Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.3% → 97.3%

Time: 16.1s

Alternatives: 3

Speedup: N/A×

• could not determine a ground truth

  1. z = -4413817213353.591

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: 96.3% 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, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 \cdot \log 7.5 - 0.06666666666666667\\ t_1 := 0.1288888888888889 + 0.5 \cdot {t\_0}^{2}\\ t_2 := 436.8961725563396 \cdot {7.5}^{0.5}\\ t_3 := 545.0353078428827 \cdot {7.5}^{0.5}\\ t_4 := 263.3831869810514 \cdot {7.5}^{0.5}\\ t_5 := {2}^{0.5} \cdot \mathsf{fma}\left(t\_4, t\_0, t\_2\right)\\ t_6 := e^{-7.5} \cdot t\_5\\ t_7 := {2}^{0.5} \cdot \mathsf{fma}\left(t\_4, t\_1, \mathsf{fma}\left(t\_2, t\_0, t\_3\right)\right)\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \mathsf{fma}\left(263.3831869810514 \cdot e^{\log \pi \cdot 0.5}, e^{-7.5} \cdot \sqrt{15}, z \cdot \mathsf{fma}\left(z, \mathsf{fma}\left({\pi}^{0.5}, \mathsf{fma}\left(131.6915934905257 \cdot e^{-7.5}, \sqrt{15}, \mathsf{fma}\left(e^{-7.5}, t\_7, t\_6\right)\right), \left(z \cdot \mathsf{fma}\left(0.5 \cdot e^{-7.5}, t\_5, \mathsf{fma}\left(43.89719783017524 \cdot e^{-7.5}, \sqrt{15}, \mathsf{fma}\left(e^{-7.5}, {2}^{0.5} \cdot \mathsf{fma}\left(t\_4, 0.008493827160493827 + \mathsf{fma}\left(0.1288888888888889, t\_0, 0.16666666666666666 \cdot {t\_0}^{3}\right), \mathsf{fma}\left(t\_2, t\_1, \mathsf{fma}\left(t\_3, t\_0, 606.6766809167608 \cdot {7.5}^{0.5}\right)\right)\right), e^{-7.5} \cdot t\_7\right)\right)\right)\right) \cdot {\pi}^{0.5}\right), {\pi}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot e^{-7.5}, \sqrt{15}, t\_6\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (* -1.0 (log 7.5)) 0.06666666666666667))
        (t_1 (+ 0.1288888888888889 (* 0.5 (pow t_0 2.0))))
        (t_2 (* 436.8961725563396 (pow 7.5 0.5)))
        (t_3 (* 545.0353078428827 (pow 7.5 0.5)))
        (t_4 (* 263.3831869810514 (pow 7.5 0.5)))
        (t_5 (* (pow 2.0 0.5) (fma t_4 t_0 t_2)))
        (t_6 (* (exp -7.5) t_5))
        (t_7 (* (pow 2.0 0.5) (fma t_4 t_1 (fma t_2 t_0 t_3)))))
   (*
    (/ PI (sin (* PI z)))
    (fma
     (* 263.3831869810514 (exp (* (log PI) 0.5)))
     (* (exp -7.5) (sqrt 15.0))
     (*
      z
      (fma
       z
       (fma
        (pow PI 0.5)
        (fma
         (* 131.6915934905257 (exp -7.5))
         (sqrt 15.0)
         (fma (exp -7.5) t_7 t_6))
        (*
         (*
          z
          (fma
           (* 0.5 (exp -7.5))
           t_5
           (fma
            (* 43.89719783017524 (exp -7.5))
            (sqrt 15.0)
            (fma
             (exp -7.5)
             (*
              (pow 2.0 0.5)
              (fma
               t_4
               (+
                0.008493827160493827
                (fma
                 0.1288888888888889
                 t_0
                 (* 0.16666666666666666 (pow t_0 3.0))))
               (fma
                t_2
                t_1
                (fma t_3 t_0 (* 606.6766809167608 (pow 7.5 0.5))))))
             (* (exp -7.5) t_7)))))
         (pow PI 0.5)))
       (*
        (pow PI 0.5)
        (fma (* 263.3831869810514 (exp -7.5)) (sqrt 15.0) t_6))))))))

C

double code(double z) {
	double t_0 = (-1.0 * log(7.5)) - 0.06666666666666667;
	double t_1 = 0.1288888888888889 + (0.5 * pow(t_0, 2.0));
	double t_2 = 436.8961725563396 * pow(7.5, 0.5);
	double t_3 = 545.0353078428827 * pow(7.5, 0.5);
	double t_4 = 263.3831869810514 * pow(7.5, 0.5);
	double t_5 = pow(2.0, 0.5) * fma(t_4, t_0, t_2);
	double t_6 = exp(-7.5) * t_5;
	double t_7 = pow(2.0, 0.5) * fma(t_4, t_1, fma(t_2, t_0, t_3));
	return (((double) M_PI) / sin((((double) M_PI) * z))) * fma((263.3831869810514 * exp((log(((double) M_PI)) * 0.5))), (exp(-7.5) * sqrt(15.0)), (z * fma(z, fma(pow(((double) M_PI), 0.5), fma((131.6915934905257 * exp(-7.5)), sqrt(15.0), fma(exp(-7.5), t_7, t_6)), ((z * fma((0.5 * exp(-7.5)), t_5, fma((43.89719783017524 * exp(-7.5)), sqrt(15.0), fma(exp(-7.5), (pow(2.0, 0.5) * fma(t_4, (0.008493827160493827 + fma(0.1288888888888889, t_0, (0.16666666666666666 * pow(t_0, 3.0)))), fma(t_2, t_1, fma(t_3, t_0, (606.6766809167608 * pow(7.5, 0.5)))))), (exp(-7.5) * t_7))))) * pow(((double) M_PI), 0.5))), (pow(((double) M_PI), 0.5) * fma((263.3831869810514 * exp(-7.5)), sqrt(15.0), t_6)))));
}

Julia

function code(z)
	t_0 = Float64(Float64(-1.0 * log(7.5)) - 0.06666666666666667)
	t_1 = Float64(0.1288888888888889 + Float64(0.5 * (t_0 ^ 2.0)))
	t_2 = Float64(436.8961725563396 * (7.5 ^ 0.5))
	t_3 = Float64(545.0353078428827 * (7.5 ^ 0.5))
	t_4 = Float64(263.3831869810514 * (7.5 ^ 0.5))
	t_5 = Float64((2.0 ^ 0.5) * fma(t_4, t_0, t_2))
	t_6 = Float64(exp(-7.5) * t_5)
	t_7 = Float64((2.0 ^ 0.5) * fma(t_4, t_1, fma(t_2, t_0, t_3)))
	return Float64(Float64(pi / sin(Float64(pi * z))) * fma(Float64(263.3831869810514 * exp(Float64(log(pi) * 0.5))), Float64(exp(-7.5) * sqrt(15.0)), Float64(z * fma(z, fma((pi ^ 0.5), fma(Float64(131.6915934905257 * exp(-7.5)), sqrt(15.0), fma(exp(-7.5), t_7, t_6)), Float64(Float64(z * fma(Float64(0.5 * exp(-7.5)), t_5, fma(Float64(43.89719783017524 * exp(-7.5)), sqrt(15.0), fma(exp(-7.5), Float64((2.0 ^ 0.5) * fma(t_4, Float64(0.008493827160493827 + fma(0.1288888888888889, t_0, Float64(0.16666666666666666 * (t_0 ^ 3.0)))), fma(t_2, t_1, fma(t_3, t_0, Float64(606.6766809167608 * (7.5 ^ 0.5)))))), Float64(exp(-7.5) * t_7))))) * (pi ^ 0.5))), Float64((pi ^ 0.5) * fma(Float64(263.3831869810514 * exp(-7.5)), sqrt(15.0), t_6))))))
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(-1.0 * N[Log[7.5], $MachinePrecision]), $MachinePrecision] - 0.06666666666666667), $MachinePrecision]}, Block[{t$95$1 = N[(0.1288888888888889 + N[(0.5 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(436.8961725563396 * N[Power[7.5, 0.5], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(545.0353078428827 * N[Power[7.5, 0.5], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(263.3831869810514 * N[Power[7.5, 0.5], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[2.0, 0.5], $MachinePrecision] * N[(t$95$4 * t$95$0 + t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Exp[-7.5], $MachinePrecision] * t$95$5), $MachinePrecision]}, Block[{t$95$7 = N[(N[Power[2.0, 0.5], $MachinePrecision] * N[(t$95$4 * t$95$1 + N[(t$95$2 * t$95$0 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(263.3831869810514 * N[Exp[N[(N[Log[Pi], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] * N[Sqrt[15.0], $MachinePrecision]), $MachinePrecision] + N[(z * N[(z * N[(N[Power[Pi, 0.5], $MachinePrecision] * N[(N[(131.6915934905257 * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision] * N[Sqrt[15.0], $MachinePrecision] + N[(N[Exp[-7.5], $MachinePrecision] * t$95$7 + t$95$6), $MachinePrecision]), $MachinePrecision] + N[(N[(z * N[(N[(0.5 * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision] * t$95$5 + N[(N[(43.89719783017524 * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision] * N[Sqrt[15.0], $MachinePrecision] + N[(N[Exp[-7.5], $MachinePrecision] * N[(N[Power[2.0, 0.5], $MachinePrecision] * N[(t$95$4 * N[(0.008493827160493827 + N[(0.1288888888888889 * t$95$0 + N[(0.16666666666666666 * N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * t$95$1 + N[(t$95$3 * t$95$0 + N[(606.6766809167608 * N[Power[7.5, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Exp[-7.5], $MachinePrecision] * t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[Pi, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[Pi, 0.5], $MachinePrecision] * N[(N[(263.3831869810514 * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision] * N[Sqrt[15.0], $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Initial program 96.2%

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

3. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)\right)\right) + z \cdot \left(z \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(\frac{1106209385320415913103082059}{8400000000000000000000000} \cdot \left(e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)\right) + \left(e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(\frac{29}{225} + \frac{1}{2} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{2}\right)\right) + \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)\right) + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \sqrt{\frac{15}{2}}\right)\right)\right) + e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)\right) + \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \sqrt{\frac{15}{2}}\right)\right)\right)\right) + \left(z \cdot \left(\frac{1}{2} \cdot \left(e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)\right) + \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \sqrt{\frac{15}{2}}\right)\right)\right) + \left(\frac{1106209385320415913103082059}{25200000000000000000000000} \cdot \left(e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)\right) + \left(e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(\frac{86}{10125} + \left(\frac{29}{225} \cdot \left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right) + \frac{1}{6} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{3}\right)\right)\right) + \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(\frac{29}{225} + \frac{1}{2} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{2}\right)\right) + \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)\right) + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \sqrt{\frac{15}{2}}\right)\right)\right)\right) + e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(\frac{29}{225} + \frac{1}{2} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{2}\right)\right) + \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)\right) + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \sqrt{\frac{15}{2}}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) + \sqrt{\mathsf{PI}\left(\right)} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)\right) + e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)\right) + \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \sqrt{\frac{15}{2}}\right)\right)\right)\right)\right)} \]

5. Applied rewrites 97.1%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\mathsf{fma}\left(263.3831869810514 \cdot {\pi}^{0.5}, e^{-7.5} \cdot \sqrt{15}, z \cdot \mathsf{fma}\left(z, \mathsf{fma}\left({\pi}^{0.5}, \mathsf{fma}\left(131.6915934905257 \cdot e^{-7.5}, \sqrt{15}, \mathsf{fma}\left(e^{-7.5}, {2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, 0.1288888888888889 + 0.5 \cdot {\left(-1 \cdot \log 7.5 - 0.06666666666666667\right)}^{2}, \mathsf{fma}\left(436.8961725563396 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 545.0353078428827 \cdot {7.5}^{0.5}\right)\right), e^{-7.5} \cdot \left({2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 436.8961725563396 \cdot {7.5}^{0.5}\right)\right)\right)\right), \left(z \cdot \mathsf{fma}\left(0.5 \cdot e^{-7.5}, {2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 436.8961725563396 \cdot {7.5}^{0.5}\right), \mathsf{fma}\left(43.89719783017524 \cdot e^{-7.5}, \sqrt{15}, \mathsf{fma}\left(e^{-7.5}, {2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, 0.008493827160493827 + \mathsf{fma}\left(0.1288888888888889, -1 \cdot \log 7.5 - 0.06666666666666667, 0.16666666666666666 \cdot {\left(-1 \cdot \log 7.5 - 0.06666666666666667\right)}^{3}\right), \mathsf{fma}\left(436.8961725563396 \cdot {7.5}^{0.5}, 0.1288888888888889 + 0.5 \cdot {\left(-1 \cdot \log 7.5 - 0.06666666666666667\right)}^{2}, \mathsf{fma}\left(545.0353078428827 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 606.6766809167608 \cdot {7.5}^{0.5}\right)\right)\right), e^{-7.5} \cdot \left({2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, 0.1288888888888889 + 0.5 \cdot {\left(-1 \cdot \log 7.5 - 0.06666666666666667\right)}^{2}, \mathsf{fma}\left(436.8961725563396 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 545.0353078428827 \cdot {7.5}^{0.5}\right)\right)\right)\right)\right)\right)\right) \cdot {\pi}^{0.5}\right), {\pi}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot e^{-7.5}, \sqrt{15}, e^{-7.5} \cdot \left({2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 436.8961725563396 \cdot {7.5}^{0.5}\right)\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{2}}, e^{\frac{-15}{2}} \cdot \sqrt{15}, z \cdot \mathsf{fma}\left(z, \mathsf{fma}\left({\pi}^{\frac{1}{2}}, \mathsf{fma}\left(\frac{1106209385320415913103082059}{8400000000000000000000000} \cdot e^{\frac{-15}{2}}, \sqrt{15}, \mathsf{fma}\left(e^{\frac{-15}{2}}, {2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, \frac{29}{225} + \frac{1}{2} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{2}, \mathsf{fma}\left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right), e^{\frac{-15}{2}} \cdot \left({2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)\right)\right), \left(z \cdot \mathsf{fma}\left(\frac{1}{2} \cdot e^{\frac{-15}{2}}, {2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right), \mathsf{fma}\left(\frac{1106209385320415913103082059}{25200000000000000000000000} \cdot e^{\frac{-15}{2}}, \sqrt{15}, \mathsf{fma}\left(e^{\frac{-15}{2}}, {2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, \frac{86}{10125} + \mathsf{fma}\left(\frac{29}{225}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{1}{6} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{3}\right), \mathsf{fma}\left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, \frac{29}{225} + \frac{1}{2} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{2}, \mathsf{fma}\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)\right), e^{\frac{-15}{2}} \cdot \left({2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, \frac{29}{225} + \frac{1}{2} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{2}, \mathsf{fma}\left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)\right)\right)\right)\right)\right) \cdot {\pi}^{\frac{1}{2}}\right), {\pi}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot e^{\frac{-15}{2}}, \sqrt{15}, e^{\frac{-15}{2}} \cdot \left({2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)\right)\right)\right) \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{2}}, e^{\frac{-15}{2}} \cdot \sqrt{15}, z \cdot \mathsf{fma}\left(z, \mathsf{fma}\left({\pi}^{\frac{1}{2}}, \mathsf{fma}\left(\frac{1106209385320415913103082059}{8400000000000000000000000} \cdot e^{\frac{-15}{2}}, \sqrt{15}, \mathsf{fma}\left(e^{\frac{-15}{2}}, {2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, \frac{29}{225} + \frac{1}{2} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{2}, \mathsf{fma}\left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right), e^{\frac{-15}{2}} \cdot \left({2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)\right)\right), \left(z \cdot \mathsf{fma}\left(\frac{1}{2} \cdot e^{\frac{-15}{2}}, {2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right), \mathsf{fma}\left(\frac{1106209385320415913103082059}{25200000000000000000000000} \cdot e^{\frac{-15}{2}}, \sqrt{15}, \mathsf{fma}\left(e^{\frac{-15}{2}}, {2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, \frac{86}{10125} + \mathsf{fma}\left(\frac{29}{225}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{1}{6} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{3}\right), \mathsf{fma}\left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, \frac{29}{225} + \frac{1}{2} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{2}, \mathsf{fma}\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)\right), e^{\frac{-15}{2}} \cdot \left({2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, \frac{29}{225} + \frac{1}{2} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{2}, \mathsf{fma}\left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)\right)\right)\right)\right)\right) \cdot {\pi}^{\frac{1}{2}}\right), {\pi}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot e^{\frac{-15}{2}}, \sqrt{15}, e^{\frac{-15}{2}} \cdot \left({2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)\right)\right)\right) \]

   3. pow-to-exp N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot e^{\log \mathsf{PI}\left(\right) \cdot \frac{1}{2}}, e^{\frac{-15}{2}} \cdot \sqrt{15}, z \cdot \mathsf{fma}\left(z, \mathsf{fma}\left({\pi}^{\frac{1}{2}}, \mathsf{fma}\left(\frac{1106209385320415913103082059}{8400000000000000000000000} \cdot e^{\frac{-15}{2}}, \sqrt{15}, \mathsf{fma}\left(e^{\frac{-15}{2}}, {2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, \frac{29}{225} + \frac{1}{2} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{2}, \mathsf{fma}\left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right), e^{\frac{-15}{2}} \cdot \left({2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)\right)\right), \left(z \cdot \mathsf{fma}\left(\frac{1}{2} \cdot e^{\frac{-15}{2}}, {2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right), \mathsf{fma}\left(\frac{1106209385320415913103082059}{25200000000000000000000000} \cdot e^{\frac{-15}{2}}, \sqrt{15}, \mathsf{fma}\left(e^{\frac{-15}{2}}, {2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, \frac{86}{10125} + \mathsf{fma}\left(\frac{29}{225}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{1}{6} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{3}\right), \mathsf{fma}\left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, \frac{29}{225} + \frac{1}{2} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{2}, \mathsf{fma}\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)\right), e^{\frac{-15}{2}} \cdot \left({2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, \frac{29}{225} + \frac{1}{2} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{2}, \mathsf{fma}\left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)\right)\right)\right)\right)\right) \cdot {\pi}^{\frac{1}{2}}\right), {\pi}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot e^{\frac{-15}{2}}, \sqrt{15}, e^{\frac{-15}{2}} \cdot \left({2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)\right)\right)\right) \]

   4. lower-exp.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot e^{\log \mathsf{PI}\left(\right) \cdot \frac{1}{2}}, e^{\frac{-15}{2}} \cdot \sqrt{15}, z \cdot \mathsf{fma}\left(z, \mathsf{fma}\left({\pi}^{\frac{1}{2}}, \mathsf{fma}\left(\frac{1106209385320415913103082059}{8400000000000000000000000} \cdot e^{\frac{-15}{2}}, \sqrt{15}, \mathsf{fma}\left(e^{\frac{-15}{2}}, {2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, \frac{29}{225} + \frac{1}{2} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{2}, \mathsf{fma}\left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right), e^{\frac{-15}{2}} \cdot \left({2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)\right)\right), \left(z \cdot \mathsf{fma}\left(\frac{1}{2} \cdot e^{\frac{-15}{2}}, {2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right), \mathsf{fma}\left(\frac{1106209385320415913103082059}{25200000000000000000000000} \cdot e^{\frac{-15}{2}}, \sqrt{15}, \mathsf{fma}\left(e^{\frac{-15}{2}}, {2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, \frac{86}{10125} + \mathsf{fma}\left(\frac{29}{225}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{1}{6} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{3}\right), \mathsf{fma}\left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, \frac{29}{225} + \frac{1}{2} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{2}, \mathsf{fma}\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)\right), e^{\frac{-15}{2}} \cdot \left({2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, \frac{29}{225} + \frac{1}{2} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{2}, \mathsf{fma}\left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)\right)\right)\right)\right)\right) \cdot {\pi}^{\frac{1}{2}}\right), {\pi}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot e^{\frac{-15}{2}}, \sqrt{15}, e^{\frac{-15}{2}} \cdot \left({2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)\right)\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot e^{\log \mathsf{PI}\left(\right) \cdot \frac{1}{2}}, e^{\frac{-15}{2}} \cdot \sqrt{15}, z \cdot \mathsf{fma}\left(z, \mathsf{fma}\left({\pi}^{\frac{1}{2}}, \mathsf{fma}\left(\frac{1106209385320415913103082059}{8400000000000000000000000} \cdot e^{\frac{-15}{2}}, \sqrt{15}, \mathsf{fma}\left(e^{\frac{-15}{2}}, {2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, \frac{29}{225} + \frac{1}{2} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{2}, \mathsf{fma}\left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right), e^{\frac{-15}{2}} \cdot \left({2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)\right)\right), \left(z \cdot \mathsf{fma}\left(\frac{1}{2} \cdot e^{\frac{-15}{2}}, {2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right), \mathsf{fma}\left(\frac{1106209385320415913103082059}{25200000000000000000000000} \cdot e^{\frac{-15}{2}}, \sqrt{15}, \mathsf{fma}\left(e^{\frac{-15}{2}}, {2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, \frac{86}{10125} + \mathsf{fma}\left(\frac{29}{225}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{1}{6} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{3}\right), \mathsf{fma}\left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, \frac{29}{225} + \frac{1}{2} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{2}, \mathsf{fma}\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)\right), e^{\frac{-15}{2}} \cdot \left({2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, \frac{29}{225} + \frac{1}{2} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{2}, \mathsf{fma}\left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)\right)\right)\right)\right)\right) \cdot {\pi}^{\frac{1}{2}}\right), {\pi}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot e^{\frac{-15}{2}}, \sqrt{15}, e^{\frac{-15}{2}} \cdot \left({2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)\right)\right)\right) \]

   6. lower-log.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot e^{\log \mathsf{PI}\left(\right) \cdot \frac{1}{2}}, e^{\frac{-15}{2}} \cdot \sqrt{15}, z \cdot \mathsf{fma}\left(z, \mathsf{fma}\left({\pi}^{\frac{1}{2}}, \mathsf{fma}\left(\frac{1106209385320415913103082059}{8400000000000000000000000} \cdot e^{\frac{-15}{2}}, \sqrt{15}, \mathsf{fma}\left(e^{\frac{-15}{2}}, {2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, \frac{29}{225} + \frac{1}{2} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{2}, \mathsf{fma}\left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right), e^{\frac{-15}{2}} \cdot \left({2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)\right)\right), \left(z \cdot \mathsf{fma}\left(\frac{1}{2} \cdot e^{\frac{-15}{2}}, {2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right), \mathsf{fma}\left(\frac{1106209385320415913103082059}{25200000000000000000000000} \cdot e^{\frac{-15}{2}}, \sqrt{15}, \mathsf{fma}\left(e^{\frac{-15}{2}}, {2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, \frac{86}{10125} + \mathsf{fma}\left(\frac{29}{225}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{1}{6} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{3}\right), \mathsf{fma}\left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, \frac{29}{225} + \frac{1}{2} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{2}, \mathsf{fma}\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)\right), e^{\frac{-15}{2}} \cdot \left({2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, \frac{29}{225} + \frac{1}{2} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{2}, \mathsf{fma}\left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)\right)\right)\right)\right)\right) \cdot {\pi}^{\frac{1}{2}}\right), {\pi}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot e^{\frac{-15}{2}}, \sqrt{15}, e^{\frac{-15}{2}} \cdot \left({2}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}, -1 \cdot \log \frac{15}{2} - \frac{1}{15}, \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)\right)\right)\right)\right) \]

   7. lift-PI.f64 97.9

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \mathsf{fma}\left(263.3831869810514 \cdot e^{\log \pi \cdot 0.5}, e^{-7.5} \cdot \sqrt{15}, z \cdot \mathsf{fma}\left(z, \mathsf{fma}\left({\pi}^{0.5}, \mathsf{fma}\left(131.6915934905257 \cdot e^{-7.5}, \sqrt{15}, \mathsf{fma}\left(e^{-7.5}, {2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, 0.1288888888888889 + 0.5 \cdot {\left(-1 \cdot \log 7.5 - 0.06666666666666667\right)}^{2}, \mathsf{fma}\left(436.8961725563396 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 545.0353078428827 \cdot {7.5}^{0.5}\right)\right), e^{-7.5} \cdot \left({2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 436.8961725563396 \cdot {7.5}^{0.5}\right)\right)\right)\right), \left(z \cdot \mathsf{fma}\left(0.5 \cdot e^{-7.5}, {2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 436.8961725563396 \cdot {7.5}^{0.5}\right), \mathsf{fma}\left(43.89719783017524 \cdot e^{-7.5}, \sqrt{15}, \mathsf{fma}\left(e^{-7.5}, {2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, 0.008493827160493827 + \mathsf{fma}\left(0.1288888888888889, -1 \cdot \log 7.5 - 0.06666666666666667, 0.16666666666666666 \cdot {\left(-1 \cdot \log 7.5 - 0.06666666666666667\right)}^{3}\right), \mathsf{fma}\left(436.8961725563396 \cdot {7.5}^{0.5}, 0.1288888888888889 + 0.5 \cdot {\left(-1 \cdot \log 7.5 - 0.06666666666666667\right)}^{2}, \mathsf{fma}\left(545.0353078428827 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 606.6766809167608 \cdot {7.5}^{0.5}\right)\right)\right), e^{-7.5} \cdot \left({2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, 0.1288888888888889 + 0.5 \cdot {\left(-1 \cdot \log 7.5 - 0.06666666666666667\right)}^{2}, \mathsf{fma}\left(436.8961725563396 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 545.0353078428827 \cdot {7.5}^{0.5}\right)\right)\right)\right)\right)\right)\right) \cdot {\pi}^{0.5}\right), {\pi}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot e^{-7.5}, \sqrt{15}, e^{-7.5} \cdot \left({2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 436.8961725563396 \cdot {7.5}^{0.5}\right)\right)\right)\right)\right) \]

7. Applied rewrites 97.9%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \mathsf{fma}\left(263.3831869810514 \cdot e^{\log \pi \cdot 0.5}, e^{-7.5} \cdot \sqrt{15}, z \cdot \mathsf{fma}\left(z, \mathsf{fma}\left({\pi}^{0.5}, \mathsf{fma}\left(131.6915934905257 \cdot e^{-7.5}, \sqrt{15}, \mathsf{fma}\left(e^{-7.5}, {2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, 0.1288888888888889 + 0.5 \cdot {\left(-1 \cdot \log 7.5 - 0.06666666666666667\right)}^{2}, \mathsf{fma}\left(436.8961725563396 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 545.0353078428827 \cdot {7.5}^{0.5}\right)\right), e^{-7.5} \cdot \left({2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 436.8961725563396 \cdot {7.5}^{0.5}\right)\right)\right)\right), \left(z \cdot \mathsf{fma}\left(0.5 \cdot e^{-7.5}, {2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 436.8961725563396 \cdot {7.5}^{0.5}\right), \mathsf{fma}\left(43.89719783017524 \cdot e^{-7.5}, \sqrt{15}, \mathsf{fma}\left(e^{-7.5}, {2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, 0.008493827160493827 + \mathsf{fma}\left(0.1288888888888889, -1 \cdot \log 7.5 - 0.06666666666666667, 0.16666666666666666 \cdot {\left(-1 \cdot \log 7.5 - 0.06666666666666667\right)}^{3}\right), \mathsf{fma}\left(436.8961725563396 \cdot {7.5}^{0.5}, 0.1288888888888889 + 0.5 \cdot {\left(-1 \cdot \log 7.5 - 0.06666666666666667\right)}^{2}, \mathsf{fma}\left(545.0353078428827 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 606.6766809167608 \cdot {7.5}^{0.5}\right)\right)\right), e^{-7.5} \cdot \left({2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, 0.1288888888888889 + 0.5 \cdot {\left(-1 \cdot \log 7.5 - 0.06666666666666667\right)}^{2}, \mathsf{fma}\left(436.8961725563396 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 545.0353078428827 \cdot {7.5}^{0.5}\right)\right)\right)\right)\right)\right)\right) \cdot {\pi}^{0.5}\right), {\pi}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot e^{-7.5}, \sqrt{15}, e^{-7.5} \cdot \left({2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 436.8961725563396 \cdot {7.5}^{0.5}\right)\right)\right)\right)\right) \]
8. Add Preprocessing

Alternative 2: 96.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 \cdot \log 7.5 - 0.06666666666666667\\ t_1 := e^{-7.5} \cdot \sqrt{15}\\ t_2 := \frac{t\_1}{\pi}\\ t_3 := 436.8961725563396 \cdot {7.5}^{0.5}\\ t_4 := 545.0353078428827 \cdot {7.5}^{0.5}\\ t_5 := 263.3831869810514 \cdot {7.5}^{0.5}\\ t_6 := {\left({\pi}^{3}\right)}^{0.5}\\ t_7 := 0.1288888888888889 + 0.5 \cdot {t\_0}^{2}\\ t_8 := {2}^{0.5} \cdot \mathsf{fma}\left(t\_5, t\_7, \mathsf{fma}\left(t\_3, t\_0, t\_4\right)\right)\\ t_9 := e^{-7.5} \cdot \left({2}^{0.5} \cdot \mathsf{fma}\left(t\_5, t\_0, t\_3\right)\right)\\ t_10 := \frac{t\_9}{\pi}\\ t_11 := \mathsf{fma}\left(263.3831869810514, t\_2, t\_10\right)\\ \frac{\mathsf{fma}\left(263.3831869810514 \cdot {\pi}^{0.5}, t\_1, z \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(t\_6, \mathsf{fma}\left(131.6915934905257, t\_2, \frac{\mathsf{fma}\left(e^{-7.5}, t\_8, t\_9\right)}{\pi} - \left(-43.89719783017524 \cdot \pi\right) \cdot t\_1\right), \left(z \cdot \left(\mathsf{fma}\left(0.5, t\_10, \mathsf{fma}\left(43.89719783017524, t\_2, \frac{\mathsf{fma}\left(e^{-7.5}, {2}^{0.5} \cdot \mathsf{fma}\left(t\_5, 0.008493827160493827 + \mathsf{fma}\left(0.1288888888888889, t\_0, 0.16666666666666666 \cdot {t\_0}^{3}\right), \mathsf{fma}\left(t\_3, t\_7, \mathsf{fma}\left(t\_4, t\_0, 606.6766809167608 \cdot {7.5}^{0.5}\right)\right)\right), e^{-7.5} \cdot t\_8\right)}{\pi}\right)\right) - \left(-0.16666666666666666 \cdot \left(\pi \cdot \pi\right)\right) \cdot t\_11\right)\right) \cdot t\_6\right), t\_6 \cdot t\_11\right)\right)}{z} \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (* -1.0 (log 7.5)) 0.06666666666666667))
        (t_1 (* (exp -7.5) (sqrt 15.0)))
        (t_2 (/ t_1 PI))
        (t_3 (* 436.8961725563396 (pow 7.5 0.5)))
        (t_4 (* 545.0353078428827 (pow 7.5 0.5)))
        (t_5 (* 263.3831869810514 (pow 7.5 0.5)))
        (t_6 (pow (pow PI 3.0) 0.5))
        (t_7 (+ 0.1288888888888889 (* 0.5 (pow t_0 2.0))))
        (t_8 (* (pow 2.0 0.5) (fma t_5 t_7 (fma t_3 t_0 t_4))))
        (t_9 (* (exp -7.5) (* (pow 2.0 0.5) (fma t_5 t_0 t_3))))
        (t_10 (/ t_9 PI))
        (t_11 (fma 263.3831869810514 t_2 t_10)))
   (/
    (fma
     (* 263.3831869810514 (pow PI 0.5))
     t_1
     (*
      z
      (fma
       z
       (fma
        t_6
        (fma
         131.6915934905257
         t_2
         (- (/ (fma (exp -7.5) t_8 t_9) PI) (* (* -43.89719783017524 PI) t_1)))
        (*
         (*
          z
          (-
           (fma
            0.5
            t_10
            (fma
             43.89719783017524
             t_2
             (/
              (fma
               (exp -7.5)
               (*
                (pow 2.0 0.5)
                (fma
                 t_5
                 (+
                  0.008493827160493827
                  (fma
                   0.1288888888888889
                   t_0
                   (* 0.16666666666666666 (pow t_0 3.0))))
                 (fma
                  t_3
                  t_7
                  (fma t_4 t_0 (* 606.6766809167608 (pow 7.5 0.5))))))
               (* (exp -7.5) t_8))
              PI)))
           (* (* -0.16666666666666666 (* PI PI)) t_11)))
         t_6))
       (* t_6 t_11))))
    z)))

C

double code(double z) {
	double t_0 = (-1.0 * log(7.5)) - 0.06666666666666667;
	double t_1 = exp(-7.5) * sqrt(15.0);
	double t_2 = t_1 / ((double) M_PI);
	double t_3 = 436.8961725563396 * pow(7.5, 0.5);
	double t_4 = 545.0353078428827 * pow(7.5, 0.5);
	double t_5 = 263.3831869810514 * pow(7.5, 0.5);
	double t_6 = pow(pow(((double) M_PI), 3.0), 0.5);
	double t_7 = 0.1288888888888889 + (0.5 * pow(t_0, 2.0));
	double t_8 = pow(2.0, 0.5) * fma(t_5, t_7, fma(t_3, t_0, t_4));
	double t_9 = exp(-7.5) * (pow(2.0, 0.5) * fma(t_5, t_0, t_3));
	double t_10 = t_9 / ((double) M_PI);
	double t_11 = fma(263.3831869810514, t_2, t_10);
	return fma((263.3831869810514 * pow(((double) M_PI), 0.5)), t_1, (z * fma(z, fma(t_6, fma(131.6915934905257, t_2, ((fma(exp(-7.5), t_8, t_9) / ((double) M_PI)) - ((-43.89719783017524 * ((double) M_PI)) * t_1))), ((z * (fma(0.5, t_10, fma(43.89719783017524, t_2, (fma(exp(-7.5), (pow(2.0, 0.5) * fma(t_5, (0.008493827160493827 + fma(0.1288888888888889, t_0, (0.16666666666666666 * pow(t_0, 3.0)))), fma(t_3, t_7, fma(t_4, t_0, (606.6766809167608 * pow(7.5, 0.5)))))), (exp(-7.5) * t_8)) / ((double) M_PI)))) - ((-0.16666666666666666 * (((double) M_PI) * ((double) M_PI))) * t_11))) * t_6)), (t_6 * t_11)))) / z;
}

Julia

function code(z)
	t_0 = Float64(Float64(-1.0 * log(7.5)) - 0.06666666666666667)
	t_1 = Float64(exp(-7.5) * sqrt(15.0))
	t_2 = Float64(t_1 / pi)
	t_3 = Float64(436.8961725563396 * (7.5 ^ 0.5))
	t_4 = Float64(545.0353078428827 * (7.5 ^ 0.5))
	t_5 = Float64(263.3831869810514 * (7.5 ^ 0.5))
	t_6 = (pi ^ 3.0) ^ 0.5
	t_7 = Float64(0.1288888888888889 + Float64(0.5 * (t_0 ^ 2.0)))
	t_8 = Float64((2.0 ^ 0.5) * fma(t_5, t_7, fma(t_3, t_0, t_4)))
	t_9 = Float64(exp(-7.5) * Float64((2.0 ^ 0.5) * fma(t_5, t_0, t_3)))
	t_10 = Float64(t_9 / pi)
	t_11 = fma(263.3831869810514, t_2, t_10)
	return Float64(fma(Float64(263.3831869810514 * (pi ^ 0.5)), t_1, Float64(z * fma(z, fma(t_6, fma(131.6915934905257, t_2, Float64(Float64(fma(exp(-7.5), t_8, t_9) / pi) - Float64(Float64(-43.89719783017524 * pi) * t_1))), Float64(Float64(z * Float64(fma(0.5, t_10, fma(43.89719783017524, t_2, Float64(fma(exp(-7.5), Float64((2.0 ^ 0.5) * fma(t_5, Float64(0.008493827160493827 + fma(0.1288888888888889, t_0, Float64(0.16666666666666666 * (t_0 ^ 3.0)))), fma(t_3, t_7, fma(t_4, t_0, Float64(606.6766809167608 * (7.5 ^ 0.5)))))), Float64(exp(-7.5) * t_8)) / pi))) - Float64(Float64(-0.16666666666666666 * Float64(pi * pi)) * t_11))) * t_6)), Float64(t_6 * t_11)))) / z)
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(-1.0 * N[Log[7.5], $MachinePrecision]), $MachinePrecision] - 0.06666666666666667), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[-7.5], $MachinePrecision] * N[Sqrt[15.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / Pi), $MachinePrecision]}, Block[{t$95$3 = N[(436.8961725563396 * N[Power[7.5, 0.5], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(545.0353078428827 * N[Power[7.5, 0.5], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(263.3831869810514 * N[Power[7.5, 0.5], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Power[N[Power[Pi, 3.0], $MachinePrecision], 0.5], $MachinePrecision]}, Block[{t$95$7 = N[(0.1288888888888889 + N[(0.5 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(N[Power[2.0, 0.5], $MachinePrecision] * N[(t$95$5 * t$95$7 + N[(t$95$3 * t$95$0 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[(N[Exp[-7.5], $MachinePrecision] * N[(N[Power[2.0, 0.5], $MachinePrecision] * N[(t$95$5 * t$95$0 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$10 = N[(t$95$9 / Pi), $MachinePrecision]}, Block[{t$95$11 = N[(263.3831869810514 * t$95$2 + t$95$10), $MachinePrecision]}, N[(N[(N[(263.3831869810514 * N[Power[Pi, 0.5], $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(z * N[(z * N[(t$95$6 * N[(131.6915934905257 * t$95$2 + N[(N[(N[(N[Exp[-7.5], $MachinePrecision] * t$95$8 + t$95$9), $MachinePrecision] / Pi), $MachinePrecision] - N[(N[(-43.89719783017524 * Pi), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z * N[(N[(0.5 * t$95$10 + N[(43.89719783017524 * t$95$2 + N[(N[(N[Exp[-7.5], $MachinePrecision] * N[(N[Power[2.0, 0.5], $MachinePrecision] * N[(t$95$5 * N[(0.008493827160493827 + N[(0.1288888888888889 * t$95$0 + N[(0.16666666666666666 * N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * t$95$7 + N[(t$95$4 * t$95$0 + N[(606.6766809167608 * N[Power[7.5, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Exp[-7.5], $MachinePrecision] * t$95$8), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.16666666666666666 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * t$95$11), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$6), $MachinePrecision]), $MachinePrecision] + N[(t$95$6 * t$95$11), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Initial program 96.2%

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

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)\right)\right) + z \cdot \left(z \cdot \left(\sqrt{{\mathsf{PI}\left(\right)}^{3}} \cdot \left(\left(\frac{1106209385320415913103082059}{8400000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{\mathsf{PI}\left(\right)} + \left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(\frac{29}{225} + \frac{1}{2} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{2}\right)\right) + \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)\right) + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \sqrt{\frac{15}{2}}\right)\right)\right)}{\mathsf{PI}\left(\right)} + \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)\right) + \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \sqrt{\frac{15}{2}}\right)\right)}{\mathsf{PI}\left(\right)}\right)\right) - \frac{-1106209385320415913103082059}{25200000000000000000000000} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)\right)\right)\right) + \left(z \cdot \left(\left(\frac{1}{2} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)\right) + \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \sqrt{\frac{15}{2}}\right)\right)}{\mathsf{PI}\left(\right)} + \left(\frac{1106209385320415913103082059}{25200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{\mathsf{PI}\left(\right)} + \left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(\frac{86}{10125} + \left(\frac{29}{225} \cdot \left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right) + \frac{1}{6} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{3}\right)\right)\right) + \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(\frac{29}{225} + \frac{1}{2} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{2}\right)\right) + \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)\right) + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \sqrt{\frac{15}{2}}\right)\right)\right)\right)}{\mathsf{PI}\left(\right)} + \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(\frac{29}{225} + \frac{1}{2} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{2}\right)\right) + \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)\right) + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \sqrt{\frac{15}{2}}\right)\right)\right)}{\mathsf{PI}\left(\right)}\right)\right)\right) - \frac{-1}{6} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{\mathsf{PI}\left(\right)} + \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)\right) + \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \sqrt{\frac{15}{2}}\right)\right)}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \cdot \sqrt{{\mathsf{PI}\left(\right)}^{3}}\right) + \sqrt{{\mathsf{PI}\left(\right)}^{3}} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{\mathsf{PI}\left(\right)} + \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)\right) + \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \sqrt{\frac{15}{2}}\right)\right)}{\mathsf{PI}\left(\right)}\right)\right)}{z}} \]

5. Applied rewrites 97.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(263.3831869810514 \cdot {\pi}^{0.5}, e^{-7.5} \cdot \sqrt{15}, z \cdot \mathsf{fma}\left(z, \mathsf{fma}\left({\left({\pi}^{3}\right)}^{0.5}, \mathsf{fma}\left(131.6915934905257, \frac{e^{-7.5} \cdot \sqrt{15}}{\pi}, \frac{\mathsf{fma}\left(e^{-7.5}, {2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, 0.1288888888888889 + 0.5 \cdot {\left(-1 \cdot \log 7.5 - 0.06666666666666667\right)}^{2}, \mathsf{fma}\left(436.8961725563396 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 545.0353078428827 \cdot {7.5}^{0.5}\right)\right), e^{-7.5} \cdot \left({2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 436.8961725563396 \cdot {7.5}^{0.5}\right)\right)\right)}{\pi} - \left(-43.89719783017524 \cdot \pi\right) \cdot \left(e^{-7.5} \cdot \sqrt{15}\right)\right), \left(z \cdot \left(\mathsf{fma}\left(0.5, \frac{e^{-7.5} \cdot \left({2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 436.8961725563396 \cdot {7.5}^{0.5}\right)\right)}{\pi}, \mathsf{fma}\left(43.89719783017524, \frac{e^{-7.5} \cdot \sqrt{15}}{\pi}, \frac{\mathsf{fma}\left(e^{-7.5}, {2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, 0.008493827160493827 + \mathsf{fma}\left(0.1288888888888889, -1 \cdot \log 7.5 - 0.06666666666666667, 0.16666666666666666 \cdot {\left(-1 \cdot \log 7.5 - 0.06666666666666667\right)}^{3}\right), \mathsf{fma}\left(436.8961725563396 \cdot {7.5}^{0.5}, 0.1288888888888889 + 0.5 \cdot {\left(-1 \cdot \log 7.5 - 0.06666666666666667\right)}^{2}, \mathsf{fma}\left(545.0353078428827 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 606.6766809167608 \cdot {7.5}^{0.5}\right)\right)\right), e^{-7.5} \cdot \left({2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, 0.1288888888888889 + 0.5 \cdot {\left(-1 \cdot \log 7.5 - 0.06666666666666667\right)}^{2}, \mathsf{fma}\left(436.8961725563396 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 545.0353078428827 \cdot {7.5}^{0.5}\right)\right)\right)\right)}{\pi}\right)\right) - \left(-0.16666666666666666 \cdot \left(\pi \cdot \pi\right)\right) \cdot \mathsf{fma}\left(263.3831869810514, \frac{e^{-7.5} \cdot \sqrt{15}}{\pi}, \frac{e^{-7.5} \cdot \left({2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 436.8961725563396 \cdot {7.5}^{0.5}\right)\right)}{\pi}\right)\right)\right) \cdot {\left({\pi}^{3}\right)}^{0.5}\right), {\left({\pi}^{3}\right)}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514, \frac{e^{-7.5} \cdot \sqrt{15}}{\pi}, \frac{e^{-7.5} \cdot \left({2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 436.8961725563396 \cdot {7.5}^{0.5}\right)\right)}{\pi}\right)\right)\right)}{z}} \]
6. Add Preprocessing

Alternative 3: 95.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \left(263.3831869810514 \cdot \frac{e^{-7.5} \cdot \sqrt{15}}{z}\right) \cdot e^{\log \pi \cdot 0.5} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (* 263.3831869810514 (/ (* (exp -7.5) (sqrt 15.0)) z))
  (exp (* (log PI) 0.5))))

C

double code(double z) {
	return (263.3831869810514 * ((exp(-7.5) * sqrt(15.0)) / z)) * exp((log(((double) M_PI)) * 0.5));
}

Java

public static double code(double z) {
	return (263.3831869810514 * ((Math.exp(-7.5) * Math.sqrt(15.0)) / z)) * Math.exp((Math.log(Math.PI) * 0.5));
}

Python

def code(z):
	return (263.3831869810514 * ((math.exp(-7.5) * math.sqrt(15.0)) / z)) * math.exp((math.log(math.pi) * 0.5))

Julia

function code(z)
	return Float64(Float64(263.3831869810514 * Float64(Float64(exp(-7.5) * sqrt(15.0)) / z)) * exp(Float64(log(pi) * 0.5)))
end

MATLAB

function tmp = code(z)
	tmp = (263.3831869810514 * ((exp(-7.5) * sqrt(15.0)) / z)) * exp((log(pi) * 0.5));
end

Wolfram

code[z_] := N[(N[(263.3831869810514 * N[(N[(N[Exp[-7.5], $MachinePrecision] * N[Sqrt[15.0], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Log[Pi], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

3. Taylor expanded in z around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 96.5%

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

   1. lift-PI.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow-to-exp N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-log.f64 N/A

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

   7. lift-PI.f64 97.0

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

7. Applied rewrites 97.0%

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

Reproduce

herbie shell --seed 2025064 
(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))))))