Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.5% → 97.6%

Time: 9.1s

Alternatives: 11

Speedup: 2.1×

• could not determine a ground truth

  1. z = -171014392155.8604

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: 96.5% 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: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := t\_0 + 7\\ t_2 := t\_1 + 0.5\\ \frac{1 + \left(z \cdot z\right) \cdot \left(-1 \cdot \left(\left(z \cdot z\right) \cdot \mathsf{fma}\left(-0.027777777777777776, {\pi}^{4}, 0.008333333333333333 \cdot {\pi}^{4}\right)\right) - -0.16666666666666666 \cdot \left(\pi \cdot \pi\right)\right)}{z} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {t\_2}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{-t\_2}\right) \cdot \left(\left(\left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 - -606.656776085461 \cdot z\right)\right)\right) + \frac{12.507343278686905}{t\_0 + 5}\right) + \frac{-0.13857109526572012}{t\_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_1}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\right)\right) \end{array} \end{array} \]

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)))
   (*
    (/
     (+
      1.0
      (*
       (* z z)
       (-
        (*
         -1.0
         (*
          (* z z)
          (fma
           -0.027777777777777776
           (pow PI 4.0)
           (* 0.008333333333333333 (pow PI 4.0)))))
        (* -0.16666666666666666 (* PI PI)))))
     z)
    (*
     (* (* (* (sqrt PI) (sqrt 2.0)) (pow t_2 (+ t_0 0.5))) (exp (- t_2)))
     (+
      (+
       (+
        (+
         (+
          260.9048120626994
          (*
           z
           (+
            436.3997278161676
            (* z (- 544.9358906000987 (* -606.656776085461 z))))))
         (/ 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 ((1.0 + ((z * z) * ((-1.0 * ((z * z) * fma(-0.027777777777777776, pow(((double) M_PI), 4.0), (0.008333333333333333 * pow(((double) M_PI), 4.0))))) - (-0.16666666666666666 * (((double) M_PI) * ((double) M_PI)))))) / z) * ((((sqrt(((double) M_PI)) * sqrt(2.0)) * pow(t_2, (t_0 + 0.5))) * exp(-t_2)) * (((((260.9048120626994 + (z * (436.3997278161676 + (z * (544.9358906000987 - (-606.656776085461 * z)))))) + (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(Float64(1.0 + Float64(Float64(z * z) * Float64(Float64(-1.0 * Float64(Float64(z * z) * fma(-0.027777777777777776, (pi ^ 4.0), Float64(0.008333333333333333 * (pi ^ 4.0))))) - Float64(-0.16666666666666666 * Float64(pi * pi))))) / z) * Float64(Float64(Float64(Float64(sqrt(pi) * sqrt(2.0)) * (t_2 ^ Float64(t_0 + 0.5))) * exp(Float64(-t_2))) * Float64(Float64(Float64(Float64(Float64(260.9048120626994 + Float64(z * Float64(436.3997278161676 + Float64(z * Float64(544.9358906000987 - Float64(-606.656776085461 * z)))))) + 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

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[(N[(1.0 + N[(N[(z * z), $MachinePrecision] * N[(N[(-1.0 * N[(N[(z * z), $MachinePrecision] * N[(-0.027777777777777776 * N[Power[Pi, 4.0], $MachinePrecision] + N[(0.008333333333333333 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-0.16666666666666666 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * N[(N[(N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[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[(260.9048120626994 + N[(z * N[(436.3997278161676 + N[(z * N[(544.9358906000987 - N[(-606.656776085461 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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]]]]

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} + \frac{62898174544540606049}{103680000000000000} \cdot z\right)\right)\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + \color{blue}{z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} + \frac{62898174544540606049}{103680000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \color{blue}{\left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} + \frac{62898174544540606049}{103680000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + \color{blue}{z \cdot \left(\frac{4708246094784852251}{8640000000000000} + \frac{62898174544540606049}{103680000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \color{blue}{\left(\frac{4708246094784852251}{8640000000000000} + \frac{62898174544540606049}{103680000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. fp-cancel-sign-sub-inv N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{62898174544540606049}{103680000000000000}\right)\right) \cdot z}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower--.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{62898174544540606049}{103680000000000000}\right)\right) \cdot z}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \left(\mathsf{neg}\left(\frac{62898174544540606049}{103680000000000000}\right)\right) \cdot \color{blue}{z}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. metadata-eval 96.5

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 - -606.656776085461 \cdot z\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 96.5%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 - -606.656776085461 \cdot z\right)\right)\right)} + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\sqrt{\pi \cdot 2}} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lift-PI.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)} \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right) \cdot 2}} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. sqrt-prod N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lift-PI.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\color{blue}{\pi}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lift-sqrt.f64 97.3

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \color{blue}{\sqrt{2}}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 - -606.656776085461 \cdot z\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Applied rewrites 97.3%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 - -606.656776085461 \cdot z\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

7. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1 + {z}^{2} \cdot \left(-1 \cdot \left({z}^{2} \cdot \left(\frac{-1}{36} \cdot {\mathsf{PI}\left(\right)}^{4} + \frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right) - \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{z}} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1 + {z}^{2} \cdot \left(-1 \cdot \left({z}^{2} \cdot \left(\frac{-1}{36} \cdot {\mathsf{PI}\left(\right)}^{4} + \frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right) - \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{\color{blue}{z}} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

9. Applied rewrites 97.6%

   \[\leadsto \color{blue}{\frac{1 + \left(z \cdot z\right) \cdot \left(-1 \cdot \left(\left(z \cdot z\right) \cdot \mathsf{fma}\left(-0.027777777777777776, {\pi}^{4}, 0.008333333333333333 \cdot {\pi}^{4}\right)\right) - -0.16666666666666666 \cdot \left(\pi \cdot \pi\right)\right)}{z}} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 - -606.656776085461 \cdot z\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
10. Add Preprocessing

Alternative 2: 97.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := t\_0 + 7\\ t_2 := t\_1 + 0.5\\ \frac{\pi}{z \cdot \left(\pi + -0.16666666666666666 \cdot \left(\left(z \cdot z\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right)\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {t\_2}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{-t\_2}\right) \cdot \left(\left(\left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 - -606.656776085461 \cdot z\right)\right)\right) + \frac{12.507343278686905}{t\_0 + 5}\right) + \frac{-0.13857109526572012}{t\_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_1}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\right)\right) \end{array} \end{array} \]

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 (* z (+ PI (* -0.16666666666666666 (* (* z z) (* (* PI PI) PI))))))
    (*
     (* (* (* (sqrt PI) (sqrt 2.0)) (pow t_2 (+ t_0 0.5))) (exp (- t_2)))
     (+
      (+
       (+
        (+
         (+
          260.9048120626994
          (*
           z
           (+
            436.3997278161676
            (* z (- 544.9358906000987 (* -606.656776085461 z))))))
         (/ 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) / (z * (((double) M_PI) + (-0.16666666666666666 * ((z * z) * ((((double) M_PI) * ((double) M_PI)) * ((double) M_PI))))))) * ((((sqrt(((double) M_PI)) * sqrt(2.0)) * pow(t_2, (t_0 + 0.5))) * exp(-t_2)) * (((((260.9048120626994 + (z * (436.3997278161676 + (z * (544.9358906000987 - (-606.656776085461 * z)))))) + (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 / (z * (Math.PI + (-0.16666666666666666 * ((z * z) * ((Math.PI * Math.PI) * Math.PI)))))) * ((((Math.sqrt(Math.PI) * Math.sqrt(2.0)) * Math.pow(t_2, (t_0 + 0.5))) * Math.exp(-t_2)) * (((((260.9048120626994 + (z * (436.3997278161676 + (z * (544.9358906000987 - (-606.656776085461 * z)))))) + (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 / (z * (math.pi + (-0.16666666666666666 * ((z * z) * ((math.pi * math.pi) * math.pi)))))) * ((((math.sqrt(math.pi) * math.sqrt(2.0)) * math.pow(t_2, (t_0 + 0.5))) * math.exp(-t_2)) * (((((260.9048120626994 + (z * (436.3997278161676 + (z * (544.9358906000987 - (-606.656776085461 * z)))))) + (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 / Float64(z * Float64(pi + Float64(-0.16666666666666666 * Float64(Float64(z * z) * Float64(Float64(pi * pi) * pi)))))) * Float64(Float64(Float64(Float64(sqrt(pi) * sqrt(2.0)) * (t_2 ^ Float64(t_0 + 0.5))) * exp(Float64(-t_2))) * Float64(Float64(Float64(Float64(Float64(260.9048120626994 + Float64(z * Float64(436.3997278161676 + Float64(z * Float64(544.9358906000987 - Float64(-606.656776085461 * z)))))) + 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 / (z * (pi + (-0.16666666666666666 * ((z * z) * ((pi * pi) * pi)))))) * ((((sqrt(pi) * sqrt(2.0)) * (t_2 ^ (t_0 + 0.5))) * exp(-t_2)) * (((((260.9048120626994 + (z * (436.3997278161676 + (z * (544.9358906000987 - (-606.656776085461 * z)))))) + (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[(z * N[(Pi + N[(-0.16666666666666666 * N[(N[(z * z), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[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[(260.9048120626994 + N[(z * N[(436.3997278161676 + N[(z * N[(544.9358906000987 - N[(-606.656776085461 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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]]]]

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} + \frac{62898174544540606049}{103680000000000000} \cdot z\right)\right)\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + \color{blue}{z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} + \frac{62898174544540606049}{103680000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \color{blue}{\left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} + \frac{62898174544540606049}{103680000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + \color{blue}{z \cdot \left(\frac{4708246094784852251}{8640000000000000} + \frac{62898174544540606049}{103680000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \color{blue}{\left(\frac{4708246094784852251}{8640000000000000} + \frac{62898174544540606049}{103680000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. fp-cancel-sign-sub-inv N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{62898174544540606049}{103680000000000000}\right)\right) \cdot z}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower--.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{62898174544540606049}{103680000000000000}\right)\right) \cdot z}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \left(\mathsf{neg}\left(\frac{62898174544540606049}{103680000000000000}\right)\right) \cdot \color{blue}{z}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. metadata-eval 96.5

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 - -606.656776085461 \cdot z\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 96.5%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 - -606.656776085461 \cdot z\right)\right)\right)} + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\sqrt{\pi \cdot 2}} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lift-PI.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)} \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right) \cdot 2}} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. sqrt-prod N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lift-PI.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\color{blue}{\pi}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lift-sqrt.f64 97.3

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \color{blue}{\sqrt{2}}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 - -606.656776085461 \cdot z\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Applied rewrites 97.3%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 - -606.656776085461 \cdot z\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

7. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\color{blue}{z \cdot \left(\mathsf{PI}\left(\right) + \frac{-1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)}} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{z \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \frac{-1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)}} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{z \cdot \left(\mathsf{PI}\left(\right) + \color{blue}{\frac{-1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)}\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lift-PI.f64 N/A

      \[\leadsto \frac{\pi}{z \cdot \left(\pi + \color{blue}{\frac{-1}{6}} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{z \cdot \left(\pi + \frac{-1}{6} \cdot \color{blue}{\left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)}\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{z \cdot \left(\pi + \frac{-1}{6} \cdot \left({z}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{3}}\right)\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. pow2 N/A

      \[\leadsto \frac{\pi}{z \cdot \left(\pi + \frac{-1}{6} \cdot \left(\left(z \cdot z\right) \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{3}\right)\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\pi}{z \cdot \left(\pi + \frac{-1}{6} \cdot \left(\left(z \cdot z\right) \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{3}\right)\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. unpow3 N/A

      \[\leadsto \frac{\pi}{z \cdot \left(\pi + \frac{-1}{6} \cdot \left(\left(z \cdot z\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   9. pow2 N/A

      \[\leadsto \frac{\pi}{z \cdot \left(\pi + \frac{-1}{6} \cdot \left(\left(z \cdot z\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\pi}{z \cdot \left(\pi + \frac{-1}{6} \cdot \left(\left(z \cdot z\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   11. pow2 N/A

       \[\leadsto \frac{\pi}{z \cdot \left(\pi + \frac{-1}{6} \cdot \left(\left(z \cdot z\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\pi}{z \cdot \left(\pi + \frac{-1}{6} \cdot \left(\left(z \cdot z\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   13. lift-PI.f64 N/A

       \[\leadsto \frac{\pi}{z \cdot \left(\pi + \frac{-1}{6} \cdot \left(\left(z \cdot z\right) \cdot \left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   14. lift-PI.f64 N/A

       \[\leadsto \frac{\pi}{z \cdot \left(\pi + \frac{-1}{6} \cdot \left(\left(z \cdot z\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   15. lift-PI.f64 97.4

       \[\leadsto \frac{\pi}{z \cdot \left(\pi + -0.16666666666666666 \cdot \left(\left(z \cdot z\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right)\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 - -606.656776085461 \cdot z\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

9. Applied rewrites 97.4%

   \[\leadsto \frac{\pi}{\color{blue}{z \cdot \left(\pi + -0.16666666666666666 \cdot \left(\left(z \cdot z\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right)\right)}} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 - -606.656776085461 \cdot z\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
10. Add Preprocessing

Alternative 3: 97.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := \left(t\_0 + 7\right) + 0.5\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {t\_1}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{-t\_1}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)) (t_1 (+ (+ t_0 7.0) 0.5)))
   (*
    (/ PI (sin (* PI z)))
    (*
     (* (* (* (sqrt PI) (sqrt 2.0)) (pow t_1 (+ t_0 0.5))) (exp (- t_1)))
     (+
      263.3831869810514
      (*
       z
       (+
        436.8961725563396
        (* z (+ 545.0353078428827 (* 606.6766809167608 z))))))))))

C

double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = (t_0 + 7.0) + 0.5;
	return (((double) M_PI) / sin((((double) M_PI) * z))) * ((((sqrt(((double) M_PI)) * sqrt(2.0)) * pow(t_1, (t_0 + 0.5))) * exp(-t_1)) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z)))))));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = (t_0 + 7.0) + 0.5;
	return (Math.PI / Math.sin((Math.PI * z))) * ((((Math.sqrt(Math.PI) * Math.sqrt(2.0)) * Math.pow(t_1, (t_0 + 0.5))) * Math.exp(-t_1)) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z)))))));
}

Python

def code(z):
	t_0 = (1.0 - z) - 1.0
	t_1 = (t_0 + 7.0) + 0.5
	return (math.pi / math.sin((math.pi * z))) * ((((math.sqrt(math.pi) * math.sqrt(2.0)) * math.pow(t_1, (t_0 + 0.5))) * math.exp(-t_1)) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z)))))))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - 1.0)
	t_1 = Float64(Float64(t_0 + 7.0) + 0.5)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(Float64(sqrt(pi) * sqrt(2.0)) * (t_1 ^ Float64(t_0 + 0.5))) * exp(Float64(-t_1))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * Float64(545.0353078428827 + Float64(606.6766809167608 * z))))))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - 1.0;
	t_1 = (t_0 + 7.0) + 0.5;
	tmp = (pi / sin((pi * z))) * ((((sqrt(pi) * sqrt(2.0)) * (t_1 ^ (t_0 + 0.5))) * exp(-t_1)) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z)))))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = 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[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$1, N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$1)], $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * N[(545.0353078428827 + N[(606.6766809167608 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} + \frac{62898174544540606049}{103680000000000000} \cdot z\right)\right)\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + \color{blue}{z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} + \frac{62898174544540606049}{103680000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \color{blue}{\left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} + \frac{62898174544540606049}{103680000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + \color{blue}{z \cdot \left(\frac{4708246094784852251}{8640000000000000} + \frac{62898174544540606049}{103680000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \color{blue}{\left(\frac{4708246094784852251}{8640000000000000} + \frac{62898174544540606049}{103680000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. fp-cancel-sign-sub-inv N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{62898174544540606049}{103680000000000000}\right)\right) \cdot z}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower--.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{62898174544540606049}{103680000000000000}\right)\right) \cdot z}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \left(\mathsf{neg}\left(\frac{62898174544540606049}{103680000000000000}\right)\right) \cdot \color{blue}{z}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. metadata-eval 96.5

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 - -606.656776085461 \cdot z\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 96.5%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 - -606.656776085461 \cdot z\right)\right)\right)} + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\sqrt{\pi \cdot 2}} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lift-PI.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)} \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right) \cdot 2}} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. sqrt-prod N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lift-PI.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\color{blue}{\pi}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lift-sqrt.f64 97.3

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \color{blue}{\sqrt{2}}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 - -606.656776085461 \cdot z\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Applied rewrites 97.3%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 - -606.656776085461 \cdot z\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

7. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)}\right) \]
8. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + \color{blue}{z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \color{blue}{\left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \color{blue}{z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)}\right)\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \color{blue}{\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)}\right)\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \color{blue}{\frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z}\right)\right)\right)\right) \]

   6. lower-*.f64 97.3

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot \color{blue}{z}\right)\right)\right)\right) \]

9. Applied rewrites 97.3%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)}\right) \]
10. Add Preprocessing

Alternative 4: 96.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = (t_0 + 7.0) + 0.5;
	return (((double) M_PI) / sin((((double) M_PI) * z))) * ((((sqrt(((double) M_PI)) * sqrt(2.0)) * pow(t_1, (t_0 + 0.5))) * exp(-t_1)) * (263.3831869810514 + (436.8961725563396 * z)));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = (t_0 + 7.0) + 0.5;
	return (Math.PI / Math.sin((Math.PI * z))) * ((((Math.sqrt(Math.PI) * Math.sqrt(2.0)) * Math.pow(t_1, (t_0 + 0.5))) * Math.exp(-t_1)) * (263.3831869810514 + (436.8961725563396 * z)));
}

Python

def code(z):
	t_0 = (1.0 - z) - 1.0
	t_1 = (t_0 + 7.0) + 0.5
	return (math.pi / math.sin((math.pi * z))) * ((((math.sqrt(math.pi) * math.sqrt(2.0)) * math.pow(t_1, (t_0 + 0.5))) * math.exp(-t_1)) * (263.3831869810514 + (436.8961725563396 * z)))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - 1.0)
	t_1 = Float64(Float64(t_0 + 7.0) + 0.5)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(Float64(sqrt(pi) * sqrt(2.0)) * (t_1 ^ Float64(t_0 + 0.5))) * exp(Float64(-t_1))) * Float64(263.3831869810514 + Float64(436.8961725563396 * z))))
end

MATLAB

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

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = 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[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$1, N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$1)], $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(436.8961725563396 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} + \frac{62898174544540606049}{103680000000000000} \cdot z\right)\right)\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + \color{blue}{z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} + \frac{62898174544540606049}{103680000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \color{blue}{\left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} + \frac{62898174544540606049}{103680000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + \color{blue}{z \cdot \left(\frac{4708246094784852251}{8640000000000000} + \frac{62898174544540606049}{103680000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \color{blue}{\left(\frac{4708246094784852251}{8640000000000000} + \frac{62898174544540606049}{103680000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. fp-cancel-sign-sub-inv N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{62898174544540606049}{103680000000000000}\right)\right) \cdot z}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower--.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{62898174544540606049}{103680000000000000}\right)\right) \cdot z}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \left(\mathsf{neg}\left(\frac{62898174544540606049}{103680000000000000}\right)\right) \cdot \color{blue}{z}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. metadata-eval 96.5

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 - -606.656776085461 \cdot z\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 96.5%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 - -606.656776085461 \cdot z\right)\right)\right)} + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\sqrt{\pi \cdot 2}} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lift-PI.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)} \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right) \cdot 2}} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. sqrt-prod N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lift-PI.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\color{blue}{\pi}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} - \frac{-62898174544540606049}{103680000000000000} \cdot z\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lift-sqrt.f64 97.3

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\pi} \cdot \color{blue}{\sqrt{2}}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 - -606.656776085461 \cdot z\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Applied rewrites 97.3%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 - -606.656776085461 \cdot z\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

7. Taylor expanded in z around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 96.8

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

9. Applied rewrites 96.8%

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

Alternative 5: 96.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left(e^{\log \left(7.5 - z\right) \cdot \left(0.5 - z\right)} \cdot \left(e^{z - 7.5} \cdot \sqrt{2}\right)\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* PI z)))
  (*
   (*
    (sqrt PI)
    (* (exp (* (log (- 7.5 z)) (- 0.5 z))) (* (exp (- z 7.5)) (sqrt 2.0))))
   (+
    263.3831869810514
    (*
     z
     (+
      436.8961725563396
      (* z (+ 545.0353078428827 (* 606.6766809167608 z)))))))))

C

double code(double z) {
	return (((double) M_PI) / sin((((double) M_PI) * z))) * ((sqrt(((double) M_PI)) * (exp((log((7.5 - z)) * (0.5 - z))) * (exp((z - 7.5)) * sqrt(2.0)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z)))))));
}

Java

public static double code(double z) {
	return (Math.PI / Math.sin((Math.PI * z))) * ((Math.sqrt(Math.PI) * (Math.exp((Math.log((7.5 - z)) * (0.5 - z))) * (Math.exp((z - 7.5)) * Math.sqrt(2.0)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z)))))));
}

Python

def code(z):
	return (math.pi / math.sin((math.pi * z))) * ((math.sqrt(math.pi) * (math.exp((math.log((7.5 - z)) * (0.5 - z))) * (math.exp((z - 7.5)) * math.sqrt(2.0)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z)))))))

Julia

function code(z)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(sqrt(pi) * Float64(exp(Float64(log(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(z * Float64(545.0353078428827 + Float64(606.6766809167608 * z))))))))
end

MATLAB

function tmp = code(z)
	tmp = (pi / sin((pi * z))) * ((sqrt(pi) * (exp((log((7.5 - z)) * (0.5 - z))) * (exp((z - 7.5)) * sqrt(2.0)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z)))))));
end

Wolfram

code[z_] := N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[Exp[N[(N[Log[N[(7.5 - z), $MachinePrecision]], $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[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * N[(545.0353078428827 + N[(606.6766809167608 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in z around inf

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(\color{blue}{e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lift-PI.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left(e^{\color{blue}{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \color{blue}{\left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 96.5%

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

5. Taylor expanded in z around inf

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lift--.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - \color{blue}{1}\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lift-*.f64 96.7

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

7. Applied rewrites 96.7%

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

8. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)}\right) \]
9. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + \color{blue}{z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \color{blue}{\left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \color{blue}{z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)}\right)\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \color{blue}{\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)}\right)\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \color{blue}{\frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z}\right)\right)\right)\right) \]

   6. lower-*.f64 96.5

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

10. Applied rewrites 96.5%

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

Alternative 6: 96.5% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* PI z)))
  (*
   (*
    (sqrt PI)
    (* (exp (* (log (- 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) / sin((((double) M_PI) * z))) * ((sqrt(((double) M_PI)) * (exp((log((7.5 - z)) * (0.5 - z))) * (exp((z - 7.5)) * sqrt(2.0)))) * (263.3831869810514 + (z * (436.8961725563396 + (545.0353078428827 * z)))));
}

Java

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

Python

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

Julia

function code(z)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(sqrt(pi) * Float64(exp(Float64(log(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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in z around inf

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(\color{blue}{e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lift-PI.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left(e^{\color{blue}{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \color{blue}{\left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 96.5%

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

5. Taylor expanded in z around inf

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lift--.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - \color{blue}{1}\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lift-*.f64 96.7

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

7. Applied rewrites 96.7%

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

8. Taylor expanded in z around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 96.6

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

10. Applied rewrites 96.6%

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

Alternative 7: 96.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-7.5} \cdot \sqrt{15}\\ \frac{1}{z} \cdot \left(z \cdot \mathsf{fma}\left(263.3831869810514, \frac{t\_0}{z} \cdot \sqrt{\pi}, \sqrt{\pi} \cdot \mathsf{fma}\left(263.3831869810514, t\_0, e^{-7.5} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(263.3831869810514, \sqrt{7.5} \cdot \left(\log 0.13333333333333333 - 0.06666666666666667\right), 436.8961725563396 \cdot \sqrt{7.5}\right)\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (* (exp -7.5) (sqrt 15.0))))
   (*
    (/ 1.0 z)
    (*
     z
     (fma
      263.3831869810514
      (* (/ t_0 z) (sqrt PI))
      (*
       (sqrt PI)
       (fma
        263.3831869810514
        t_0
        (*
         (exp -7.5)
         (*
          (sqrt 2.0)
          (fma
           263.3831869810514
           (* (sqrt 7.5) (- (log 0.13333333333333333) 0.06666666666666667))
           (* 436.8961725563396 (sqrt 7.5))))))))))))

C

double code(double z) {
	double t_0 = exp(-7.5) * sqrt(15.0);
	return (1.0 / z) * (z * fma(263.3831869810514, ((t_0 / z) * sqrt(((double) M_PI))), (sqrt(((double) M_PI)) * fma(263.3831869810514, t_0, (exp(-7.5) * (sqrt(2.0) * fma(263.3831869810514, (sqrt(7.5) * (log(0.13333333333333333) - 0.06666666666666667)), (436.8961725563396 * sqrt(7.5)))))))));
}

Julia

function code(z)
	t_0 = Float64(exp(-7.5) * sqrt(15.0))
	return Float64(Float64(1.0 / z) * Float64(z * fma(263.3831869810514, Float64(Float64(t_0 / z) * sqrt(pi)), Float64(sqrt(pi) * fma(263.3831869810514, t_0, Float64(exp(-7.5) * Float64(sqrt(2.0) * fma(263.3831869810514, Float64(sqrt(7.5) * Float64(log(0.13333333333333333) - 0.06666666666666667)), Float64(436.8961725563396 * sqrt(7.5))))))))))
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[Exp[-7.5], $MachinePrecision] * N[Sqrt[15.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 / z), $MachinePrecision] * N[(z * N[(263.3831869810514 * N[(N[(t$95$0 / z), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[Pi], $MachinePrecision] * N[(263.3831869810514 * t$95$0 + N[(N[Exp[-7.5], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(263.3831869810514 * N[(N[Sqrt[7.5], $MachinePrecision] * N[(N[Log[0.13333333333333333], $MachinePrecision] - 0.06666666666666667), $MachinePrecision]), $MachinePrecision] + N[(436.8961725563396 * N[Sqrt[7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in z around 0

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

4. Applied rewrites 96.1%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\mathsf{fma}\left(263.3831869810514, \sqrt{\pi} \cdot \left(e^{-7.5} \cdot \sqrt{15}\right), \left(z \cdot \mathsf{fma}\left(263.3831869810514, e^{-7.5} \cdot \sqrt{15}, e^{-7.5} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(263.3831869810514, \sqrt{7.5} \cdot \left(\log 0.13333333333333333 - 0.06666666666666667\right), 436.8961725563396 \cdot \sqrt{7.5}\right)\right)\right)\right) \cdot \sqrt{\pi}\right)} \]

5. Taylor expanded in z around inf

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

7. Applied rewrites 96.3%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(z \cdot \color{blue}{\mathsf{fma}\left(263.3831869810514, \frac{e^{-7.5} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}, \sqrt{\pi} \cdot \mathsf{fma}\left(263.3831869810514, e^{-7.5} \cdot \sqrt{15}, e^{-7.5} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(263.3831869810514, \sqrt{7.5} \cdot \left(\log 0.13333333333333333 - 0.06666666666666667\right), 436.8961725563396 \cdot \sqrt{7.5}\right)\right)\right)\right)}\right) \]

8. Taylor expanded in z around 0

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

   1. lift-/.f64 96.3

      \[\leadsto \frac{1}{\color{blue}{z}} \cdot \left(z \cdot \mathsf{fma}\left(263.3831869810514, \frac{e^{-7.5} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}, \sqrt{\pi} \cdot \mathsf{fma}\left(263.3831869810514, e^{-7.5} \cdot \sqrt{15}, e^{-7.5} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(263.3831869810514, \sqrt{7.5} \cdot \left(\log 0.13333333333333333 - 0.06666666666666667\right), 436.8961725563396 \cdot \sqrt{7.5}\right)\right)\right)\right)\right) \]

10. Applied rewrites 96.3%

    \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(z \cdot \mathsf{fma}\left(263.3831869810514, \frac{e^{-7.5} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}, \sqrt{\pi} \cdot \mathsf{fma}\left(263.3831869810514, e^{-7.5} \cdot \sqrt{15}, e^{-7.5} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(263.3831869810514, \sqrt{7.5} \cdot \left(\log 0.13333333333333333 - 0.06666666666666667\right), 436.8961725563396 \cdot \sqrt{7.5}\right)\right)\right)\right)\right) \]
11. Add Preprocessing

Alternative 8: 96.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-7.5} \cdot \sqrt{15}\\ \frac{\pi}{z \cdot \pi} \cdot \mathsf{fma}\left(263.3831869810514, \sqrt{\pi} \cdot t\_0, \left(z \cdot \mathsf{fma}\left(263.3831869810514, t\_0, e^{-7.5} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(263.3831869810514, \sqrt{7.5} \cdot \left(\log 0.13333333333333333 - 0.06666666666666667\right), 436.8961725563396 \cdot \sqrt{7.5}\right)\right)\right)\right) \cdot \sqrt{\pi}\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (* (exp -7.5) (sqrt 15.0))))
   (*
    (/ PI (* z PI))
    (fma
     263.3831869810514
     (* (sqrt PI) t_0)
     (*
      (*
       z
       (fma
        263.3831869810514
        t_0
        (*
         (exp -7.5)
         (*
          (sqrt 2.0)
          (fma
           263.3831869810514
           (* (sqrt 7.5) (- (log 0.13333333333333333) 0.06666666666666667))
           (* 436.8961725563396 (sqrt 7.5)))))))
      (sqrt PI))))))

C

double code(double z) {
	double t_0 = exp(-7.5) * sqrt(15.0);
	return (((double) M_PI) / (z * ((double) M_PI))) * fma(263.3831869810514, (sqrt(((double) M_PI)) * t_0), ((z * fma(263.3831869810514, t_0, (exp(-7.5) * (sqrt(2.0) * fma(263.3831869810514, (sqrt(7.5) * (log(0.13333333333333333) - 0.06666666666666667)), (436.8961725563396 * sqrt(7.5))))))) * sqrt(((double) M_PI))));
}

Julia

function code(z)
	t_0 = Float64(exp(-7.5) * sqrt(15.0))
	return Float64(Float64(pi / Float64(z * pi)) * fma(263.3831869810514, Float64(sqrt(pi) * t_0), Float64(Float64(z * fma(263.3831869810514, t_0, Float64(exp(-7.5) * Float64(sqrt(2.0) * fma(263.3831869810514, Float64(sqrt(7.5) * Float64(log(0.13333333333333333) - 0.06666666666666667)), Float64(436.8961725563396 * sqrt(7.5))))))) * sqrt(pi))))
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[Exp[-7.5], $MachinePrecision] * N[Sqrt[15.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(Pi / N[(z * Pi), $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 * N[(N[Sqrt[Pi], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[(z * N[(263.3831869810514 * t$95$0 + N[(N[Exp[-7.5], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(263.3831869810514 * N[(N[Sqrt[7.5], $MachinePrecision] * N[(N[Log[0.13333333333333333], $MachinePrecision] - 0.06666666666666667), $MachinePrecision]), $MachinePrecision] + N[(436.8961725563396 * N[Sqrt[7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in z around 0

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

4. Applied rewrites 96.1%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\mathsf{fma}\left(263.3831869810514, \sqrt{\pi} \cdot \left(e^{-7.5} \cdot \sqrt{15}\right), \left(z \cdot \mathsf{fma}\left(263.3831869810514, e^{-7.5} \cdot \sqrt{15}, e^{-7.5} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(263.3831869810514, \sqrt{7.5} \cdot \left(\log 0.13333333333333333 - 0.06666666666666667\right), 436.8961725563396 \cdot \sqrt{7.5}\right)\right)\right)\right) \cdot \sqrt{\pi}\right)} \]

5. Taylor expanded in z around 0

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

   1. lower-*.f64 N/A

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

   2. lift-PI.f64 96.1

      \[\leadsto \frac{\pi}{z \cdot \pi} \cdot \mathsf{fma}\left(263.3831869810514, \sqrt{\pi} \cdot \left(e^{-7.5} \cdot \sqrt{15}\right), \left(z \cdot \mathsf{fma}\left(263.3831869810514, e^{-7.5} \cdot \sqrt{15}, e^{-7.5} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(263.3831869810514, \sqrt{7.5} \cdot \left(\log 0.13333333333333333 - 0.06666666666666667\right), 436.8961725563396 \cdot \sqrt{7.5}\right)\right)\right)\right) \cdot \sqrt{\pi}\right) \]

7. Applied rewrites 96.1%

   \[\leadsto \frac{\pi}{\color{blue}{z \cdot \pi}} \cdot \mathsf{fma}\left(263.3831869810514, \sqrt{\pi} \cdot \left(e^{-7.5} \cdot \sqrt{15}\right), \left(z \cdot \mathsf{fma}\left(263.3831869810514, e^{-7.5} \cdot \sqrt{15}, e^{-7.5} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(263.3831869810514, \sqrt{7.5} \cdot \left(\log 0.13333333333333333 - 0.06666666666666667\right), 436.8961725563396 \cdot \sqrt{7.5}\right)\right)\right)\right) \cdot \sqrt{\pi}\right) \]
8. Add Preprocessing

Alternative 9: 96.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-7.5} \cdot \sqrt{15}\\ \frac{1}{z} \cdot \mathsf{fma}\left(263.3831869810514, \sqrt{\pi} \cdot t\_0, \left(z \cdot \mathsf{fma}\left(263.3831869810514, t\_0, e^{-7.5} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(263.3831869810514, \sqrt{7.5} \cdot \left(\log 0.13333333333333333 - 0.06666666666666667\right), 436.8961725563396 \cdot \sqrt{7.5}\right)\right)\right)\right) \cdot \sqrt{\pi}\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (* (exp -7.5) (sqrt 15.0))))
   (*
    (/ 1.0 z)
    (fma
     263.3831869810514
     (* (sqrt PI) t_0)
     (*
      (*
       z
       (fma
        263.3831869810514
        t_0
        (*
         (exp -7.5)
         (*
          (sqrt 2.0)
          (fma
           263.3831869810514
           (* (sqrt 7.5) (- (log 0.13333333333333333) 0.06666666666666667))
           (* 436.8961725563396 (sqrt 7.5)))))))
      (sqrt PI))))))

C

double code(double z) {
	double t_0 = exp(-7.5) * sqrt(15.0);
	return (1.0 / z) * fma(263.3831869810514, (sqrt(((double) M_PI)) * t_0), ((z * fma(263.3831869810514, t_0, (exp(-7.5) * (sqrt(2.0) * fma(263.3831869810514, (sqrt(7.5) * (log(0.13333333333333333) - 0.06666666666666667)), (436.8961725563396 * sqrt(7.5))))))) * sqrt(((double) M_PI))));
}

Julia

function code(z)
	t_0 = Float64(exp(-7.5) * sqrt(15.0))
	return Float64(Float64(1.0 / z) * fma(263.3831869810514, Float64(sqrt(pi) * t_0), Float64(Float64(z * fma(263.3831869810514, t_0, Float64(exp(-7.5) * Float64(sqrt(2.0) * fma(263.3831869810514, Float64(sqrt(7.5) * Float64(log(0.13333333333333333) - 0.06666666666666667)), Float64(436.8961725563396 * sqrt(7.5))))))) * sqrt(pi))))
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[Exp[-7.5], $MachinePrecision] * N[Sqrt[15.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 / z), $MachinePrecision] * N[(263.3831869810514 * N[(N[Sqrt[Pi], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[(z * N[(263.3831869810514 * t$95$0 + N[(N[Exp[-7.5], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(263.3831869810514 * N[(N[Sqrt[7.5], $MachinePrecision] * N[(N[Log[0.13333333333333333], $MachinePrecision] - 0.06666666666666667), $MachinePrecision]), $MachinePrecision] + N[(436.8961725563396 * N[Sqrt[7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in z around 0

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

4. Applied rewrites 96.1%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\mathsf{fma}\left(263.3831869810514, \sqrt{\pi} \cdot \left(e^{-7.5} \cdot \sqrt{15}\right), \left(z \cdot \mathsf{fma}\left(263.3831869810514, e^{-7.5} \cdot \sqrt{15}, e^{-7.5} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(263.3831869810514, \sqrt{7.5} \cdot \left(\log 0.13333333333333333 - 0.06666666666666667\right), 436.8961725563396 \cdot \sqrt{7.5}\right)\right)\right)\right) \cdot \sqrt{\pi}\right)} \]

5. Taylor expanded in z around 0

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

   1. lift-/.f64 96.1

      \[\leadsto \frac{1}{\color{blue}{z}} \cdot \mathsf{fma}\left(263.3831869810514, \sqrt{\pi} \cdot \left(e^{-7.5} \cdot \sqrt{15}\right), \left(z \cdot \mathsf{fma}\left(263.3831869810514, e^{-7.5} \cdot \sqrt{15}, e^{-7.5} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(263.3831869810514, \sqrt{7.5} \cdot \left(\log 0.13333333333333333 - 0.06666666666666667\right), 436.8961725563396 \cdot \sqrt{7.5}\right)\right)\right)\right) \cdot \sqrt{\pi}\right) \]

7. Applied rewrites 96.1%

   \[\leadsto \color{blue}{\frac{1}{z}} \cdot \mathsf{fma}\left(263.3831869810514, \sqrt{\pi} \cdot \left(e^{-7.5} \cdot \sqrt{15}\right), \left(z \cdot \mathsf{fma}\left(263.3831869810514, e^{-7.5} \cdot \sqrt{15}, e^{-7.5} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(263.3831869810514, \sqrt{7.5} \cdot \left(\log 0.13333333333333333 - 0.06666666666666667\right), 436.8961725563396 \cdot \sqrt{7.5}\right)\right)\right)\right) \cdot \sqrt{\pi}\right) \]
8. Add Preprocessing

Alternative 10: 96.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in z around inf

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(\color{blue}{e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lift-PI.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left(e^{\color{blue}{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \color{blue}{\left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 96.5%

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

5. Taylor expanded in z around inf

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lift--.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - \color{blue}{1}\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lift-*.f64 96.7

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

7. Applied rewrites 96.7%

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

8. Taylor expanded in z around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 96.0

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

10. Applied rewrites 96.0%

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

Alternative 11: 95.7% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

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 95.7%

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

Reproduce

herbie shell --seed 2025118 
(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))))))