5203681218851 / (1.0 - z)) + ((-0.13857109526572012 / (6.0 - z)) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z))))) + ((-1259.1392167224028 / (2.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))) + ((z * -10.53814559148631) - 41.65228863479777))

Julia

function code(z)
	return Float64(Float64(exp(Float64(z + -7.5)) * Float64(Float64(sqrt(Float64(2.0 * pi)) * Float64(pi * (Float64(7.5 - z) ^ Float64(0.5 - z)))) / sin(Float64(z * pi)))) * Float64(Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(0.9999999999998099 + Float64(771.3234287776531 / Float64(3.0 - z))))) + Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) + Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))))) + Float64(Float64(z * -10.53814559148631) - 41.65228863479777)))
end

MATLAB

function tmp = code(z)
	tmp = (exp((z + -7.5)) * ((sqrt((2.0 * pi)) * (pi * ((7.5 - z) ^ (0.5 - z)))) / sin((z * pi)))) * ((((676.5203681218851 / (1.0 - z)) + ((-0.13857109526572012 / (6.0 - z)) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z))))) + ((-1259.1392167224028 / (2.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))) + ((z * -10.53814559148631) - 41.65228863479777));
end

Wolfram

code[z_] := N[(N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision] * N[(Pi * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z * -10.53814559148631), $MachinePrecision] - 41.65228863479777), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.9%

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

3. Simplified 95.1%

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

   1. expm1-log1p-u 95.0%

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

   2. expm1-udef 95.0%

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

5. Applied egg-rr 95.1%

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

   1. expm1-def 95.1%

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

   2. expm1-log1p 95.1%

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

   3. associate-+l+ 95.8%

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

   4. associate-+l+ 97.8%

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

7. Simplified 97.8%

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

8. Taylor expanded in z around 0 97.6%

   \[\leadsto \left(\color{blue}{\left(-10.53814559148631 \cdot z - 41.65228863479777\right)} + \left(\left(\frac{676.5203681218851}{1 - z} + \left(\frac{-0.13857109526572012}{6 - z} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

9. Final simplification 97.6%

   \[\leadsto \left(e^{z + -7.5} \cdot \frac{\sqrt{2 \cdot \pi} \cdot \left(\pi \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\left(\frac{676.5203681218851}{1 - z} + \left(\frac{-0.13857109526572012}{6 - z} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) + \left(z \cdot -10.53814559148631 - 41.65228863479777\right)\right) \]

Alternative 5: 96.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \left(e^{z + -7.5} \cdot \frac{\sqrt{2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot {\pi}^{1.5}\right)}{\sin \left(z \cdot \pi\right)}\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (+ 263.3831869810514 (* z 436.8961725563396))
  (*
   (exp (+ z -7.5))
   (/
    (* (sqrt 2.0) (* (pow (- 7.5 z) (- 0.5 z)) (pow PI 1.5)))
    (sin (* z PI))))))

C

double code(double z) {
	return (263.3831869810514 + (z * 436.8961725563396)) * (exp((z + -7.5)) * ((sqrt(2.0) * (pow((7.5 - z), (0.5 - z)) * pow(((double) M_PI), 1.5))) / sin((z * ((double) M_PI)))));
}

Java

public static double code(double z) {
	return (263.3831869810514 + (z * 436.8961725563396)) * (Math.exp((z + -7.5)) * ((Math.sqrt(2.0) * (Math.pow((7.5 - z), (0.5 - z)) * Math.pow(Math.PI, 1.5))) / Math.sin((z * Math.PI))));
}

Python

def code(z):
	return (263.3831869810514 + (z * 436.8961725563396)) * (math.exp((z + -7.5)) * ((math.sqrt(2.0) * (math.pow((7.5 - z), (0.5 - z)) * math.pow(math.pi, 1.5))) / math.sin((z * math.pi))))

Julia

function code(z)
	return Float64(Float64(263.3831869810514 + Float64(z * 436.8961725563396)) * Float64(exp(Float64(z + -7.5)) * Float64(Float64(sqrt(2.0) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * (pi ^ 1.5))) / sin(Float64(z * pi)))))
end

MATLAB

function tmp = code(z)
	tmp = (263.3831869810514 + (z * 436.8961725563396)) * (exp((z + -7.5)) * ((sqrt(2.0) * (((7.5 - z) ^ (0.5 - z)) * (pi ^ 1.5))) / sin((z * pi))));
end

Wolfram

code[z_] := N[(N[(263.3831869810514 + N[(z * 436.8961725563396), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Power[Pi, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.9%

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

3. Simplified 95.1%

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

4. Taylor expanded in z around 0 97.4%

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

   1. *-commutative 97.4%

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

6. Simplified 97.4%

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

7. Taylor expanded in z around inf 96.8%

   \[\leadsto \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \left(e^{z + -7.5} \cdot \frac{\color{blue}{\sqrt{{\pi}^{3}} \cdot \left(e^{\log \left(7.5 - z\right) \cdot \left(0.5 - z\right)} \cdot \sqrt{2}\right)}}{\sin \left(\pi \cdot z\right)}\right) \]
8. Step-by-step derivation

   1. *-commutative 96.8%

      \[\leadsto \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \left(e^{z + -7.5} \cdot \frac{\sqrt{{\pi}^{3}} \cdot \color{blue}{\left(\sqrt{2} \cdot e^{\log \left(7.5 - z\right) \cdot \left(0.5 - z\right)}\right)}}{\sin \left(\pi \cdot z\right)}\right) \]

   2. exp-to-pow 96.8%

      \[\leadsto \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \left(e^{z + -7.5} \cdot \frac{\sqrt{{\pi}^{3}} \cdot \left(\sqrt{2} \cdot \color{blue}{{\left(7.5 - z\right)}^{\left(0.5 - z\right)}}\right)}{\sin \left(\pi \cdot z\right)}\right) \]

   3. sub-neg 96.8%

      \[\leadsto \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \left(e^{z + -7.5} \cdot \frac{\sqrt{{\pi}^{3}} \cdot \left(\sqrt{2} \cdot {\color{blue}{\left(7.5 + \left(-z\right)\right)}}^{\left(0.5 - z\right)}\right)}{\sin \left(\pi \cdot z\right)}\right) \]

   4. +-commutative 96.8%

      \[\leadsto \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \left(e^{z + -7.5} \cdot \frac{\sqrt{{\pi}^{3}} \cdot \left(\sqrt{2} \cdot {\color{blue}{\left(\left(-z\right) + 7.5\right)}}^{\left(0.5 - z\right)}\right)}{\sin \left(\pi \cdot z\right)}\right) \]

   5. mul-1-neg 96.8%

      \[\leadsto \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \left(e^{z + -7.5} \cdot \frac{\sqrt{{\pi}^{3}} \cdot \left(\sqrt{2} \cdot {\left(\color{blue}{-1 \cdot z} + 7.5\right)}^{\left(0.5 - z\right)}\right)}{\sin \left(\pi \cdot z\right)}\right) \]

   6. fma-udef 96.8%

      \[\leadsto \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \left(e^{z + -7.5} \cdot \frac{\sqrt{{\pi}^{3}} \cdot \left(\sqrt{2} \cdot {\color{blue}{\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}}^{\left(0.5 - z\right)}\right)}{\sin \left(\pi \cdot z\right)}\right) \]

   7. *-commutative 96.8%

      \[\leadsto \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \left(e^{z + -7.5} \cdot \frac{\color{blue}{\left(\sqrt{2} \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{{\pi}^{3}}}}{\sin \left(\pi \cdot z\right)}\right) \]

   8. associate-*l* 97.4%

      \[\leadsto \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \left(e^{z + -7.5} \cdot \frac{\color{blue}{\sqrt{2} \cdot \left({\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)} \cdot \sqrt{{\pi}^{3}}\right)}}{\sin \left(\pi \cdot z\right)}\right) \]

   9. fma-udef 97.4%

      \[\leadsto \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \left(e^{z + -7.5} \cdot \frac{\sqrt{2} \cdot \left({\color{blue}{\left(-1 \cdot z + 7.5\right)}}^{\left(0.5 - z\right)} \cdot \sqrt{{\pi}^{3}}\right)}{\sin \left(\pi \cdot z\right)}\right) \]

   10. mul-1-neg 97.4%

       \[\leadsto \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \left(e^{z + -7.5} \cdot \frac{\sqrt{2} \cdot \left({\left(\color{blue}{\left(-z\right)} + 7.5\right)}^{\left(0.5 - z\right)} \cdot \sqrt{{\pi}^{3}}\right)}{\sin \left(\pi \cdot z\right)}\right) \]

   11. +-commutative 97.4%

       \[\leadsto \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \left(e^{z + -7.5} \cdot \frac{\sqrt{2} \cdot \left({\color{blue}{\left(7.5 + \left(-z\right)\right)}}^{\left(0.5 - z\right)} \cdot \sqrt{{\pi}^{3}}\right)}{\sin \left(\pi \cdot z\right)}\right) \]

   12. sub-neg 97.4%

       \[\leadsto \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \left(e^{z + -7.5} \cdot \frac{\sqrt{2} \cdot \left({\color{blue}{\left(7.5 - z\right)}}^{\left(0.5 - z\right)} \cdot \sqrt{{\pi}^{3}}\right)}{\sin \left(\pi \cdot z\right)}\right) \]

   13. sqr-pow 97.4%

       \[\leadsto \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \left(e^{z + -7.5} \cdot \frac{\sqrt{2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\color{blue}{{\pi}^{\left(\frac{3}{2}\right)} \cdot {\pi}^{\left(\frac{3}{2}\right)}}}\right)}{\sin \left(\pi \cdot z\right)}\right) \]

   14. rem-sqrt-square 97.4%

       \[\leadsto \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \left(e^{z + -7.5} \cdot \frac{\sqrt{2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \color{blue}{\left|{\pi}^{\left(\frac{3}{2}\right)}\right|}\right)}{\sin \left(\pi \cdot z\right)}\right) \]

   15. sqr-pow 97.4%

       \[\leadsto \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \left(e^{z + -7.5} \cdot \frac{\sqrt{2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left|\color{blue}{{\pi}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\pi}^{\left(\frac{\frac{3}{2}}{2}\right)}}\right|\right)}{\sin \left(\pi \cdot z\right)}\right) \]

   16. fabs-sqr 97.4%

       \[\leadsto \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \left(e^{z + -7.5} \cdot \frac{\sqrt{2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \color{blue}{\left({\pi}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\pi}^{\left(\frac{\frac{3}{2}}{2}\right)}\right)}\right)}{\sin \left(\pi \cdot z\right)}\right) \]

   17. sqr-pow 97.4%

       \[\leadsto \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \left(e^{z + -7.5} \cdot \frac{\sqrt{2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \color{blue}{{\pi}^{\left(\frac{3}{2}\right)}}\right)}{\sin \left(\pi \cdot z\right)}\right) \]

9. Simplified 97.4%

   \[\leadsto \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \left(e^{z + -7.5} \cdot \frac{\color{blue}{\sqrt{2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot {\pi}^{1.5}\right)}}{\sin \left(\pi \cdot z\right)}\right) \]

10. Final simplification 97.4%

    \[\leadsto \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \left(e^{z + -7.5} \cdot \frac{\sqrt{2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot {\pi}^{1.5}\right)}{\sin \left(z \cdot \pi\right)}\right) \]

Alternative 6: 95.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.9%

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

3. Simplified 95.1%

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

4. Taylor expanded in z around 0 94.0%

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

   1. *-commutative 94.0%

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

   2. associate-/l* 93.9%

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

6. Simplified 93.9%

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

7. Taylor expanded in z around 0 96.4%

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

   1. associate-/r/ 96.4%

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

   2. sqrt-unprod 96.4%

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

   3. metadata-eval 96.4%

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

9. Applied egg-rr 96.4%

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

10. Final simplification 96.4%

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

Alternative 7: 96.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.9%

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

3. Simplified 95.1%

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

4. Taylor expanded in z around 0 94.0%

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

   1. *-commutative 94.0%

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

   2. associate-/l* 93.9%

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

6. Simplified 93.9%

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

7. Taylor expanded in z around 0 96.4%

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

   1. associate-*r/ 96.7%

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

   2. sqrt-unprod 96.7%

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

   3. metadata-eval 96.7%

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

9. Applied egg-rr 96.7%

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

10. Final simplification 96.7%

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

Alternative 8: 24.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{69370.70318429549 \cdot \left(\pi \cdot \left(15 \cdot \frac{e^{-15}}{z \cdot z}\right)\right)} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (sqrt (* 69370.70318429549 (* PI (* 15.0 (/ (exp -15.0) (* z z)))))))

C

double code(double z) {
	return sqrt((69370.70318429549 * (((double) M_PI) * (15.0 * (exp(-15.0) / (z * z))))));
}

Java

public static double code(double z) {
	return Math.sqrt((69370.70318429549 * (Math.PI * (15.0 * (Math.exp(-15.0) / (z * z))))));
}

Python

def code(z):
	return math.sqrt((69370.70318429549 * (math.pi * (15.0 * (math.exp(-15.0) / (z * z))))))

Julia

function code(z)
	return sqrt(Float64(69370.70318429549 * Float64(pi * Float64(15.0 * Float64(exp(-15.0) / Float64(z * z))))))
end

MATLAB

function tmp = code(z)
	tmp = sqrt((69370.70318429549 * (pi * (15.0 * (exp(-15.0) / (z * z))))));
end

Wolfram

code[z_] := N[Sqrt[N[(69370.70318429549 * N[(Pi * N[(15.0 * N[(N[Exp[-15.0], $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 95.9%

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

3. Simplified 95.1%

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

4. Taylor expanded in z around 0 94.0%

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

   1. *-commutative 94.0%

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

   2. associate-/l* 93.9%

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

6. Simplified 93.9%

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

7. Taylor expanded in z around 0 96.4%

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

   1. associate-/r/ 96.4%

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

   2. sqrt-unprod 96.4%

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

   3. metadata-eval 96.4%

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

9. Applied egg-rr 96.4%

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

    1. add-sqr-sqrt 48.6%

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

    2. pow1/2 48.6%

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

    3. pow1/2 48.6%

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

    4. pow-prod-down 26.5%

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

11. Applied egg-rr 26.6%

    \[\leadsto \color{blue}{{\left(69370.70318429549 \cdot \left(\pi \cdot \left(15 \cdot \frac{e^{-15}}{z \cdot z}\right)\right)\right)}^{0.5}} \]
12. Step-by-step derivation

    1. unpow1/2 26.6%

       \[\leadsto \color{blue}{\sqrt{69370.70318429549 \cdot \left(\pi \cdot \left(15 \cdot \frac{e^{-15}}{z \cdot z}\right)\right)}} \]

    2. associate-*r/ 26.5%

       \[\leadsto \sqrt{69370.70318429549 \cdot \left(\pi \cdot \color{blue}{\frac{15 \cdot e^{-15}}{z \cdot z}}\right)} \]

    3. associate-*r/ 26.6%

       \[\leadsto \sqrt{69370.70318429549 \cdot \left(\pi \cdot \color{blue}{\left(15 \cdot \frac{e^{-15}}{z \cdot z}\right)}\right)} \]

13. Simplified 26.6%

    \[\leadsto \color{blue}{\sqrt{69370.70318429549 \cdot \left(\pi \cdot \left(15 \cdot \frac{e^{-15}}{z \cdot z}\right)\right)}} \]

14. Final simplification 26.6%

    \[\leadsto \sqrt{69370.70318429549 \cdot \left(\pi \cdot \left(15 \cdot \frac{e^{-15}}{z \cdot z}\right)\right)} \]

Alternative 9: 24.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\pi \cdot \left(\left(15 \cdot \frac{e^{-15}}{z \cdot z}\right) \cdot 69370.70318429549\right)} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (sqrt (* PI (* (* 15.0 (/ (exp -15.0) (* z z))) 69370.70318429549))))

C

double code(double z) {
	return sqrt((((double) M_PI) * ((15.0 * (exp(-15.0) / (z * z))) * 69370.70318429549)));
}

Java

public static double code(double z) {
	return Math.sqrt((Math.PI * ((15.0 * (Math.exp(-15.0) / (z * z))) * 69370.70318429549)));
}

Python

def code(z):
	return math.sqrt((math.pi * ((15.0 * (math.exp(-15.0) / (z * z))) * 69370.70318429549)))

Julia

function code(z)
	return sqrt(Float64(pi * Float64(Float64(15.0 * Float64(exp(-15.0) / Float64(z * z))) * 69370.70318429549)))
end

MATLAB

function tmp = code(z)
	tmp = sqrt((pi * ((15.0 * (exp(-15.0) / (z * z))) * 69370.70318429549)));
end

Wolfram

code[z_] := N[Sqrt[N[(Pi * N[(N[(15.0 * N[(N[Exp[-15.0], $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 69370.70318429549), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 95.9%

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

3. Simplified 95.1%

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

4. Taylor expanded in z around 0 94.0%

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

   1. *-commutative 94.0%

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

   2. associate-/l* 93.9%

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

6. Simplified 93.9%

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

7. Taylor expanded in z around 0 96.4%

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

   1. associate-/r/ 96.4%

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

   2. sqrt-unprod 96.4%

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

   3. metadata-eval 96.4%

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

9. Applied egg-rr 96.4%

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

    1. add-sqr-sqrt 48.6%

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

    2. pow1/2 48.6%

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

    3. pow1/2 48.6%

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

    4. pow-prod-down 26.5%

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

11. Applied egg-rr 26.6%

    \[\leadsto \color{blue}{{\left(69370.70318429549 \cdot \left(\pi \cdot \left(15 \cdot \frac{e^{-15}}{z \cdot z}\right)\right)\right)}^{0.5}} \]
12. Step-by-step derivation

    1. unpow1/2 26.6%

       \[\leadsto \color{blue}{\sqrt{69370.70318429549 \cdot \left(\pi \cdot \left(15 \cdot \frac{e^{-15}}{z \cdot z}\right)\right)}} \]

    2. *-commutative 26.6%

       \[\leadsto \sqrt{\color{blue}{\left(\pi \cdot \left(15 \cdot \frac{e^{-15}}{z \cdot z}\right)\right) \cdot 69370.70318429549}} \]

    3. associate-*l* 26.6%

       \[\leadsto \sqrt{\color{blue}{\pi \cdot \left(\left(15 \cdot \frac{e^{-15}}{z \cdot z}\right) \cdot 69370.70318429549\right)}} \]

13. Simplified 26.6%

    \[\leadsto \color{blue}{\sqrt{\pi \cdot \left(\left(15 \cdot \frac{e^{-15}}{z \cdot z}\right) \cdot 69370.70318429549\right)}} \]

14. Final simplification 26.6%

    \[\leadsto \sqrt{\pi \cdot \left(\left(15 \cdot \frac{e^{-15}}{z \cdot z}\right) \cdot 69370.70318429549\right)} \]

Alternative 10: 24.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\pi \cdot \frac{1040560.5477644323}{\frac{z \cdot z}{e^{-15}}}} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (sqrt (* PI (/ 1040560.5477644323 (/ (* z z) (exp -15.0))))))

C

double code(double z) {
	return sqrt((((double) M_PI) * (1040560.5477644323 / ((z * z) / exp(-15.0)))));
}

Java

public static double code(double z) {
	return Math.sqrt((Math.PI * (1040560.5477644323 / ((z * z) / Math.exp(-15.0)))));
}

Python

def code(z):
	return math.sqrt((math.pi * (1040560.5477644323 / ((z * z) / math.exp(-15.0)))))

Julia

function code(z)
	return sqrt(Float64(pi * Float64(1040560.5477644323 / Float64(Float64(z * z) / exp(-15.0)))))
end

MATLAB

function tmp = code(z)
	tmp = sqrt((pi * (1040560.5477644323 / ((z * z) / exp(-15.0)))));
end

Wolfram

code[z_] := N[Sqrt[N[(Pi * N[(1040560.5477644323 / N[(N[(z * z), $MachinePrecision] / N[Exp[-15.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 95.9%

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

3. Simplified 95.1%

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

4. Taylor expanded in z around 0 94.0%

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

   1. *-commutative 94.0%

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

   2. associate-/l* 93.9%

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

6. Simplified 93.9%

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

7. Taylor expanded in z around 0 96.4%

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

   1. associate-/r/ 96.4%

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

   2. sqrt-unprod 96.4%

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

   3. metadata-eval 96.4%

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

9. Applied egg-rr 96.4%

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

    1. add-sqr-sqrt 48.6%

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

    2. pow1/2 48.6%

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

    3. pow1/2 48.6%

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

    4. pow-prod-down 26.5%

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

11. Applied egg-rr 26.6%

    \[\leadsto \color{blue}{{\left(69370.70318429549 \cdot \left(\pi \cdot \left(15 \cdot \frac{e^{-15}}{z \cdot z}\right)\right)\right)}^{0.5}} \]
12. Step-by-step derivation

    1. unpow1/2 26.6%

       \[\leadsto \color{blue}{\sqrt{69370.70318429549 \cdot \left(\pi \cdot \left(15 \cdot \frac{e^{-15}}{z \cdot z}\right)\right)}} \]

    2. *-commutative 26.6%

       \[\leadsto \sqrt{69370.70318429549 \cdot \color{blue}{\left(\left(15 \cdot \frac{e^{-15}}{z \cdot z}\right) \cdot \pi\right)}} \]

    3. associate-*r* 26.6%

       \[\leadsto \sqrt{\color{blue}{\left(69370.70318429549 \cdot \left(15 \cdot \frac{e^{-15}}{z \cdot z}\right)\right) \cdot \pi}} \]

    4. associate-*r/ 26.5%

       \[\leadsto \sqrt{\left(69370.70318429549 \cdot \color{blue}{\frac{15 \cdot e^{-15}}{z \cdot z}}\right) \cdot \pi} \]

    5. associate-/l* 26.6%

       \[\leadsto \sqrt{\left(69370.70318429549 \cdot \color{blue}{\frac{15}{\frac{z \cdot z}{e^{-15}}}}\right) \cdot \pi} \]

    6. associate-*r/ 26.6%

       \[\leadsto \sqrt{\color{blue}{\frac{69370.70318429549 \cdot 15}{\frac{z \cdot z}{e^{-15}}}} \cdot \pi} \]

    7. metadata-eval 26.5%

       \[\leadsto \sqrt{\frac{\color{blue}{1040560.5477644323}}{\frac{z \cdot z}{e^{-15}}} \cdot \pi} \]

13. Simplified 26.5%

    \[\leadsto \color{blue}{\sqrt{\frac{1040560.5477644323}{\frac{z \cdot z}{e^{-15}}} \cdot \pi}} \]

14. Final simplification 26.5%

    \[\leadsto \sqrt{\pi \cdot \frac{1040560.5477644323}{\frac{z \cdot z}{e^{-15}}}} \]

Reproduce

herbie shell --seed 2023299 
(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))))))