Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.4% → 98.4%

Time: 9.2s

Alternatives: 7

Speedup: 3.2×

• could not determine a ground truth

  1. z = -230510629406431.6

Specification

\[z \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(z)
	return Float64(Float64(Float64(pi / sin(Float64(z * pi))) * Float64(exp(fma(Float64(0.5 - z), log(Float64(7.5 - z)), Float64(z - 7.5))) * sqrt(Float64(pi + pi)))) * fma(fma(545.0353078428827, z, 436.8961725563396), z, 263.3831869810514))
end

Wolfram

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

Derivation

1. Initial program 96.4%

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

2. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)}\right) \]
3. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + \color{blue}{z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \color{blue}{\left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \color{blue}{\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z}\right)\right)\right) \]

   4. lower-*.f64 96.6

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

4. Applied rewrites 96.6%

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

5. Taylor expanded in z around inf

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\color{blue}{\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \color{blue}{\left(e^{z - \frac{15}{2}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   2. lower-exp.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(\color{blue}{e^{z - \frac{15}{2}}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{\color{blue}{z - \frac{15}{2}}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   4. lower-log.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{\color{blue}{z} - \frac{15}{2}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   5. lower--.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   6. lower--.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \color{blue}{\frac{15}{2}}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \color{blue}{\sqrt{2 \cdot \mathsf{PI}\left(\right)}}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   8. lower-exp.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{\color{blue}{2 \cdot \mathsf{PI}\left(\right)}}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   9. lower--.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{\color{blue}{2} \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   12. lower-PI.f64 96.6

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

7. Applied rewrites 96.6%

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

9. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(e^{\mathsf{fma}\left(0.5 - z, \log \left(7.5 - z\right), z - 7.5\right)} \cdot \sqrt{\pi + \pi}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(545.0353078428827, z, 436.8961725563396\right), z, 263.3831869810514\right)} \]
10. Add Preprocessing

Alternative 2: 96.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\pi}{z \cdot \pi} \cdot \left(\left(\left(z - -1\right) \cdot e^{-7.5}\right) \cdot \left({\left(\left(\left(1 - z\right) - -6\right) - -0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{\pi + \pi}\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(545.0353078428827, z, 436.8961725563396\right), z, 263.3831869810514\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (/ PI (* z PI))
   (*
    (* (- z -1.0) (exp -7.5))
    (* (pow (- (- (- 1.0 z) -6.0) -0.5) (- (- 1.0 z) 0.5)) (sqrt (+ PI PI)))))
  (fma (fma 545.0353078428827 z 436.8961725563396) z 263.3831869810514)))

C

double code(double z) {
	return ((((double) M_PI) / (z * ((double) M_PI))) * (((z - -1.0) * exp(-7.5)) * (pow((((1.0 - z) - -6.0) - -0.5), ((1.0 - z) - 0.5)) * sqrt((((double) M_PI) + ((double) M_PI)))))) * fma(fma(545.0353078428827, z, 436.8961725563396), z, 263.3831869810514);
}

Julia

function code(z)
	return Float64(Float64(Float64(pi / Float64(z * pi)) * Float64(Float64(Float64(z - -1.0) * exp(-7.5)) * Float64((Float64(Float64(Float64(1.0 - z) - -6.0) - -0.5) ^ Float64(Float64(1.0 - z) - 0.5)) * sqrt(Float64(pi + pi))))) * fma(fma(545.0353078428827, z, 436.8961725563396), z, 263.3831869810514))
end

Wolfram

code[z_] := N[(N[(N[(Pi / N[(z * Pi), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(z - -1.0), $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision] - -0.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi + Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(545.0353078428827 * z + 436.8961725563396), $MachinePrecision] * z + 263.3831869810514), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.4%

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

2. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)}\right) \]
3. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + \color{blue}{z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \color{blue}{\left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \color{blue}{\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z}\right)\right)\right) \]

   4. lower-*.f64 96.6

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

4. Applied rewrites 96.6%

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

5. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\color{blue}{z \cdot \mathsf{PI}\left(\right)}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{z \cdot \color{blue}{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   2. lower-PI.f64 96.2

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

7. Applied rewrites 96.2%

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

8. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{z \cdot \pi} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \color{blue}{\left(e^{\frac{-15}{2}} + z \cdot e^{\frac{-15}{2}}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]
9. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{z \cdot \pi} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \color{blue}{z \cdot e^{\frac{-15}{2}}}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   2. lower-exp.f64 N/A

      \[\leadsto \frac{\pi}{z \cdot \pi} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \color{blue}{z} \cdot e^{\frac{-15}{2}}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{z \cdot \pi} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + z \cdot \color{blue}{e^{\frac{-15}{2}}}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   4. lower-exp.f64 96.6

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

10. Applied rewrites 96.6%

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

12. Applied rewrites 96.9%

    \[\leadsto \color{blue}{\left(\frac{\pi}{z \cdot \pi} \cdot \left(\left(\left(z - -1\right) \cdot e^{-7.5}\right) \cdot \left({\left(\left(\left(1 - z\right) - -6\right) - -0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{\pi + \pi}\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(545.0353078428827, z, 436.8961725563396\right), z, 263.3831869810514\right)} \]
13. Add Preprocessing

Alternative 3: 96.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.4%

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

2. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)}\right) \]
3. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + \color{blue}{z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \color{blue}{\left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \color{blue}{\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z}\right)\right)\right) \]

   4. lower-*.f64 96.6

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

4. Applied rewrites 96.6%

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

5. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\color{blue}{z \cdot \mathsf{PI}\left(\right)}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{z \cdot \color{blue}{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   2. lower-PI.f64 96.2

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

7. Applied rewrites 96.2%

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

8. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{z \cdot \pi} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\color{blue}{\frac{15}{2}}}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]
9. Step-by-step derivation

10. Applied rewrites 95.5%

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

11. Taylor expanded in z around 0

    \[\leadsto \frac{\pi}{z \cdot \pi} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\frac{15}{2}}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\color{blue}{\frac{15}{2}}}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]
12. Step-by-step derivation

13. Applied rewrites 95.8%

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

Alternative 4: 96.2% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.4%

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

2. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{z}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \color{blue}{\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{z}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{\color{blue}{z}} \]

4. Applied rewrites 95.7%

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

   1. lift-*.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \color{blue}{\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \pi} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{z}} \]

   2. *-commutative N/A

      \[\leadsto \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \pi} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{z} \cdot \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000}} \]

   3. lower-*.f64 95.7

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

6. Applied rewrites 95.7%

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

7. Taylor expanded in z around 0

   \[\leadsto \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}{z} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}{z} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}{z} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

   3. lower-exp.f64 N/A

      \[\leadsto \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}{z} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}{z} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}{z} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

   6. lower-PI.f64 95.7

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

9. Applied rewrites 95.7%

   \[\leadsto \frac{e^{-7.5} \cdot \sqrt{15 \cdot \pi}}{z} \cdot 263.3831869810514 \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \pi}}{z} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

    2. *-commutative N/A

       \[\leadsto \frac{\sqrt{15 \cdot \pi} \cdot e^{\frac{-15}{2}}}{z} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

11. Applied rewrites 96.4%

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

Alternative 5: 95.8% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{0.0037967495627271876 \cdot \frac{z}{e^{-7.5} \cdot \sqrt{15 \cdot \pi}}} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (/ 1.0 (* 0.0037967495627271876 (/ z (* (exp -7.5) (sqrt (* 15.0 PI)))))))

C

double code(double z) {
	return 1.0 / (0.0037967495627271876 * (z / (exp(-7.5) * sqrt((15.0 * ((double) M_PI))))));
}

Java

public static double code(double z) {
	return 1.0 / (0.0037967495627271876 * (z / (Math.exp(-7.5) * Math.sqrt((15.0 * Math.PI)))));
}

Python

def code(z):
	return 1.0 / (0.0037967495627271876 * (z / (math.exp(-7.5) * math.sqrt((15.0 * math.pi)))))

Julia

function code(z)
	return Float64(1.0 / Float64(0.0037967495627271876 * Float64(z / Float64(exp(-7.5) * sqrt(Float64(15.0 * pi))))))
end

MATLAB

function tmp = code(z)
	tmp = 1.0 / (0.0037967495627271876 * (z / (exp(-7.5) * sqrt((15.0 * pi)))));
end

Wolfram

code[z_] := N[(1.0 / N[(0.0037967495627271876 * N[(z / N[(N[Exp[-7.5], $MachinePrecision] * N[Sqrt[N[(15.0 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.4%

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

2. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right) + z \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right) + e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right) \cdot {\frac{15}{2}}^{\frac{1}{2}}\right) + \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)\right)}{z}} \]

4. Applied rewrites 96.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(263.3831869810514, e^{-7.5} \cdot \left(\sqrt{2 \cdot \pi} \cdot {7.5}^{0.5}\right), z \cdot \mathsf{fma}\left(263.3831869810514, e^{-7.5} \cdot \left(\sqrt{2 \cdot \pi} \cdot {7.5}^{0.5}\right), e^{-7.5} \cdot \left(\sqrt{2 \cdot \pi} \cdot \mathsf{fma}\left(263.3831869810514, \left(-1 \cdot \log 7.5 - 0.06666666666666667\right) \cdot {7.5}^{0.5}, 436.8961725563396 \cdot {7.5}^{0.5}\right)\right)\right)\right)}{z}} \]

6. Applied rewrites 96.2%

   \[\leadsto \frac{1}{\color{blue}{\frac{z}{\mathsf{fma}\left(263.3831869810514 \cdot e^{-7.5}, \sqrt{7.5 \cdot \left(\pi + \pi\right)}, \mathsf{fma}\left(263.3831869810514 \cdot e^{-7.5}, \sqrt{7.5 \cdot \left(\pi + \pi\right)}, \left(\sqrt{\pi + \pi} \cdot \mathsf{fma}\left(263.3831869810514 \cdot \left(\log 0.13333333333333333 - 0.06666666666666667\right), \sqrt{7.5}, 436.8961725563396 \cdot \sqrt{7.5}\right)\right) \cdot e^{-7.5}\right) \cdot z\right)}}} \]

7. Taylor expanded in z around 0

   \[\leadsto \frac{1}{\frac{4200000000000000000000000}{1106209385320415913103082059} \cdot \color{blue}{\frac{z}{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}}} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{\frac{4200000000000000000000000}{1106209385320415913103082059} \cdot \frac{z}{\color{blue}{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{\frac{4200000000000000000000000}{1106209385320415913103082059} \cdot \frac{z}{e^{\frac{-15}{2}} \cdot \color{blue}{\sqrt{15 \cdot \mathsf{PI}\left(\right)}}}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{\frac{4200000000000000000000000}{1106209385320415913103082059} \cdot \frac{z}{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}} \]

   4. lower-exp.f64 N/A

      \[\leadsto \frac{1}{\frac{4200000000000000000000000}{1106209385320415913103082059} \cdot \frac{z}{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{\frac{4200000000000000000000000}{1106209385320415913103082059} \cdot \frac{z}{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{1}{\frac{4200000000000000000000000}{1106209385320415913103082059} \cdot \frac{z}{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}} \]

   7. lower-PI.f64 96.2

      \[\leadsto \frac{1}{0.0037967495627271876 \cdot \frac{z}{e^{-7.5} \cdot \sqrt{15 \cdot \pi}}} \]

9. Applied rewrites 96.2%

   \[\leadsto \frac{1}{0.0037967495627271876 \cdot \color{blue}{\frac{z}{e^{-7.5} \cdot \sqrt{15 \cdot \pi}}}} \]
10. Add Preprocessing

Alternative 6: 95.7% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{z}{\sqrt{15 \cdot \pi} \cdot e^{-7.5}}} \cdot 263.3831869810514 \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (* (/ 1.0 (/ z (* (sqrt (* 15.0 PI)) (exp -7.5)))) 263.3831869810514))

C

double code(double z) {
	return (1.0 / (z / (sqrt((15.0 * ((double) M_PI))) * exp(-7.5)))) * 263.3831869810514;
}

Java

public static double code(double z) {
	return (1.0 / (z / (Math.sqrt((15.0 * Math.PI)) * Math.exp(-7.5)))) * 263.3831869810514;
}

Python

def code(z):
	return (1.0 / (z / (math.sqrt((15.0 * math.pi)) * math.exp(-7.5)))) * 263.3831869810514

Julia

function code(z)
	return Float64(Float64(1.0 / Float64(z / Float64(sqrt(Float64(15.0 * pi)) * exp(-7.5)))) * 263.3831869810514)
end

MATLAB

function tmp = code(z)
	tmp = (1.0 / (z / (sqrt((15.0 * pi)) * exp(-7.5)))) * 263.3831869810514;
end

Wolfram

code[z_] := N[(N[(1.0 / N[(z / N[(N[Sqrt[N[(15.0 * Pi), $MachinePrecision]], $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 263.3831869810514), $MachinePrecision]

Derivation

1. Initial program 96.4%

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

2. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{z}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \color{blue}{\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{z}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{\color{blue}{z}} \]

4. Applied rewrites 95.7%

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

   1. lift-*.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \color{blue}{\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \pi} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{z}} \]

   2. *-commutative N/A

      \[\leadsto \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \pi} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{z} \cdot \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000}} \]

   3. lower-*.f64 95.7

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

6. Applied rewrites 95.7%

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

7. Taylor expanded in z around 0

   \[\leadsto \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}{z} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}{z} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}{z} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

   3. lower-exp.f64 N/A

      \[\leadsto \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}{z} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}{z} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}{z} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

   6. lower-PI.f64 95.7

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

9. Applied rewrites 95.7%

   \[\leadsto \frac{e^{-7.5} \cdot \sqrt{15 \cdot \pi}}{z} \cdot 263.3831869810514 \]
10. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \pi}}{z} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

    2. div-flip N/A

       \[\leadsto \frac{1}{\frac{z}{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \pi}}} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

    3. lower-unsound-/.f64 N/A

       \[\leadsto \frac{1}{\frac{z}{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \pi}}} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

    4. lower-unsound-/.f64 95.7

       \[\leadsto \frac{1}{\frac{z}{e^{-7.5} \cdot \sqrt{15 \cdot \pi}}} \cdot 263.3831869810514 \]

    5. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{z}{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \pi}}} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

    6. *-commutative N/A

       \[\leadsto \frac{1}{\frac{z}{\sqrt{15 \cdot \pi} \cdot e^{\frac{-15}{2}}}} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

    7. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{z}{\sqrt{15 \cdot \pi} \cdot e^{\frac{-15}{2}}}} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

    8. *-commutative N/A

       \[\leadsto \frac{1}{\frac{z}{\sqrt{\pi \cdot 15} \cdot e^{\frac{-15}{2}}}} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

    9. metadata-eval N/A

       \[\leadsto \frac{1}{\frac{z}{\sqrt{\pi \cdot \left(\frac{15}{2} + \frac{15}{2}\right)} \cdot e^{\frac{-15}{2}}}} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

    10. distribute-lft-out N/A

        \[\leadsto \frac{1}{\frac{z}{\sqrt{\pi \cdot \frac{15}{2} + \pi \cdot \frac{15}{2}} \cdot e^{\frac{-15}{2}}}} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

    11. distribute-rgt-in N/A

        \[\leadsto \frac{1}{\frac{z}{\sqrt{\frac{15}{2} \cdot \left(\pi + \pi\right)} \cdot e^{\frac{-15}{2}}}} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

    12. lift-+.f64 N/A

        \[\leadsto \frac{1}{\frac{z}{\sqrt{\frac{15}{2} \cdot \left(\pi + \pi\right)} \cdot e^{\frac{-15}{2}}}} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

    13. *-commutative N/A

        \[\leadsto \frac{1}{\frac{z}{\sqrt{\left(\pi + \pi\right) \cdot \frac{15}{2}} \cdot e^{\frac{-15}{2}}}} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

    14. lift-*.f64 N/A

        \[\leadsto \frac{1}{\frac{z}{\sqrt{\left(\pi + \pi\right) \cdot \frac{15}{2}} \cdot e^{\frac{-15}{2}}}} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

    15. lower-*.f64 95.7

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

11. Applied rewrites 95.7%

    \[\leadsto \frac{1}{\frac{z}{\sqrt{15 \cdot \pi} \cdot e^{-7.5}}} \cdot 263.3831869810514 \]
12. Add Preprocessing

Alternative 7: 95.7% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.4%

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

2. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{z}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \color{blue}{\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{z}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{\color{blue}{z}} \]

4. Applied rewrites 95.7%

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

   1. lift-*.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \color{blue}{\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \pi} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{z}} \]

   2. *-commutative N/A

      \[\leadsto \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \pi} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{z} \cdot \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000}} \]

   3. lower-*.f64 95.7

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

6. Applied rewrites 95.7%

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

7. Taylor expanded in z around 0

   \[\leadsto \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}{z} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}{z} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}{z} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

   3. lower-exp.f64 N/A

      \[\leadsto \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}{z} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}{z} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}{z} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

   6. lower-PI.f64 95.7

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

9. Applied rewrites 95.7%

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

Reproduce

herbie shell --seed 2025154 
(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))))))