Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.4% → 97.3%

Time: 9.9s

Alternatives: 9

Speedup: 2.6×

• could not determine a ground truth

  1. z = -12632469983.68273

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: 97.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\\ t_2 := \left(t\_0 + 7\right) + 0.5\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {t\_2}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{-t\_2}\right) \cdot \frac{69370.70318429549 - t\_1 \cdot t\_1}{263.3831869810514 - t\_1}\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0))
        (t_1 (* z (+ 436.8961725563396 (* 545.0353078428827 z))))
        (t_2 (+ (+ t_0 7.0) 0.5)))
   (*
    (/ PI (sin (* PI z)))
    (*
     (* (* (sqrt (* PI 2.0)) (pow t_2 (+ t_0 0.5))) (exp (- t_2)))
     (/ (- 69370.70318429549 (* t_1 t_1)) (- 263.3831869810514 t_1))))))

C

double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = z * (436.8961725563396 + (545.0353078428827 * z));
	double t_2 = (t_0 + 7.0) + 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)) * ((69370.70318429549 - (t_1 * t_1)) / (263.3831869810514 - t_1)));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = z * (436.8961725563396 + (545.0353078428827 * z));
	double t_2 = (t_0 + 7.0) + 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)) * ((69370.70318429549 - (t_1 * t_1)) / (263.3831869810514 - t_1)));
}

Python

def code(z):
	t_0 = (1.0 - z) - 1.0
	t_1 = z * (436.8961725563396 + (545.0353078428827 * z))
	t_2 = (t_0 + 7.0) + 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)) * ((69370.70318429549 - (t_1 * t_1)) / (263.3831869810514 - t_1)))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - 1.0)
	t_1 = Float64(z * Float64(436.8961725563396 + Float64(545.0353078428827 * z)))
	t_2 = Float64(Float64(t_0 + 7.0) + 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(69370.70318429549 - Float64(t_1 * t_1)) / Float64(263.3831869810514 - t_1))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - 1.0;
	t_1 = z * (436.8961725563396 + (545.0353078428827 * z));
	t_2 = (t_0 + 7.0) + 0.5;
	tmp = (pi / sin((pi * z))) * (((sqrt((pi * 2.0)) * (t_2 ^ (t_0 + 0.5))) * exp(-t_2)) * ((69370.70318429549 - (t_1 * t_1)) / (263.3831869810514 - t_1)));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(436.8961725563396 + N[(545.0353078428827 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 + 7.0), $MachinePrecision] + 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[(69370.70318429549 - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(263.3831869810514 - t$95$1), $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 96.4

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

4. Applied rewrites 96.4%

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

   1. lift-+.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. lift-*.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. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. flip-+ N/A

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

   6. lower-special-/ N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 97.3%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \frac{69370.70318429549 - \left(z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right) \cdot \left(z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right)}{\color{blue}{263.3831869810514 - z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)}}\right) \]
7. Add Preprocessing

Alternative 2: 96.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[z_] := N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(z - 7.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $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 96.4

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

4. Applied rewrites 96.4%

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

5. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift--.f64 N/A

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

   7. exp-to-pow N/A

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

   8. lift-pow.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-sqrt.f64 96.4

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

7. Applied rewrites 96.4%

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

9. Applied rewrites 96.7%

   \[\leadsto \color{blue}{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z - 7.5} \cdot \sqrt{2}\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(545.0353078428827, z, 436.8961725563396\right), z, 263.3831869810514\right)} \]
10. Add Preprocessing

Alternative 3: 96.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{\mathsf{fma}\left(-0.16666666666666666 \cdot \left(z \cdot z\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi\right) \cdot z} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z - 7.5} \cdot \sqrt{2}\right)\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ PI (* (fma (* -0.16666666666666666 (* z z)) (* (* PI PI) PI) PI) z))
  (*
   (* (sqrt PI) (* (pow (- 7.5 z) (- 0.5 z)) (* (exp (- z 7.5)) (sqrt 2.0))))
   (+ 263.3831869810514 (* z (+ 436.8961725563396 (* 545.0353078428827 z)))))))

C

double code(double z) {
	return (((double) M_PI) / (fma((-0.16666666666666666 * (z * z)), ((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)), ((double) M_PI)) * z)) * ((sqrt(((double) M_PI)) * (pow((7.5 - z), (0.5 - z)) * (exp((z - 7.5)) * sqrt(2.0)))) * (263.3831869810514 + (z * (436.8961725563396 + (545.0353078428827 * z)))));
}

Julia

function code(z)
	return Float64(Float64(pi / Float64(fma(Float64(-0.16666666666666666 * Float64(z * z)), Float64(Float64(pi * pi) * pi), pi) * z)) * Float64(Float64(sqrt(pi) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * Float64(exp(Float64(z - 7.5)) * sqrt(2.0)))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(545.0353078428827 * z))))))
end

Wolfram

code[z_] := N[(N[(Pi / N[(N[(N[(-0.16666666666666666 * N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision] + Pi), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(z - 7.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $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 96.4

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

4. Applied rewrites 96.4%

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

5. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift--.f64 N/A

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

   7. exp-to-pow N/A

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

   8. lift-pow.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-sqrt.f64 96.4

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

7. Applied rewrites 96.4%

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

8. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\color{blue}{z \cdot \left(\mathsf{PI}\left(\right) + \frac{-1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)}} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\pi}{\left(\mathsf{PI}\left(\right) + \frac{-1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \color{blue}{z}} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\left(\mathsf{PI}\left(\right) + \frac{-1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \color{blue}{z}} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \frac{\pi}{\left(\frac{-1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \frac{\pi}{\left(\left(\frac{-1}{6} \cdot {z}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3} + \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\pi}{\mathsf{fma}\left(\frac{-1}{6} \cdot {z}^{2}, {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\mathsf{fma}\left(\frac{-1}{6} \cdot {z}^{2}, {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   7. pow2 N/A

      \[\leadsto \frac{\pi}{\mathsf{fma}\left(\frac{-1}{6} \cdot \left(z \cdot z\right), {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\pi}{\mathsf{fma}\left(\frac{-1}{6} \cdot \left(z \cdot z\right), {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   9. unpow3 N/A

      \[\leadsto \frac{\pi}{\mathsf{fma}\left(\frac{-1}{6} \cdot \left(z \cdot z\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \frac{\pi}{\mathsf{fma}\left(\frac{-1}{6} \cdot \left(z \cdot z\right), {\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\pi}{\mathsf{fma}\left(\frac{-1}{6} \cdot \left(z \cdot z\right), {\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \frac{\pi}{\mathsf{fma}\left(\frac{-1}{6} \cdot \left(z \cdot z\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\pi}{\mathsf{fma}\left(\frac{-1}{6} \cdot \left(z \cdot z\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   14. lift-PI.f64 N/A

       \[\leadsto \frac{\pi}{\mathsf{fma}\left(\frac{-1}{6} \cdot \left(z \cdot z\right), \left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   15. lift-PI.f64 N/A

       \[\leadsto \frac{\pi}{\mathsf{fma}\left(\frac{-1}{6} \cdot \left(z \cdot z\right), \left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   16. lift-PI.f64 N/A

       \[\leadsto \frac{\pi}{\mathsf{fma}\left(\frac{-1}{6} \cdot \left(z \cdot z\right), \left(\pi \cdot \pi\right) \cdot \pi, \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

   17. lift-PI.f64 96.3

       \[\leadsto \frac{\pi}{\mathsf{fma}\left(-0.16666666666666666 \cdot \left(z \cdot z\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi\right) \cdot z} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z - 7.5} \cdot \sqrt{2}\right)\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right)\right) \]

10. Applied rewrites 96.3%

    \[\leadsto \frac{\pi}{\color{blue}{\mathsf{fma}\left(-0.16666666666666666 \cdot \left(z \cdot z\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi\right) \cdot z}} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z - 7.5} \cdot \sqrt{2}\right)\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right)\right) \]
11. Add Preprocessing

Alternative 4: 96.3% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[z_] := N[(N[(Pi / N[(Pi * z), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[(N[(z + 1.0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 96.4

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

4. Applied rewrites 96.4%

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

5. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift--.f64 N/A

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

   7. exp-to-pow N/A

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

   8. lift-pow.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-sqrt.f64 96.4

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

7. Applied rewrites 96.4%

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

8. Taylor expanded in z around 0

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

   1. distribute-lft1-in N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-exp.f64 95.9

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

10. Applied rewrites 95.9%

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

11. Taylor expanded in z around 0

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

    1. *-commutative N/A

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

    2. lift-*.f64 N/A

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

    3. lift-PI.f64 96.6

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

13. Applied rewrites 96.6%

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

Alternative 5: 95.9% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[z_] := N[(N[(1.0 / z), $MachinePrecision] * N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(z - 7.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * 436.8961725563396), $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 96.4

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

4. Applied rewrites 96.4%

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

5. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift--.f64 N/A

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

   7. exp-to-pow N/A

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

   8. lift-pow.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-sqrt.f64 96.4

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

7. Applied rewrites 96.4%

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

8. Taylor expanded in z around 0

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

   1. lift-/.f64 96.0

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

10. Applied rewrites 96.0%

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

11. Taylor expanded in z around 0

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

13. Applied rewrites 95.9%

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

Alternative 6: 95.9% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 95.5%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-*.f64 N/A

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

   4. lift-exp.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. metadata-eval N/A

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

   7. sqrt-unprod N/A

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

   8. lower-*.f64 N/A

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

   9. lift-exp.f64 N/A

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

   10. sqrt-unprod N/A

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

   11. metadata-eval N/A

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

   12. lift-sqrt.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lower-/.f64 95.4

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

6. Applied rewrites 95.4%

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

   1. lift-PI.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. pow1/2 N/A

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

   4. metadata-eval N/A

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

   5. pow-sub N/A

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

   6. metadata-eval N/A

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

   7. pow-pow N/A

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

   8. pow1 N/A

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

   9. pow1 N/A

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

   10. metadata-eval N/A

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

   11. pow-pow N/A

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

   12. pow1 N/A

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

   13. pow1/2 N/A

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

   14. lower-special-/ N/A

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

   15. lower-/.f64 N/A

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

   16. lift-PI.f64 N/A

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

   17. lift-sqrt.f64 N/A

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

   18. lift-PI.f64 95.9

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

8. Applied rewrites 95.9%

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

Alternative 7: 95.8% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 95.5%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-*.f64 N/A

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

   4. lift-exp.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. metadata-eval N/A

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

   7. sqrt-unprod N/A

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

   8. lower-*.f64 N/A

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

   9. lift-exp.f64 N/A

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

   10. sqrt-unprod N/A

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

   11. metadata-eval N/A

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

   12. lift-sqrt.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lower-/.f64 95.4

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

6. Applied rewrites 95.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

8. Applied rewrites 95.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-exp.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. lift-exp.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lift-sqrt.f64 N/A

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

   18. lift-PI.f64 95.8

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

10. Applied rewrites 95.8%

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

Alternative 8: 95.6% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 95.5%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-*.f64 N/A

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

   4. lift-exp.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. metadata-eval N/A

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

   7. sqrt-unprod N/A

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

   8. lower-*.f64 N/A

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

   9. lift-exp.f64 N/A

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

   10. sqrt-unprod N/A

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

   11. metadata-eval N/A

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

   12. lift-sqrt.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lower-/.f64 95.4

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

6. Applied rewrites 95.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

8. Applied rewrites 95.6%

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

Alternative 9: 95.6% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 95.5%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-*.f64 N/A

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

   4. lift-exp.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. metadata-eval N/A

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

   7. sqrt-unprod N/A

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

   8. lower-*.f64 N/A

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

   9. lift-exp.f64 N/A

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

   10. sqrt-unprod N/A

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

   11. metadata-eval N/A

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

   12. lift-sqrt.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lower-/.f64 95.4

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

6. Applied rewrites 95.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

8. Applied rewrites 95.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. lift-exp.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lift-sqrt.f64 N/A

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

   15. lift-PI.f64 95.6

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

10. Applied rewrites 95.6%

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

Reproduce

herbie shell --seed 2025132 
(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))))))