Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.4% → 96.6%

Time: 1.9min

Alternatives: 8

Speedup: 1.2×

• could not determine a ground truth

  1. z = -9.134293186380747e+49

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: 96.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.8%

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

3. Simplified 95.8%

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

5. Taylor expanded in z around inf 95.8%

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

   1. exp-to-pow 95.8%

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

   2. sub-neg 95.8%

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

   3. metadata-eval 95.8%

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

   4. +-commutative 95.8%

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

7. Simplified 95.8%

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

8. Taylor expanded in z around 0 94.8%

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

   1. *-commutative 94.8%

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

10. Simplified 94.8%

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

    1. add-exp-log 94.3%

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

    2. *-commutative 94.3%

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

    3. +-commutative 94.3%

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

    4. sub-neg 94.3%

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

    5. +-commutative 94.3%

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

    6. log-prod 94.3%

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

    7. add-log-exp 94.3%

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

    8. log-pow 94.3%

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

    9. +-commutative 94.3%

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

    10. sub-neg 94.3%

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

12. Applied egg-rr 94.3%

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

13. Taylor expanded in z around 0 97.7%

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

14. Final simplification 97.7%

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

Alternative 2: 96.2% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (* (exp (+ (+ z -7.5) (* (- 0.5 z) (log (- 7.5 z))))) (sqrt (* PI 2.0)))
  (/ 263.3831869810514 z)))

C

double code(double z) {
	return (exp(((z + -7.5) + ((0.5 - z) * log((7.5 - z))))) * sqrt((((double) M_PI) * 2.0))) * (263.3831869810514 / z);
}

Java

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

Python

def code(z):
	return (math.exp(((z + -7.5) + ((0.5 - z) * math.log((7.5 - z))))) * math.sqrt((math.pi * 2.0))) * (263.3831869810514 / z)

Julia

function code(z)
	return Float64(Float64(exp(Float64(Float64(z + -7.5) + Float64(Float64(0.5 - z) * log(Float64(7.5 - z))))) * sqrt(Float64(pi * 2.0))) * Float64(263.3831869810514 / z))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.8%

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

3. Simplified 95.8%

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

5. Taylor expanded in z around 0 96.1%

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

   1. *-commutative 96.1%

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

7. Simplified 96.1%

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

8. Taylor expanded in z around 0 95.4%

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

   1. add-exp-log 96.4%

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

   2. *-commutative 96.4%

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

   3. log-prod 96.4%

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

   4. add-log-exp 97.1%

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

   5. neg-mul-1 97.1%

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

   6. fma-define 97.1%

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

10. Applied egg-rr 97.1%

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

11. Taylor expanded in z around inf 97.1%

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

    1. *-commutative 97.1%

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

13. Simplified 97.1%

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

14. Final simplification 97.1%

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

Alternative 3: 95.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.8%

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

3. Simplified 95.8%

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

5. Taylor expanded in z around 0 96.1%

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

   1. *-commutative 96.1%

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

7. Simplified 96.1%

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

8. Taylor expanded in z around 0 95.4%

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

   1. associate-*r/ 95.4%

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

   2. neg-mul-1 95.4%

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

   3. fma-define 95.4%

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

10. Applied egg-rr 95.4%

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

    1. associate-/l* 95.4%

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

    2. associate-*r* 95.6%

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

    3. *-commutative 95.6%

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

    4. associate-*l* 95.5%

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

    5. fma-undefine 95.5%

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

    6. mul-1-neg 95.5%

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

    7. +-commutative 95.5%

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

    8. sub-neg 95.5%

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

12. Simplified 95.5%

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

13. Final simplification 95.5%

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

Alternative 4: 95.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.8%

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

3. Simplified 95.8%

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

5. Taylor expanded in z around 0 96.1%

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

   1. *-commutative 96.1%

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

7. Simplified 96.1%

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

8. Taylor expanded in z around 0 95.4%

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

   1. associate-*r/ 95.4%

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

   2. neg-mul-1 95.4%

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

   3. fma-define 95.4%

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

10. Applied egg-rr 95.4%

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

    1. associate-/l* 95.4%

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

    2. associate-*r* 95.6%

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

    3. *-commutative 95.6%

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

    4. associate-*l* 95.5%

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

    5. fma-undefine 95.5%

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

    6. mul-1-neg 95.5%

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

    7. +-commutative 95.5%

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

    8. sub-neg 95.5%

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

    9. exp-to-pow 95.5%

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

    10. associate-*l* 95.5%

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

    11. exp-to-pow 95.5%

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

    12. *-commutative 95.5%

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

12. Simplified 95.5%

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

13. Final simplification 95.5%

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

Alternative 5: 95.1% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (* (sqrt (* PI 2.0)) (* (exp -7.5) (sqrt 7.5)))
  (/ 1.0 (/ z 263.3831869810514))))

C

double code(double z) {
	return (sqrt((((double) M_PI) * 2.0)) * (exp(-7.5) * sqrt(7.5))) * (1.0 / (z / 263.3831869810514));
}

Java

public static double code(double z) {
	return (Math.sqrt((Math.PI * 2.0)) * (Math.exp(-7.5) * Math.sqrt(7.5))) * (1.0 / (z / 263.3831869810514));
}

Python

def code(z):
	return (math.sqrt((math.pi * 2.0)) * (math.exp(-7.5) * math.sqrt(7.5))) * (1.0 / (z / 263.3831869810514))

Julia

function code(z)
	return Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64(exp(-7.5) * sqrt(7.5))) * Float64(1.0 / Float64(z / 263.3831869810514)))
end

MATLAB

function tmp = code(z)
	tmp = (sqrt((pi * 2.0)) * (exp(-7.5) * sqrt(7.5))) * (1.0 / (z / 263.3831869810514));
end

Wolfram

code[z_] := N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] * N[Sqrt[7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(z / 263.3831869810514), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.8%

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

3. Simplified 95.8%

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

5. Taylor expanded in z around 0 96.1%

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

   1. *-commutative 96.1%

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

7. Simplified 96.1%

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

8. Taylor expanded in z around 0 95.4%

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

9. Taylor expanded in z around 0 95.2%

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

    1. clear-num 95.2%

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

    2. inv-pow 95.2%

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

11. Applied egg-rr 95.2%

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

    1. unpow-1 95.2%

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

13. Simplified 95.2%

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

14. Final simplification 95.2%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left(e^{-7.5} \cdot \sqrt{7.5}\right)\right) \cdot \frac{1}{\frac{z}{263.3831869810514}} \]
15. Add Preprocessing

Alternative 6: 0.7% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.8%

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

3. Simplified 95.8%

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

5. Taylor expanded in z around 0 96.1%

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

   1. *-commutative 96.1%

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

7. Simplified 96.1%

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

8. Taylor expanded in z around 0 95.4%

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

9. Taylor expanded in z around 0 95.2%

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

    1. associate-*r/ 95.1%

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

    2. frac-2neg 95.1%

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

11. Applied egg-rr 0.8%

    \[\leadsto \color{blue}{\frac{-\sqrt{\left(\pi \cdot 2\right) \cdot \left(7.5 \cdot e^{-15}\right)} \cdot 263.3831869810514}{z}} \]
12. Step-by-step derivation

    1. distribute-rgt-neg-in 0.8%

       \[\leadsto \frac{\color{blue}{\sqrt{\left(\pi \cdot 2\right) \cdot \left(7.5 \cdot e^{-15}\right)} \cdot \left(-263.3831869810514\right)}}{z} \]

    2. associate-/l* 0.8%

       \[\leadsto \color{blue}{\sqrt{\left(\pi \cdot 2\right) \cdot \left(7.5 \cdot e^{-15}\right)} \cdot \frac{-263.3831869810514}{z}} \]

    3. associate-*r* 0.8%

       \[\leadsto \sqrt{\color{blue}{\left(\left(\pi \cdot 2\right) \cdot 7.5\right) \cdot e^{-15}}} \cdot \frac{-263.3831869810514}{z} \]

    4. *-commutative 0.8%

       \[\leadsto \sqrt{\color{blue}{e^{-15} \cdot \left(\left(\pi \cdot 2\right) \cdot 7.5\right)}} \cdot \frac{-263.3831869810514}{z} \]

    5. associate-*l* 0.8%

       \[\leadsto \sqrt{e^{-15} \cdot \color{blue}{\left(\pi \cdot \left(2 \cdot 7.5\right)\right)}} \cdot \frac{-263.3831869810514}{z} \]

    6. metadata-eval 0.8%

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

    7. metadata-eval 0.8%

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

13. Simplified 0.8%

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

14. Final simplification 0.8%

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

Alternative 7: 94.3% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.8%

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

3. Simplified 95.8%

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

5. Taylor expanded in z around 0 96.1%

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

   1. *-commutative 96.1%

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

7. Simplified 96.1%

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

8. Taylor expanded in z around 0 95.4%

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

9. Taylor expanded in z around 0 95.2%

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

    1. associate-*r/ 95.1%

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

    2. clear-num 95.1%

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

11. Applied egg-rr 94.3%

    \[\leadsto \color{blue}{\frac{1}{\frac{z}{\sqrt{\left(\pi \cdot 2\right) \cdot \left(7.5 \cdot e^{-15}\right)} \cdot 263.3831869810514}}} \]
12. Step-by-step derivation

    1. associate-/r/ 94.3%

       \[\leadsto \color{blue}{\frac{1}{z} \cdot \left(\sqrt{\left(\pi \cdot 2\right) \cdot \left(7.5 \cdot e^{-15}\right)} \cdot 263.3831869810514\right)} \]

    2. associate-*l/ 94.5%

       \[\leadsto \color{blue}{\frac{1 \cdot \left(\sqrt{\left(\pi \cdot 2\right) \cdot \left(7.5 \cdot e^{-15}\right)} \cdot 263.3831869810514\right)}{z}} \]

    3. *-lft-identity 94.5%

       \[\leadsto \frac{\color{blue}{\sqrt{\left(\pi \cdot 2\right) \cdot \left(7.5 \cdot e^{-15}\right)} \cdot 263.3831869810514}}{z} \]

    4. *-commutative 94.5%

       \[\leadsto \frac{\color{blue}{263.3831869810514 \cdot \sqrt{\left(\pi \cdot 2\right) \cdot \left(7.5 \cdot e^{-15}\right)}}}{z} \]

    5. associate-*l/ 94.4%

       \[\leadsto \color{blue}{\frac{263.3831869810514}{z} \cdot \sqrt{\left(\pi \cdot 2\right) \cdot \left(7.5 \cdot e^{-15}\right)}} \]

    6. associate-*r* 94.4%

       \[\leadsto \frac{263.3831869810514}{z} \cdot \sqrt{\color{blue}{\left(\left(\pi \cdot 2\right) \cdot 7.5\right) \cdot e^{-15}}} \]

    7. *-commutative 94.4%

       \[\leadsto \frac{263.3831869810514}{z} \cdot \sqrt{\color{blue}{e^{-15} \cdot \left(\left(\pi \cdot 2\right) \cdot 7.5\right)}} \]

    8. associate-*l* 94.4%

       \[\leadsto \frac{263.3831869810514}{z} \cdot \sqrt{e^{-15} \cdot \color{blue}{\left(\pi \cdot \left(2 \cdot 7.5\right)\right)}} \]

    9. metadata-eval 94.4%

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

13. Simplified 94.4%

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

14. Final simplification 94.4%

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

Alternative 8: 95.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{e^{-15} \cdot \left(\pi \cdot 15\right)}}{z \cdot 0.0037967495627271876} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (/ (sqrt (* (exp -15.0) (* PI 15.0))) (* z 0.0037967495627271876)))

C

double code(double z) {
	return sqrt((exp(-15.0) * (((double) M_PI) * 15.0))) / (z * 0.0037967495627271876);
}

Java

public static double code(double z) {
	return Math.sqrt((Math.exp(-15.0) * (Math.PI * 15.0))) / (z * 0.0037967495627271876);
}

Python

def code(z):
	return math.sqrt((math.exp(-15.0) * (math.pi * 15.0))) / (z * 0.0037967495627271876)

Julia

function code(z)
	return Float64(sqrt(Float64(exp(-15.0) * Float64(pi * 15.0))) / Float64(z * 0.0037967495627271876))
end

MATLAB

function tmp = code(z)
	tmp = sqrt((exp(-15.0) * (pi * 15.0))) / (z * 0.0037967495627271876);
end

Wolfram

code[z_] := N[(N[Sqrt[N[(N[Exp[-15.0], $MachinePrecision] * N[(Pi * 15.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(z * 0.0037967495627271876), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.8%

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

3. Simplified 95.8%

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

5. Taylor expanded in z around 0 96.1%

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

   1. *-commutative 96.1%

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

7. Simplified 96.1%

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

8. Taylor expanded in z around 0 95.4%

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

9. Taylor expanded in z around 0 95.2%

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

    1. clear-num 95.2%

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

    2. un-div-inv 95.5%

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

11. Applied egg-rr 95.2%

    \[\leadsto \color{blue}{\frac{\sqrt{\left(\pi \cdot 2\right) \cdot \left(7.5 \cdot e^{-15}\right)}}{z \cdot 0.0037967495627271876}} \]
12. Step-by-step derivation

    1. associate-*r* 95.2%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(\pi \cdot 2\right) \cdot 7.5\right) \cdot e^{-15}}}}{z \cdot 0.0037967495627271876} \]

    2. *-commutative 95.2%

       \[\leadsto \frac{\sqrt{\color{blue}{e^{-15} \cdot \left(\left(\pi \cdot 2\right) \cdot 7.5\right)}}}{z \cdot 0.0037967495627271876} \]

    3. associate-*l* 95.2%

       \[\leadsto \frac{\sqrt{e^{-15} \cdot \color{blue}{\left(\pi \cdot \left(2 \cdot 7.5\right)\right)}}}{z \cdot 0.0037967495627271876} \]

    4. metadata-eval 95.2%

       \[\leadsto \frac{\sqrt{e^{-15} \cdot \left(\pi \cdot \color{blue}{15}\right)}}{z \cdot 0.0037967495627271876} \]

13. Simplified 95.2%

    \[\leadsto \color{blue}{\frac{\sqrt{e^{-15} \cdot \left(\pi \cdot 15\right)}}{z \cdot 0.0037967495627271876}} \]

14. Final simplification 95.2%

    \[\leadsto \frac{\sqrt{e^{-15} \cdot \left(\pi \cdot 15\right)}}{z \cdot 0.0037967495627271876} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024089 
(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))))))