Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.6% → 97.8%

Time: 8.5s

Alternatives: 12

Speedup: 1.6×

• could not determine a ground truth

  1. z = -43073048500.47488

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: 96.6% accurate, 1.0× speedup

Math

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

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: 97.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - -6\\ t_1 := t\_0 + 0.5\\ \left(\frac{1 + 0.16666666666666666 \cdot {\left(z \cdot \pi\right)}^{2}}{z} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {t\_1}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-t\_1}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_0}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) -6.0)) (t_1 (+ t_0 0.5)))
   (*
    (*
     (/ (+ 1.0 (* 0.16666666666666666 (pow (* z PI) 2.0))) z)
     (*
      (* (* (sqrt PI) (sqrt 2.0)) (pow t_1 (- (- 1.0 z) 0.5)))
      (exp (- t_1))))
    (+
     (+
      (+
       263.3831855358925
       (*
        z
        (+
         436.8961723502244
         (* z (+ 545.0353078134797 (* 606.6766809125655 z))))))
      (/ 9.984369578019572e-6 t_0))
     (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0))))))

C

double code(double z) {
	double t_0 = (1.0 - z) - -6.0;
	double t_1 = t_0 + 0.5;
	return (((1.0 + (0.16666666666666666 * pow((z * ((double) M_PI)), 2.0))) / z) * (((sqrt(((double) M_PI)) * sqrt(2.0)) * pow(t_1, ((1.0 - z) - 0.5))) * exp(-t_1))) * (((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (606.6766809125655 * z)))))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - -6.0;
	double t_1 = t_0 + 0.5;
	return (((1.0 + (0.16666666666666666 * Math.pow((z * Math.PI), 2.0))) / z) * (((Math.sqrt(Math.PI) * Math.sqrt(2.0)) * Math.pow(t_1, ((1.0 - z) - 0.5))) * Math.exp(-t_1))) * (((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (606.6766809125655 * z)))))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
}

Python

def code(z):
	t_0 = (1.0 - z) - -6.0
	t_1 = t_0 + 0.5
	return (((1.0 + (0.16666666666666666 * math.pow((z * math.pi), 2.0))) / z) * (((math.sqrt(math.pi) * math.sqrt(2.0)) * math.pow(t_1, ((1.0 - z) - 0.5))) * math.exp(-t_1))) * (((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (606.6766809125655 * z)))))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - -6.0)
	t_1 = Float64(t_0 + 0.5)
	return Float64(Float64(Float64(Float64(1.0 + Float64(0.16666666666666666 * (Float64(z * pi) ^ 2.0))) / z) * Float64(Float64(Float64(sqrt(pi) * sqrt(2.0)) * (t_1 ^ Float64(Float64(1.0 - z) - 0.5))) * exp(Float64(-t_1)))) * Float64(Float64(Float64(263.3831855358925 + Float64(z * Float64(436.8961723502244 + Float64(z * Float64(545.0353078134797 + Float64(606.6766809125655 * z)))))) + Float64(9.984369578019572e-6 / t_0)) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - -6.0;
	t_1 = t_0 + 0.5;
	tmp = (((1.0 + (0.16666666666666666 * ((z * pi) ^ 2.0))) / z) * (((sqrt(pi) * sqrt(2.0)) * (t_1 ^ ((1.0 - z) - 0.5))) * exp(-t_1))) * (((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (606.6766809125655 * z)))))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 0.5), $MachinePrecision]}, N[(N[(N[(N[(1.0 + N[(0.16666666666666666 * N[Power[N[(z * Pi), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * N[(N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$1, N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$1)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(263.3831855358925 + N[(z * N[(436.8961723502244 + N[(z * N[(545.0353078134797 + N[(606.6766809125655 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 96.2%

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

4. Applied rewrites 96.3%

   \[\leadsto \color{blue}{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)} \]

5. Taylor expanded in z around 0

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + \color{blue}{z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \color{blue}{\left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + \color{blue}{z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \color{blue}{\left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \color{blue}{\frac{65521081538557082921549}{108000000000000000000} \cdot z}\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   6. lower-*.f64 96.8

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot \color{blue}{z}\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

7. Applied rewrites 96.8%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right)} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]
8. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\sqrt{\pi \cdot 2}} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lift-PI.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)} \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right) \cdot 2}} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   4. sqrt-prod N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   7. lift-PI.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\pi}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   8. lower-sqrt.f64 97.3

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \color{blue}{\sqrt{2}}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

9. Applied rewrites 97.3%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

10. Taylor expanded in z around 0

    \[\leadsto \left(\color{blue}{\frac{1 + \frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{z}} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]
11. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \left(\frac{1 + \frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{\color{blue}{z}} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

    2. lower-+.f64 N/A

       \[\leadsto \left(\frac{1 + \frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{z} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

    3. lower-*.f64 N/A

       \[\leadsto \left(\frac{1 + \frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{z} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

    4. pow-prod-down N/A

       \[\leadsto \left(\frac{1 + \frac{1}{6} \cdot {\left(z \cdot \mathsf{PI}\left(\right)\right)}^{2}}{z} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

    5. lower-pow.f64 N/A

       \[\leadsto \left(\frac{1 + \frac{1}{6} \cdot {\left(z \cdot \mathsf{PI}\left(\right)\right)}^{2}}{z} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

    6. lower-*.f64 N/A

       \[\leadsto \left(\frac{1 + \frac{1}{6} \cdot {\left(z \cdot \mathsf{PI}\left(\right)\right)}^{2}}{z} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

    7. lift-PI.f64 97.4

       \[\leadsto \left(\frac{1 + 0.16666666666666666 \cdot {\left(z \cdot \pi\right)}^{2}}{z} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

12. Applied rewrites 97.4%

    \[\leadsto \left(\color{blue}{\frac{1 + 0.16666666666666666 \cdot {\left(z \cdot \pi\right)}^{2}}{z}} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]
13. Add Preprocessing

Alternative 2: 97.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - -6\\ t_1 := t\_0 + 0.5\\ \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {t\_1}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-t\_1}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_0}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) -6.0)) (t_1 (+ t_0 0.5)))
   (*
    (*
     (/ PI (sin (* PI z)))
     (*
      (* (* (sqrt PI) (sqrt 2.0)) (pow t_1 (- (- 1.0 z) 0.5)))
      (exp (- t_1))))
    (+
     (+
      (+
       263.3831855358925
       (*
        z
        (+
         436.8961723502244
         (* z (+ 545.0353078134797 (* 606.6766809125655 z))))))
      (/ 9.984369578019572e-6 t_0))
     (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0))))))

C

double code(double z) {
	double t_0 = (1.0 - z) - -6.0;
	double t_1 = t_0 + 0.5;
	return ((((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt(((double) M_PI)) * sqrt(2.0)) * pow(t_1, ((1.0 - z) - 0.5))) * exp(-t_1))) * (((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (606.6766809125655 * z)))))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - -6.0;
	double t_1 = t_0 + 0.5;
	return ((Math.PI / Math.sin((Math.PI * z))) * (((Math.sqrt(Math.PI) * Math.sqrt(2.0)) * Math.pow(t_1, ((1.0 - z) - 0.5))) * Math.exp(-t_1))) * (((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (606.6766809125655 * z)))))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
}

Python

def code(z):
	t_0 = (1.0 - z) - -6.0
	t_1 = t_0 + 0.5
	return ((math.pi / math.sin((math.pi * z))) * (((math.sqrt(math.pi) * math.sqrt(2.0)) * math.pow(t_1, ((1.0 - z) - 0.5))) * math.exp(-t_1))) * (((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (606.6766809125655 * z)))))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - -6.0)
	t_1 = Float64(t_0 + 0.5)
	return Float64(Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(sqrt(pi) * sqrt(2.0)) * (t_1 ^ Float64(Float64(1.0 - z) - 0.5))) * exp(Float64(-t_1)))) * Float64(Float64(Float64(263.3831855358925 + Float64(z * Float64(436.8961723502244 + Float64(z * Float64(545.0353078134797 + Float64(606.6766809125655 * z)))))) + Float64(9.984369578019572e-6 / t_0)) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - -6.0;
	t_1 = t_0 + 0.5;
	tmp = ((pi / sin((pi * z))) * (((sqrt(pi) * sqrt(2.0)) * (t_1 ^ ((1.0 - z) - 0.5))) * exp(-t_1))) * (((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (606.6766809125655 * z)))))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 0.5), $MachinePrecision]}, N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$1, N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$1)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(263.3831855358925 + N[(z * N[(436.8961723502244 + N[(z * N[(545.0353078134797 + N[(606.6766809125655 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 96.2%

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

4. Applied rewrites 96.3%

   \[\leadsto \color{blue}{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)} \]

5. Taylor expanded in z around 0

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + \color{blue}{z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \color{blue}{\left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + \color{blue}{z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \color{blue}{\left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \color{blue}{\frac{65521081538557082921549}{108000000000000000000} \cdot z}\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   6. lower-*.f64 96.8

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot \color{blue}{z}\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

7. Applied rewrites 96.8%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right)} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]
8. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\sqrt{\pi \cdot 2}} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lift-PI.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)} \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right) \cdot 2}} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   4. sqrt-prod N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   7. lift-PI.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\pi}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   8. lower-sqrt.f64 97.3

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \color{blue}{\sqrt{2}}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

9. Applied rewrites 97.3%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]
10. Add Preprocessing

Alternative 3: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - z\right) - -6\right) + 0.5\\ \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {t\_0}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-t\_0}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ (- (- 1.0 z) -6.0) 0.5)))
   (*
    (*
     (/ PI (sin (* PI z)))
     (*
      (* (* (sqrt PI) (sqrt 2.0)) (pow t_0 (- (- 1.0 z) 0.5)))
      (exp (- t_0))))
    (+
     263.3831869810514
     (*
      z
      (+
       436.8961725563396
       (* z (+ 545.0353078428827 (* 606.6766809167608 z)))))))))

C

double code(double z) {
	double t_0 = ((1.0 - z) - -6.0) + 0.5;
	return ((((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt(((double) M_PI)) * sqrt(2.0)) * pow(t_0, ((1.0 - z) - 0.5))) * exp(-t_0))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))));
}

Java

public static double code(double z) {
	double t_0 = ((1.0 - z) - -6.0) + 0.5;
	return ((Math.PI / Math.sin((Math.PI * z))) * (((Math.sqrt(Math.PI) * Math.sqrt(2.0)) * Math.pow(t_0, ((1.0 - z) - 0.5))) * Math.exp(-t_0))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))));
}

Python

def code(z):
	t_0 = ((1.0 - z) - -6.0) + 0.5
	return ((math.pi / math.sin((math.pi * z))) * (((math.sqrt(math.pi) * math.sqrt(2.0)) * math.pow(t_0, ((1.0 - z) - 0.5))) * math.exp(-t_0))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))))

Julia

function code(z)
	t_0 = Float64(Float64(Float64(1.0 - z) - -6.0) + 0.5)
	return Float64(Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(sqrt(pi) * sqrt(2.0)) * (t_0 ^ Float64(Float64(1.0 - z) - 0.5))) * exp(Float64(-t_0)))) * 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) - -6.0) + 0.5;
	tmp = ((pi / sin((pi * z))) * (((sqrt(pi) * sqrt(2.0)) * (t_0 ^ ((1.0 - z) - 0.5))) * exp(-t_0))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$0, N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$0)], $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]]

Derivation

1. Initial program 96.2%

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

4. Applied rewrites 96.3%

   \[\leadsto \color{blue}{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)} \]

5. Taylor expanded in z around 0

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + \color{blue}{z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \color{blue}{\left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + \color{blue}{z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \color{blue}{\left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \color{blue}{\frac{65521081538557082921549}{108000000000000000000} \cdot z}\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   6. lower-*.f64 96.8

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot \color{blue}{z}\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

7. Applied rewrites 96.8%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right)} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]
8. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\sqrt{\pi \cdot 2}} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lift-PI.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)} \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right) \cdot 2}} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   4. sqrt-prod N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   7. lift-PI.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\pi}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   8. lower-sqrt.f64 97.3

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \color{blue}{\sqrt{2}}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

9. Applied rewrites 97.3%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

10. Taylor expanded in z around 0

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)} \]
11. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + \color{blue}{z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)}\right) \]

    2. lower-*.f64 N/A

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \color{blue}{\left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)}\right) \]

    3. lower-+.f64 N/A

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \color{blue}{z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)}\right)\right) \]

    4. lower-*.f64 N/A

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \color{blue}{\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)}\right)\right) \]

    5. lower-+.f64 N/A

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \color{blue}{\frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z}\right)\right)\right) \]

    6. lower-*.f64 97.3

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot \color{blue}{z}\right)\right)\right) \]

12. Applied rewrites 97.3%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)} \]
13. Add Preprocessing

Alternative 4: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - z\right) - -6\right) + 0.5\\ \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {t\_0}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-t\_0}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ (- (- 1.0 z) -6.0) 0.5)))
   (*
    (*
     (/ PI (sin (* PI z)))
     (*
      (* (* (sqrt PI) (sqrt 2.0)) (pow t_0 (- (- 1.0 z) 0.5)))
      (exp (- t_0))))
    (+
     263.3831869810514
     (* z (+ 436.8961725563396 (* 545.0353078428827 z)))))))

C

double code(double z) {
	double t_0 = ((1.0 - z) - -6.0) + 0.5;
	return ((((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt(((double) M_PI)) * sqrt(2.0)) * pow(t_0, ((1.0 - z) - 0.5))) * exp(-t_0))) * (263.3831869810514 + (z * (436.8961725563396 + (545.0353078428827 * z))));
}

Java

public static double code(double z) {
	double t_0 = ((1.0 - z) - -6.0) + 0.5;
	return ((Math.PI / Math.sin((Math.PI * z))) * (((Math.sqrt(Math.PI) * Math.sqrt(2.0)) * Math.pow(t_0, ((1.0 - z) - 0.5))) * Math.exp(-t_0))) * (263.3831869810514 + (z * (436.8961725563396 + (545.0353078428827 * z))));
}

Python

def code(z):
	t_0 = ((1.0 - z) - -6.0) + 0.5
	return ((math.pi / math.sin((math.pi * z))) * (((math.sqrt(math.pi) * math.sqrt(2.0)) * math.pow(t_0, ((1.0 - z) - 0.5))) * math.exp(-t_0))) * (263.3831869810514 + (z * (436.8961725563396 + (545.0353078428827 * z))))

Julia

function code(z)
	t_0 = Float64(Float64(Float64(1.0 - z) - -6.0) + 0.5)
	return Float64(Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(sqrt(pi) * sqrt(2.0)) * (t_0 ^ Float64(Float64(1.0 - z) - 0.5))) * exp(Float64(-t_0)))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(545.0353078428827 * z)))))
end

MATLAB

function tmp = code(z)
	t_0 = ((1.0 - z) - -6.0) + 0.5;
	tmp = ((pi / sin((pi * z))) * (((sqrt(pi) * sqrt(2.0)) * (t_0 ^ ((1.0 - z) - 0.5))) * exp(-t_0))) * (263.3831869810514 + (z * (436.8961725563396 + (545.0353078428827 * z))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$0, N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$0)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(545.0353078428827 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 96.2%

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

4. Applied rewrites 96.3%

   \[\leadsto \color{blue}{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)} \]

5. Taylor expanded in z around 0

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + \color{blue}{z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \color{blue}{\left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + \color{blue}{z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \color{blue}{\left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \color{blue}{\frac{65521081538557082921549}{108000000000000000000} \cdot z}\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   6. lower-*.f64 96.8

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot \color{blue}{z}\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

7. Applied rewrites 96.8%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right)} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]
8. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\sqrt{\pi \cdot 2}} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lift-PI.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)} \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right) \cdot 2}} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   4. sqrt-prod N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   7. lift-PI.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\pi}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   8. lower-sqrt.f64 97.3

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \color{blue}{\sqrt{2}}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

9. Applied rewrites 97.3%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

10. Taylor expanded in z around 0

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

    1. lower-+.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. lower-+.f64 N/A

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

    4. lower-*.f64 97.0

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot \color{blue}{z}\right)\right) \]

12. Applied rewrites 97.0%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right)} \]
13. Add Preprocessing

Alternative 5: 97.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - z\right) - -6\right) + 0.5\\ \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {t\_0}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-t\_0}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ (- (- 1.0 z) -6.0) 0.5)))
   (*
    (*
     (/ PI (sin (* PI z)))
     (* (* (sqrt (* PI 2.0)) (pow t_0 (- (- 1.0 z) 0.5))) (exp (- t_0))))
    (+
     263.3831869810514
     (*
      z
      (+
       436.8961725563396
       (* z (+ 545.0353078428827 (* 606.6766809167608 z)))))))))

C

double code(double z) {
	double t_0 = ((1.0 - z) - -6.0) + 0.5;
	return ((((double) M_PI) / sin((((double) M_PI) * z))) * ((sqrt((((double) M_PI) * 2.0)) * pow(t_0, ((1.0 - z) - 0.5))) * exp(-t_0))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))));
}

Java

public static double code(double z) {
	double t_0 = ((1.0 - z) - -6.0) + 0.5;
	return ((Math.PI / Math.sin((Math.PI * z))) * ((Math.sqrt((Math.PI * 2.0)) * Math.pow(t_0, ((1.0 - z) - 0.5))) * Math.exp(-t_0))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))));
}

Python

def code(z):
	t_0 = ((1.0 - z) - -6.0) + 0.5
	return ((math.pi / math.sin((math.pi * z))) * ((math.sqrt((math.pi * 2.0)) * math.pow(t_0, ((1.0 - z) - 0.5))) * math.exp(-t_0))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))))

Julia

function code(z)
	t_0 = Float64(Float64(Float64(1.0 - z) - -6.0) + 0.5)
	return Float64(Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(sqrt(Float64(pi * 2.0)) * (t_0 ^ Float64(Float64(1.0 - z) - 0.5))) * exp(Float64(-t_0)))) * 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) - -6.0) + 0.5;
	tmp = ((pi / sin((pi * z))) * ((sqrt((pi * 2.0)) * (t_0 ^ ((1.0 - z) - 0.5))) * exp(-t_0))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$0, N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$0)], $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]]

Derivation

1. Initial program 96.2%

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

4. Applied rewrites 96.3%

   \[\leadsto \color{blue}{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)} \]

5. Taylor expanded in z around 0

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)} \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + \color{blue}{z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \color{blue}{\left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)}\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \color{blue}{z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \color{blue}{\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)}\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \color{blue}{\frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z}\right)\right)\right) \]

   6. lower-*.f64 96.8

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot \color{blue}{z}\right)\right)\right) \]

7. Applied rewrites 96.8%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)} \]
8. Add Preprocessing

Alternative 6: 97.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - -6\\ t_1 := t\_0 + 0.5\\ \left(\frac{\pi}{z \cdot \pi} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {t\_1}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-t\_1}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_0}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) -6.0)) (t_1 (+ t_0 0.5)))
   (*
    (*
     (/ PI (* z PI))
     (*
      (* (* (sqrt PI) (sqrt 2.0)) (pow t_1 (- (- 1.0 z) 0.5)))
      (exp (- t_1))))
    (+
     (+
      (+
       263.3831855358925
       (*
        z
        (+
         436.8961723502244
         (* z (+ 545.0353078134797 (* 606.6766809125655 z))))))
      (/ 9.984369578019572e-6 t_0))
     (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0))))))

C

double code(double z) {
	double t_0 = (1.0 - z) - -6.0;
	double t_1 = t_0 + 0.5;
	return ((((double) M_PI) / (z * ((double) M_PI))) * (((sqrt(((double) M_PI)) * sqrt(2.0)) * pow(t_1, ((1.0 - z) - 0.5))) * exp(-t_1))) * (((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (606.6766809125655 * z)))))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - -6.0;
	double t_1 = t_0 + 0.5;
	return ((Math.PI / (z * Math.PI)) * (((Math.sqrt(Math.PI) * Math.sqrt(2.0)) * Math.pow(t_1, ((1.0 - z) - 0.5))) * Math.exp(-t_1))) * (((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (606.6766809125655 * z)))))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
}

Python

def code(z):
	t_0 = (1.0 - z) - -6.0
	t_1 = t_0 + 0.5
	return ((math.pi / (z * math.pi)) * (((math.sqrt(math.pi) * math.sqrt(2.0)) * math.pow(t_1, ((1.0 - z) - 0.5))) * math.exp(-t_1))) * (((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (606.6766809125655 * z)))))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - -6.0)
	t_1 = Float64(t_0 + 0.5)
	return Float64(Float64(Float64(pi / Float64(z * pi)) * Float64(Float64(Float64(sqrt(pi) * sqrt(2.0)) * (t_1 ^ Float64(Float64(1.0 - z) - 0.5))) * exp(Float64(-t_1)))) * Float64(Float64(Float64(263.3831855358925 + Float64(z * Float64(436.8961723502244 + Float64(z * Float64(545.0353078134797 + Float64(606.6766809125655 * z)))))) + Float64(9.984369578019572e-6 / t_0)) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - -6.0;
	t_1 = t_0 + 0.5;
	tmp = ((pi / (z * pi)) * (((sqrt(pi) * sqrt(2.0)) * (t_1 ^ ((1.0 - z) - 0.5))) * exp(-t_1))) * (((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (606.6766809125655 * z)))))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 0.5), $MachinePrecision]}, N[(N[(N[(Pi / N[(z * Pi), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$1, N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$1)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(263.3831855358925 + N[(z * N[(436.8961723502244 + N[(z * N[(545.0353078134797 + N[(606.6766809125655 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 96.2%

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

4. Applied rewrites 96.3%

   \[\leadsto \color{blue}{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)} \]

5. Taylor expanded in z around 0

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + \color{blue}{z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \color{blue}{\left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + \color{blue}{z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \color{blue}{\left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \color{blue}{\frac{65521081538557082921549}{108000000000000000000} \cdot z}\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   6. lower-*.f64 96.8

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot \color{blue}{z}\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

7. Applied rewrites 96.8%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right)} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]
8. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\sqrt{\pi \cdot 2}} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lift-PI.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)} \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right) \cdot 2}} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   4. sqrt-prod N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   7. lift-PI.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\pi}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   8. lower-sqrt.f64 97.3

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \color{blue}{\sqrt{2}}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

9. Applied rewrites 97.3%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

10. Taylor expanded in z around 0

    \[\leadsto \left(\frac{\pi}{\color{blue}{z \cdot \mathsf{PI}\left(\right)}} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]
11. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \left(\frac{\pi}{z \cdot \color{blue}{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

    2. lift-PI.f64 96.6

       \[\leadsto \left(\frac{\pi}{z \cdot \pi} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

12. Applied rewrites 96.6%

    \[\leadsto \left(\frac{\pi}{\color{blue}{z \cdot \pi}} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]
13. Add Preprocessing

Alternative 7: 96.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - -6\\ t_1 := t\_0 + 0.5\\ \left(\frac{\pi}{z \cdot \pi} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {t\_1}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-t\_1}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_0}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) -6.0)) (t_1 (+ t_0 0.5)))
   (*
    (*
     (/ PI (* z PI))
     (* (* (sqrt (* PI 2.0)) (pow t_1 (- (- 1.0 z) 0.5))) (exp (- t_1))))
    (+
     (+
      (+
       263.3831855358925
       (*
        z
        (+
         436.8961723502244
         (* z (+ 545.0353078134797 (* 606.6766809125655 z))))))
      (/ 9.984369578019572e-6 t_0))
     (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0))))))

C

double code(double z) {
	double t_0 = (1.0 - z) - -6.0;
	double t_1 = t_0 + 0.5;
	return ((((double) M_PI) / (z * ((double) M_PI))) * ((sqrt((((double) M_PI) * 2.0)) * pow(t_1, ((1.0 - z) - 0.5))) * exp(-t_1))) * (((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (606.6766809125655 * z)))))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - -6.0;
	double t_1 = t_0 + 0.5;
	return ((Math.PI / (z * Math.PI)) * ((Math.sqrt((Math.PI * 2.0)) * Math.pow(t_1, ((1.0 - z) - 0.5))) * Math.exp(-t_1))) * (((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (606.6766809125655 * z)))))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
}

Python

def code(z):
	t_0 = (1.0 - z) - -6.0
	t_1 = t_0 + 0.5
	return ((math.pi / (z * math.pi)) * ((math.sqrt((math.pi * 2.0)) * math.pow(t_1, ((1.0 - z) - 0.5))) * math.exp(-t_1))) * (((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (606.6766809125655 * z)))))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - -6.0)
	t_1 = Float64(t_0 + 0.5)
	return Float64(Float64(Float64(pi / Float64(z * pi)) * Float64(Float64(sqrt(Float64(pi * 2.0)) * (t_1 ^ Float64(Float64(1.0 - z) - 0.5))) * exp(Float64(-t_1)))) * Float64(Float64(Float64(263.3831855358925 + Float64(z * Float64(436.8961723502244 + Float64(z * Float64(545.0353078134797 + Float64(606.6766809125655 * z)))))) + Float64(9.984369578019572e-6 / t_0)) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - -6.0;
	t_1 = t_0 + 0.5;
	tmp = ((pi / (z * pi)) * ((sqrt((pi * 2.0)) * (t_1 ^ ((1.0 - z) - 0.5))) * exp(-t_1))) * (((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (606.6766809125655 * z)))))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 0.5), $MachinePrecision]}, N[(N[(N[(Pi / N[(z * Pi), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$1, N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$1)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(263.3831855358925 + N[(z * N[(436.8961723502244 + N[(z * N[(545.0353078134797 + N[(606.6766809125655 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 96.2%

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

4. Applied rewrites 96.3%

   \[\leadsto \color{blue}{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)} \]

5. Taylor expanded in z around 0

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + \color{blue}{z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \color{blue}{\left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + \color{blue}{z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \color{blue}{\left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \color{blue}{\frac{65521081538557082921549}{108000000000000000000} \cdot z}\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   6. lower-*.f64 96.8

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot \color{blue}{z}\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

7. Applied rewrites 96.8%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right)} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

8. Taylor expanded in z around 0

   \[\leadsto \left(\frac{\pi}{\color{blue}{z \cdot \mathsf{PI}\left(\right)}} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{z \cdot \color{blue}{\mathsf{PI}\left(\right)}} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lift-PI.f64 96.0

      \[\leadsto \left(\frac{\pi}{z \cdot \pi} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

10. Applied rewrites 96.0%

    \[\leadsto \left(\frac{\pi}{\color{blue}{z \cdot \pi}} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]
11. Add Preprocessing

Alternative 8: 96.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - -6\\ t_1 := t\_0 + 0.5\\ \left(\frac{\pi}{z \cdot \pi} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {t\_1}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-t\_1}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_0}\right) + \left(1.8820409189366395 \cdot 10^{-8} + 2.3525511486707994 \cdot 10^{-9} \cdot z\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) -6.0)) (t_1 (+ t_0 0.5)))
   (*
    (*
     (/ PI (* z PI))
     (* (* (sqrt (* PI 2.0)) (pow t_1 (- (- 1.0 z) 0.5))) (exp (- t_1))))
    (+
     (+
      (+
       263.3831855358925
       (*
        z
        (+
         436.8961723502244
         (* z (+ 545.0353078134797 (* 606.6766809125655 z))))))
      (/ 9.984369578019572e-6 t_0))
     (+ 1.8820409189366395e-8 (* 2.3525511486707994e-9 z))))))

C

double code(double z) {
	double t_0 = (1.0 - z) - -6.0;
	double t_1 = t_0 + 0.5;
	return ((((double) M_PI) / (z * ((double) M_PI))) * ((sqrt((((double) M_PI) * 2.0)) * pow(t_1, ((1.0 - z) - 0.5))) * exp(-t_1))) * (((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (606.6766809125655 * z)))))) + (9.984369578019572e-6 / t_0)) + (1.8820409189366395e-8 + (2.3525511486707994e-9 * z)));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - -6.0;
	double t_1 = t_0 + 0.5;
	return ((Math.PI / (z * Math.PI)) * ((Math.sqrt((Math.PI * 2.0)) * Math.pow(t_1, ((1.0 - z) - 0.5))) * Math.exp(-t_1))) * (((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (606.6766809125655 * z)))))) + (9.984369578019572e-6 / t_0)) + (1.8820409189366395e-8 + (2.3525511486707994e-9 * z)));
}

Python

def code(z):
	t_0 = (1.0 - z) - -6.0
	t_1 = t_0 + 0.5
	return ((math.pi / (z * math.pi)) * ((math.sqrt((math.pi * 2.0)) * math.pow(t_1, ((1.0 - z) - 0.5))) * math.exp(-t_1))) * (((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (606.6766809125655 * z)))))) + (9.984369578019572e-6 / t_0)) + (1.8820409189366395e-8 + (2.3525511486707994e-9 * z)))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - -6.0)
	t_1 = Float64(t_0 + 0.5)
	return Float64(Float64(Float64(pi / Float64(z * pi)) * Float64(Float64(sqrt(Float64(pi * 2.0)) * (t_1 ^ Float64(Float64(1.0 - z) - 0.5))) * exp(Float64(-t_1)))) * Float64(Float64(Float64(263.3831855358925 + Float64(z * Float64(436.8961723502244 + Float64(z * Float64(545.0353078134797 + Float64(606.6766809125655 * z)))))) + Float64(9.984369578019572e-6 / t_0)) + Float64(1.8820409189366395e-8 + Float64(2.3525511486707994e-9 * z))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - -6.0;
	t_1 = t_0 + 0.5;
	tmp = ((pi / (z * pi)) * ((sqrt((pi * 2.0)) * (t_1 ^ ((1.0 - z) - 0.5))) * exp(-t_1))) * (((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (606.6766809125655 * z)))))) + (9.984369578019572e-6 / t_0)) + (1.8820409189366395e-8 + (2.3525511486707994e-9 * z)));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 0.5), $MachinePrecision]}, N[(N[(N[(Pi / N[(z * Pi), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$1, N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$1)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(263.3831855358925 + N[(z * N[(436.8961723502244 + N[(z * N[(545.0353078134797 + N[(606.6766809125655 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.8820409189366395e-8 + N[(2.3525511486707994e-9 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 96.2%

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

4. Applied rewrites 96.3%

   \[\leadsto \color{blue}{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)} \]

5. Taylor expanded in z around 0

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + \color{blue}{z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \color{blue}{\left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + \color{blue}{z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \color{blue}{\left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \color{blue}{\frac{65521081538557082921549}{108000000000000000000} \cdot z}\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   6. lower-*.f64 96.8

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot \color{blue}{z}\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

7. Applied rewrites 96.8%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right)} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

8. Taylor expanded in z around 0

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \color{blue}{\left(\frac{3764081837873279}{200000000000000000000000} + \frac{3764081837873279}{1600000000000000000000000} \cdot z\right)}\right) \]
9. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \left(\frac{3764081837873279}{200000000000000000000000} + \color{blue}{\frac{3764081837873279}{1600000000000000000000000} \cdot z}\right)\right) \]

   2. lower-*.f64 96.8

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \left(1.8820409189366395 \cdot 10^{-8} + 2.3525511486707994 \cdot 10^{-9} \cdot \color{blue}{z}\right)\right) \]

10. Applied rewrites 96.8%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \color{blue}{\left(1.8820409189366395 \cdot 10^{-8} + 2.3525511486707994 \cdot 10^{-9} \cdot z\right)}\right) \]

11. Taylor expanded in z around 0

    \[\leadsto \left(\frac{\pi}{\color{blue}{z \cdot \mathsf{PI}\left(\right)}} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \left(\frac{3764081837873279}{200000000000000000000000} + \frac{3764081837873279}{1600000000000000000000000} \cdot z\right)\right) \]
12. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \left(\frac{\pi}{z \cdot \color{blue}{\mathsf{PI}\left(\right)}} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \left(\frac{3764081837873279}{200000000000000000000000} + \frac{3764081837873279}{1600000000000000000000000} \cdot z\right)\right) \]

    2. lift-PI.f64 96.0

       \[\leadsto \left(\frac{\pi}{z \cdot \pi} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \left(1.8820409189366395 \cdot 10^{-8} + 2.3525511486707994 \cdot 10^{-9} \cdot z\right)\right) \]

13. Applied rewrites 96.0%

    \[\leadsto \left(\frac{\pi}{\color{blue}{z \cdot \pi}} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \left(1.8820409189366395 \cdot 10^{-8} + 2.3525511486707994 \cdot 10^{-9} \cdot z\right)\right) \]
14. Add Preprocessing

Alternative 9: 95.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \left(\frac{e^{-7.5} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(606.6766809125655 \cdot \left(z \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (* (/ (* (exp -7.5) (sqrt 15.0)) z) (sqrt PI))
  (+
   (+
    (+ 263.3831855358925 (* z (* 606.6766809125655 (* z z))))
    (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0)))
   (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0)))))

C

double code(double z) {
	return (((exp(-7.5) * sqrt(15.0)) / z) * sqrt(((double) M_PI))) * (((263.3831855358925 + (z * (606.6766809125655 * (z * z)))) + (9.984369578019572e-6 / ((1.0 - z) - -6.0))) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
}

Java

public static double code(double z) {
	return (((Math.exp(-7.5) * Math.sqrt(15.0)) / z) * Math.sqrt(Math.PI)) * (((263.3831855358925 + (z * (606.6766809125655 * (z * z)))) + (9.984369578019572e-6 / ((1.0 - z) - -6.0))) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
}

Python

def code(z):
	return (((math.exp(-7.5) * math.sqrt(15.0)) / z) * math.sqrt(math.pi)) * (((263.3831855358925 + (z * (606.6766809125655 * (z * z)))) + (9.984369578019572e-6 / ((1.0 - z) - -6.0))) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)))

Julia

function code(z)
	return Float64(Float64(Float64(Float64(exp(-7.5) * sqrt(15.0)) / z) * sqrt(pi)) * Float64(Float64(Float64(263.3831855358925 + Float64(z * Float64(606.6766809125655 * Float64(z * z)))) + Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0))) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0))))
end

MATLAB

function tmp = code(z)
	tmp = (((exp(-7.5) * sqrt(15.0)) / z) * sqrt(pi)) * (((263.3831855358925 + (z * (606.6766809125655 * (z * z)))) + (9.984369578019572e-6 / ((1.0 - z) - -6.0))) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
end

Wolfram

code[z_] := N[(N[(N[(N[(N[Exp[-7.5], $MachinePrecision] * N[Sqrt[15.0], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(263.3831855358925 + N[(z * N[(606.6766809125655 * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

4. Applied rewrites 96.3%

   \[\leadsto \color{blue}{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)} \]

5. Taylor expanded in z around 0

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + \color{blue}{z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \color{blue}{\left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + \color{blue}{z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \color{blue}{\left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \color{blue}{\frac{65521081538557082921549}{108000000000000000000} \cdot z}\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   6. lower-*.f64 96.8

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot \color{blue}{z}\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

7. Applied rewrites 96.8%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right)} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

8. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z} \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lift-exp.f64 N/A

      \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   3. sqrt-unprod N/A

      \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \sqrt{2 \cdot \frac{15}{2}}}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   4. metadata-eval N/A

      \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \sqrt{15}}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \sqrt{15}}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \sqrt{15}}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   7. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \sqrt{15}}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   8. lift-PI.f64 N/A

      \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   9. lift-*.f64 95.4

      \[\leadsto \left(\frac{e^{-7.5} \cdot \sqrt{15}}{z} \cdot \color{blue}{\sqrt{\pi}}\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

10. Applied rewrites 95.4%

    \[\leadsto \color{blue}{\left(\frac{e^{-7.5} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}\right)} \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

11. Taylor expanded in z around inf

    \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{65521081538557082921549}{108000000000000000000} \cdot \color{blue}{{z}^{2}}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]
12. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{65521081538557082921549}{108000000000000000000} \cdot {z}^{\color{blue}{2}}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

    2. unpow2 N/A

       \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{65521081538557082921549}{108000000000000000000} \cdot \left(z \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

    3. lower-*.f64 95.4

       \[\leadsto \left(\frac{e^{-7.5} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(606.6766809125655 \cdot \left(z \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

13. Applied rewrites 95.4%

    \[\leadsto \left(\frac{e^{-7.5} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(606.6766809125655 \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]
14. Add Preprocessing

Alternative 10: 95.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \left(\frac{e^{-7.5} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}\right) \cdot \left(\left(263.3831855358925 + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (* (/ (* (exp -7.5) (sqrt 15.0)) z) (sqrt PI))
  (+
   (+ 263.3831855358925 (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0)))
   (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0)))))

C

double code(double z) {
	return (((exp(-7.5) * sqrt(15.0)) / z) * sqrt(((double) M_PI))) * ((263.3831855358925 + (9.984369578019572e-6 / ((1.0 - z) - -6.0))) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
}

Java

public static double code(double z) {
	return (((Math.exp(-7.5) * Math.sqrt(15.0)) / z) * Math.sqrt(Math.PI)) * ((263.3831855358925 + (9.984369578019572e-6 / ((1.0 - z) - -6.0))) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
}

Python

def code(z):
	return (((math.exp(-7.5) * math.sqrt(15.0)) / z) * math.sqrt(math.pi)) * ((263.3831855358925 + (9.984369578019572e-6 / ((1.0 - z) - -6.0))) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)))

Julia

function code(z)
	return Float64(Float64(Float64(Float64(exp(-7.5) * sqrt(15.0)) / z) * sqrt(pi)) * Float64(Float64(263.3831855358925 + Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0))) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0))))
end

MATLAB

function tmp = code(z)
	tmp = (((exp(-7.5) * sqrt(15.0)) / z) * sqrt(pi)) * ((263.3831855358925 + (9.984369578019572e-6 / ((1.0 - z) - -6.0))) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
end

Wolfram

code[z_] := N[(N[(N[(N[(N[Exp[-7.5], $MachinePrecision] * N[Sqrt[15.0], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(263.3831855358925 + N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

4. Applied rewrites 96.3%

   \[\leadsto \color{blue}{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)} \]

5. Taylor expanded in z around 0

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + \color{blue}{z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \color{blue}{\left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + \color{blue}{z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \color{blue}{\left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \color{blue}{\frac{65521081538557082921549}{108000000000000000000} \cdot z}\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   6. lower-*.f64 96.8

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot \color{blue}{z}\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

7. Applied rewrites 96.8%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right)} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

8. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z} \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lift-exp.f64 N/A

      \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   3. sqrt-unprod N/A

      \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \sqrt{2 \cdot \frac{15}{2}}}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   4. metadata-eval N/A

      \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \sqrt{15}}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \sqrt{15}}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \sqrt{15}}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   7. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \sqrt{15}}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   8. lift-PI.f64 N/A

      \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   9. lift-*.f64 95.4

      \[\leadsto \left(\frac{e^{-7.5} \cdot \sqrt{15}}{z} \cdot \color{blue}{\sqrt{\pi}}\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

10. Applied rewrites 95.4%

    \[\leadsto \color{blue}{\left(\frac{e^{-7.5} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}\right)} \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

11. Taylor expanded in z around 0

    \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}\right) \cdot \left(\left(\frac{9876869457595968283}{37500000000000000} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]
12. Step-by-step derivation

13. Applied rewrites 95.4%

    \[\leadsto \left(\frac{e^{-7.5} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}\right) \cdot \left(\left(263.3831855358925 + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]
14. Add Preprocessing

Alternative 11: 95.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \left(\frac{e^{-7.5} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}\right) \cdot \left(263.383186962231 + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (* (/ (* (exp -7.5) (sqrt 15.0)) z) (sqrt PI))
  (+ 263.383186962231 (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0)))))

C

double code(double z) {
	return (((exp(-7.5) * sqrt(15.0)) / z) * sqrt(((double) M_PI))) * (263.383186962231 + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
}

Java

public static double code(double z) {
	return (((Math.exp(-7.5) * Math.sqrt(15.0)) / z) * Math.sqrt(Math.PI)) * (263.383186962231 + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
}

Python

def code(z):
	return (((math.exp(-7.5) * math.sqrt(15.0)) / z) * math.sqrt(math.pi)) * (263.383186962231 + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)))

Julia

function code(z)
	return Float64(Float64(Float64(Float64(exp(-7.5) * sqrt(15.0)) / z) * sqrt(pi)) * Float64(263.383186962231 + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0))))
end

MATLAB

function tmp = code(z)
	tmp = (((exp(-7.5) * sqrt(15.0)) / z) * sqrt(pi)) * (263.383186962231 + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
end

Wolfram

code[z_] := N[(N[(N[(N[(N[Exp[-7.5], $MachinePrecision] * N[Sqrt[15.0], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(263.383186962231 + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

4. Applied rewrites 96.3%

   \[\leadsto \color{blue}{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)} \]

5. Taylor expanded in z around 0

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + \color{blue}{z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \color{blue}{\left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + \color{blue}{z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \color{blue}{\left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \color{blue}{\frac{65521081538557082921549}{108000000000000000000} \cdot z}\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   6. lower-*.f64 96.8

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot \color{blue}{z}\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

7. Applied rewrites 96.8%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right)} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

8. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z} \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   2. lift-exp.f64 N/A

      \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   3. sqrt-unprod N/A

      \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \sqrt{2 \cdot \frac{15}{2}}}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   4. metadata-eval N/A

      \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \sqrt{15}}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \sqrt{15}}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \sqrt{15}}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   7. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \sqrt{15}}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   8. lift-PI.f64 N/A

      \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]

   9. lift-*.f64 95.4

      \[\leadsto \left(\frac{e^{-7.5} \cdot \sqrt{15}}{z} \cdot \color{blue}{\sqrt{\pi}}\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

10. Applied rewrites 95.4%

    \[\leadsto \color{blue}{\left(\frac{e^{-7.5} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}\right)} \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

11. Taylor expanded in z around 0

    \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}\right) \cdot \left(\color{blue}{\frac{1382761731551712743134679}{5250000000000000000000}} + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right) \]
12. Step-by-step derivation

13. Applied rewrites 95.4%

    \[\leadsto \left(\frac{e^{-7.5} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}\right) \cdot \left(\color{blue}{263.383186962231} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]
14. Add Preprocessing

Alternative 12: 95.8% accurate, 3.9× 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 96.2%

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

3. Taylor expanded in z around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 95.4%

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

6. Taylor expanded in z around 0

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

   1. lower-*.f64 N/A

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

8. Applied rewrites 95.4%

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

Reproduce

herbie shell --seed 2025083 
(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))))))