Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.6% → 98.5%

Time: 2.4min

Alternatives: 7

Speedup: 1.2×

• could not determine a ground truth

  1. z = -1.4104439624798163e+72

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: 98.5% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (+
   (+
    (+
     (+
      (+
       0.9999999999998099
       (+ (/ 676.5203681218851 (- 1.0 z)) (/ -1259.1392167224028 (- 2.0 z))))
      (/ 771.3234287776531 (- 3.0 z)))
     (/ -176.6150291621406 (- 4.0 z)))
    (/ 12.507343278686905 (- 5.0 z)))
   (+
    (/ 1.5056327351493116e-7 (- 8.0 z))
    (+ (/ -0.13857109526572012 (- 6.0 z)) (/ 9.984369578019572e-6 (- 7.0 z)))))
  (*
   PI
   (/
    (/ (* (sqrt (* 2.0 PI)) (pow (- 7.5 z) (- 0.5 z))) (exp (- 7.5 z)))
    (sin (* z PI))))))

C

double code(double z) {
	return (((((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (771.3234287776531 / (3.0 - z))) + (-176.6150291621406 / (4.0 - z))) + (12.507343278686905 / (5.0 - z))) + ((1.5056327351493116e-7 / (8.0 - z)) + ((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))))) * (((double) M_PI) * (((sqrt((2.0 * ((double) M_PI))) * pow((7.5 - z), (0.5 - z))) / exp((7.5 - z))) / sin((z * ((double) M_PI)))));
}

Java

public static double code(double z) {
	return (((((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (771.3234287776531 / (3.0 - z))) + (-176.6150291621406 / (4.0 - z))) + (12.507343278686905 / (5.0 - z))) + ((1.5056327351493116e-7 / (8.0 - z)) + ((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))))) * (Math.PI * (((Math.sqrt((2.0 * Math.PI)) * Math.pow((7.5 - z), (0.5 - z))) / Math.exp((7.5 - z))) / Math.sin((z * Math.PI))));
}

Python

def code(z):
	return (((((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (771.3234287776531 / (3.0 - z))) + (-176.6150291621406 / (4.0 - z))) + (12.507343278686905 / (5.0 - z))) + ((1.5056327351493116e-7 / (8.0 - z)) + ((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))))) * (math.pi * (((math.sqrt((2.0 * math.pi)) * math.pow((7.5 - z), (0.5 - z))) / math.exp((7.5 - z))) / math.sin((z * math.pi))))

Julia

function code(z)
	return Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(-1259.1392167224028 / Float64(2.0 - z)))) + Float64(771.3234287776531 / Float64(3.0 - z))) + Float64(-176.6150291621406 / Float64(4.0 - z))) + Float64(12.507343278686905 / Float64(5.0 - z))) + Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) + Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(9.984369578019572e-6 / Float64(7.0 - z))))) * Float64(pi * Float64(Float64(Float64(sqrt(Float64(2.0 * pi)) * (Float64(7.5 - z) ^ Float64(0.5 - z))) / exp(Float64(7.5 - z))) / sin(Float64(z * pi)))))
end

MATLAB

function tmp = code(z)
	tmp = (((((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (771.3234287776531 / (3.0 - z))) + (-176.6150291621406 / (4.0 - z))) + (12.507343278686905 / (5.0 - z))) + ((1.5056327351493116e-7 / (8.0 - z)) + ((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))))) * (pi * (((sqrt((2.0 * pi)) * ((7.5 - z) ^ (0.5 - z))) / exp((7.5 - z))) / sin((z * pi))));
end

Wolfram

code[z_] := N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(N[(N[(N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Exp[N[(7.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.0%

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

3. Simplified 97.7%

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

   1. associate-+l+ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} + \frac{\frac{-3147848041806007}{2500000000000}}{2 - z}\right)\right), \mathsf{/.f64}\left(\frac{7713234287776531}{10000000000000}, \mathsf{\_.f64}\left(3, z\right)\right)\right), \mathsf{/.f64}\left(\frac{-883075145810703}{5000000000000}, \mathsf{\_.f64}\left(4, z\right)\right)\right), \mathsf{/.f64}\left(\frac{2501468655737381}{200000000000000}, \mathsf{\_.f64}\left(5, z\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{9999999999998099}{10000000000000000}, \left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} + \frac{\frac{-3147848041806007}{2500000000000}}{2 - z}\right)\right), \mathsf{/.f64}\left(\frac{7713234287776531}{10000000000000}, \mathsf{\_.f64}\left(3, z\right)\right)\right), \mathsf{/.f64}\left(\frac{-883075145810703}{5000000000000}, \mathsf{\_.f64}\left(4, z\right)\right)\right), \mathsf{/.f64}\left(\frac{2501468655737381}{200000000000000}, \mathsf{\_.f64}\left(5, z\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{9999999999998099}{10000000000000000}, \mathsf{+.f64}\left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right), \left(\frac{\frac{-3147848041806007}{2500000000000}}{2 - z}\right)\right)\right), \mathsf{/.f64}\left(\frac{7713234287776531}{10000000000000}, \mathsf{\_.f64}\left(3, z\right)\right)\right), \mathsf{/.f64}\left(\frac{-883075145810703}{5000000000000}, \mathsf{\_.f64}\left(4, z\right)\right)\right), \mathsf{/.f64}\left(\frac{2501468655737381}{200000000000000}, \mathsf{\_.f64}\left(5, z\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{9999999999998099}{10000000000000000}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{6765203681218851}{10000000000000}, \left(1 - z\right)\right), \left(\frac{\frac{-3147848041806007}{2500000000000}}{2 - z}\right)\right)\right), \mathsf{/.f64}\left(\frac{7713234287776531}{10000000000000}, \mathsf{\_.f64}\left(3, z\right)\right)\right), \mathsf{/.f64}\left(\frac{-883075145810703}{5000000000000}, \mathsf{\_.f64}\left(4, z\right)\right)\right), \mathsf{/.f64}\left(\frac{2501468655737381}{200000000000000}, \mathsf{\_.f64}\left(5, z\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{9999999999998099}{10000000000000000}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{6765203681218851}{10000000000000}, \mathsf{\_.f64}\left(1, z\right)\right), \left(\frac{\frac{-3147848041806007}{2500000000000}}{2 - z}\right)\right)\right), \mathsf{/.f64}\left(\frac{7713234287776531}{10000000000000}, \mathsf{\_.f64}\left(3, z\right)\right)\right), \mathsf{/.f64}\left(\frac{-883075145810703}{5000000000000}, \mathsf{\_.f64}\left(4, z\right)\right)\right), \mathsf{/.f64}\left(\frac{2501468655737381}{200000000000000}, \mathsf{\_.f64}\left(5, z\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{9999999999998099}{10000000000000000}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{6765203681218851}{10000000000000}, \mathsf{\_.f64}\left(1, z\right)\right), \mathsf{/.f64}\left(\frac{-3147848041806007}{2500000000000}, \left(2 - z\right)\right)\right)\right), \mathsf{/.f64}\left(\frac{7713234287776531}{10000000000000}, \mathsf{\_.f64}\left(3, z\right)\right)\right), \mathsf{/.f64}\left(\frac{-883075145810703}{5000000000000}, \mathsf{\_.f64}\left(4, z\right)\right)\right), \mathsf{/.f64}\left(\frac{2501468655737381}{200000000000000}, \mathsf{\_.f64}\left(5, z\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   7. --lowering--.f64 98.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{9999999999998099}{10000000000000000}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{6765203681218851}{10000000000000}, \mathsf{\_.f64}\left(1, z\right)\right), \mathsf{/.f64}\left(\frac{-3147848041806007}{2500000000000}, \mathsf{\_.f64}\left(2, z\right)\right)\right)\right), \mathsf{/.f64}\left(\frac{7713234287776531}{10000000000000}, \mathsf{\_.f64}\left(3, z\right)\right)\right), \mathsf{/.f64}\left(\frac{-883075145810703}{5000000000000}, \mathsf{\_.f64}\left(4, z\right)\right)\right), \mathsf{/.f64}\left(\frac{2501468655737381}{200000000000000}, \mathsf{\_.f64}\left(5, z\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

6. Applied egg-rr 98.9%

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

7. Final simplification 98.9%

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

Alternative 2: 97.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\pi \cdot \frac{\frac{\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{e^{7.5 - z}}}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right) + \left(263.4062807184368 + z \cdot \left(436.9000215473151 + z \cdot \left(545.0359493463282 + z \cdot 606.6767878347069\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   PI
   (/
    (/ (* (sqrt (* 2.0 PI)) (pow (- 7.5 z) (- 0.5 z))) (exp (- 7.5 z)))
    (sin (* z PI))))
  (+
   (+
    (/ 1.5056327351493116e-7 (- 8.0 z))
    (+ (/ -0.13857109526572012 (- 6.0 z)) (/ 9.984369578019572e-6 (- 7.0 z))))
   (+
    263.4062807184368
    (*
     z
     (+
      436.9000215473151
      (* z (+ 545.0359493463282 (* z 606.6767878347069)))))))))

C

double code(double z) {
	return (((double) M_PI) * (((sqrt((2.0 * ((double) M_PI))) * pow((7.5 - z), (0.5 - z))) / exp((7.5 - z))) / sin((z * ((double) M_PI))))) * (((1.5056327351493116e-7 / (8.0 - z)) + ((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z)))) + (263.4062807184368 + (z * (436.9000215473151 + (z * (545.0359493463282 + (z * 606.6767878347069)))))));
}

Java

public static double code(double z) {
	return (Math.PI * (((Math.sqrt((2.0 * Math.PI)) * Math.pow((7.5 - z), (0.5 - z))) / Math.exp((7.5 - z))) / Math.sin((z * Math.PI)))) * (((1.5056327351493116e-7 / (8.0 - z)) + ((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z)))) + (263.4062807184368 + (z * (436.9000215473151 + (z * (545.0359493463282 + (z * 606.6767878347069)))))));
}

Python

def code(z):
	return (math.pi * (((math.sqrt((2.0 * math.pi)) * math.pow((7.5 - z), (0.5 - z))) / math.exp((7.5 - z))) / math.sin((z * math.pi)))) * (((1.5056327351493116e-7 / (8.0 - z)) + ((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z)))) + (263.4062807184368 + (z * (436.9000215473151 + (z * (545.0359493463282 + (z * 606.6767878347069)))))))

Julia

function code(z)
	return Float64(Float64(pi * Float64(Float64(Float64(sqrt(Float64(2.0 * pi)) * (Float64(7.5 - z) ^ Float64(0.5 - z))) / exp(Float64(7.5 - z))) / sin(Float64(z * pi)))) * Float64(Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) + Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(9.984369578019572e-6 / Float64(7.0 - z)))) + Float64(263.4062807184368 + Float64(z * Float64(436.9000215473151 + Float64(z * Float64(545.0359493463282 + Float64(z * 606.6767878347069))))))))
end

MATLAB

function tmp = code(z)
	tmp = (pi * (((sqrt((2.0 * pi)) * ((7.5 - z) ^ (0.5 - z))) / exp((7.5 - z))) / sin((z * pi)))) * (((1.5056327351493116e-7 / (8.0 - z)) + ((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z)))) + (263.4062807184368 + (z * (436.9000215473151 + (z * (545.0359493463282 + (z * 606.6767878347069)))))));
end

Wolfram

code[z_] := N[(N[(Pi * N[(N[(N[(N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Exp[N[(7.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(263.4062807184368 + N[(z * N[(436.9000215473151 + N[(z * N[(545.0359493463282 + N[(z * 606.6767878347069), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.0%

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

3. Simplified 97.7%

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

5. Taylor expanded in z around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \left(\frac{39321001939258358983}{90000000000000000} + z \cdot \left(\frac{2943194126470171931171}{5400000000000000000} + \frac{196563279258445065194677}{324000000000000000000} \cdot z\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{7902188421553103227}{30000000000000000}, \left(z \cdot \left(\frac{39321001939258358983}{90000000000000000} + z \cdot \left(\frac{2943194126470171931171}{5400000000000000000} + \frac{196563279258445065194677}{324000000000000000000} \cdot z\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{7902188421553103227}{30000000000000000}, \mathsf{*.f64}\left(z, \left(\frac{39321001939258358983}{90000000000000000} + z \cdot \left(\frac{2943194126470171931171}{5400000000000000000} + \frac{196563279258445065194677}{324000000000000000000} \cdot z\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{7902188421553103227}{30000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{39321001939258358983}{90000000000000000}, \left(z \cdot \left(\frac{2943194126470171931171}{5400000000000000000} + \frac{196563279258445065194677}{324000000000000000000} \cdot z\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{7902188421553103227}{30000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{39321001939258358983}{90000000000000000}, \mathsf{*.f64}\left(z, \left(\frac{2943194126470171931171}{5400000000000000000} + \frac{196563279258445065194677}{324000000000000000000} \cdot z\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{7902188421553103227}{30000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{39321001939258358983}{90000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{2943194126470171931171}{5400000000000000000}, \left(\frac{196563279258445065194677}{324000000000000000000} \cdot z\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{7902188421553103227}{30000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{39321001939258358983}{90000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{2943194126470171931171}{5400000000000000000}, \left(z \cdot \frac{196563279258445065194677}{324000000000000000000}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 98.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{7902188421553103227}{30000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{39321001939258358983}{90000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{2943194126470171931171}{5400000000000000000}, \mathsf{*.f64}\left(z, \frac{196563279258445065194677}{324000000000000000000}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

7. Simplified 98.9%

   \[\leadsto \left(\color{blue}{\left(263.4062807184368 + z \cdot \left(436.9000215473151 + z \cdot \left(545.0359493463282 + z \cdot 606.6767878347069\right)\right)\right)} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right)\right) \cdot \left(\pi \cdot \frac{\frac{\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{e^{7.5 - z}}}{\sin \left(\pi \cdot z\right)}\right) \]

8. Final simplification 98.9%

   \[\leadsto \left(\pi \cdot \frac{\frac{\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{e^{7.5 - z}}}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right) + \left(263.4062807184368 + z \cdot \left(436.9000215473151 + z \cdot \left(545.0359493463282 + z \cdot 606.6767878347069\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 3: 97.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\pi \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(263.4062807372572 + z \cdot \left(436.90002154966766 + z \cdot 545.0359493466223\right)\right)\right)}{\frac{\sin \left(z \cdot \pi\right)}{\frac{\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{e^{7.5 - z}}}} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (/
  (*
   PI
   (+
    (+ (/ -0.13857109526572012 (- 6.0 z)) (/ 9.984369578019572e-6 (- 7.0 z)))
    (+
     263.4062807372572
     (* z (+ 436.90002154966766 (* z 545.0359493466223))))))
  (/
   (sin (* z PI))
   (/ (* (sqrt (* 2.0 PI)) (pow (- 7.5 z) (- 0.5 z))) (exp (- 7.5 z))))))

C

double code(double z) {
	return (((double) M_PI) * (((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (263.4062807372572 + (z * (436.90002154966766 + (z * 545.0359493466223)))))) / (sin((z * ((double) M_PI))) / ((sqrt((2.0 * ((double) M_PI))) * pow((7.5 - z), (0.5 - z))) / exp((7.5 - z))));
}

Java

public static double code(double z) {
	return (Math.PI * (((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (263.4062807372572 + (z * (436.90002154966766 + (z * 545.0359493466223)))))) / (Math.sin((z * Math.PI)) / ((Math.sqrt((2.0 * Math.PI)) * Math.pow((7.5 - z), (0.5 - z))) / Math.exp((7.5 - z))));
}

Python

def code(z):
	return (math.pi * (((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (263.4062807372572 + (z * (436.90002154966766 + (z * 545.0359493466223)))))) / (math.sin((z * math.pi)) / ((math.sqrt((2.0 * math.pi)) * math.pow((7.5 - z), (0.5 - z))) / math.exp((7.5 - z))))

Julia

function code(z)
	return Float64(Float64(pi * Float64(Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(9.984369578019572e-6 / Float64(7.0 - z))) + Float64(263.4062807372572 + Float64(z * Float64(436.90002154966766 + Float64(z * 545.0359493466223)))))) / Float64(sin(Float64(z * pi)) / Float64(Float64(sqrt(Float64(2.0 * pi)) * (Float64(7.5 - z) ^ Float64(0.5 - z))) / exp(Float64(7.5 - z)))))
end

MATLAB

function tmp = code(z)
	tmp = (pi * (((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (263.4062807372572 + (z * (436.90002154966766 + (z * 545.0359493466223)))))) / (sin((z * pi)) / ((sqrt((2.0 * pi)) * ((7.5 - z) ^ (0.5 - z))) / exp((7.5 - z))));
end

Wolfram

code[z_] := N[(N[(Pi * N[(N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(263.4062807372572 + N[(z * N[(436.90002154966766 + N[(z * 545.0359493466223), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision] / N[(N[(N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Exp[N[(7.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.0%

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

3. Simplified 97.7%

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

5. Taylor expanded in z around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \left(\frac{39321001939258358983}{90000000000000000} + \frac{2943194126470171931171}{5400000000000000000} \cdot z\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{7902188421553103227}{30000000000000000}, \left(z \cdot \left(\frac{39321001939258358983}{90000000000000000} + \frac{2943194126470171931171}{5400000000000000000} \cdot z\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{7902188421553103227}{30000000000000000}, \mathsf{*.f64}\left(z, \left(\frac{39321001939258358983}{90000000000000000} + \frac{2943194126470171931171}{5400000000000000000} \cdot z\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{7902188421553103227}{30000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{39321001939258358983}{90000000000000000}, \left(\frac{2943194126470171931171}{5400000000000000000} \cdot z\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{7902188421553103227}{30000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{39321001939258358983}{90000000000000000}, \left(z \cdot \frac{2943194126470171931171}{5400000000000000000}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f64 98.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{7902188421553103227}{30000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{39321001939258358983}{90000000000000000}, \mathsf{*.f64}\left(z, \frac{2943194126470171931171}{5400000000000000000}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

7. Simplified 98.8%

   \[\leadsto \left(\color{blue}{\left(263.4062807184368 + z \cdot \left(436.9000215473151 + z \cdot 545.0359493463282\right)\right)} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right)\right) \cdot \left(\pi \cdot \frac{\frac{\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{e^{7.5 - z}}}{\sin \left(\pi \cdot z\right)}\right) \]

9. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{\left(\left(263.4062807184368 + \left(z \cdot \left(436.9000215473151 + z \cdot 545.0359493463282\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right) \cdot \pi}{\frac{\sin \left(z \cdot \pi\right)}{\frac{\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{e^{7.5 - z}}}}} \]

10. Taylor expanded in z around 0

    \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{158043768442354310053619837}{600000000000000000000000} + z \cdot \left(\frac{6291360310315214173820859511}{14400000000000000000000000} + \frac{188364424094192633804566578533}{345600000000000000000000000} \cdot z\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(z, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right)\right)\right) \]
11. Step-by-step derivation

    1. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{158043768442354310053619837}{600000000000000000000000}, \left(z \cdot \left(\frac{6291360310315214173820859511}{14400000000000000000000000} + \frac{188364424094192633804566578533}{345600000000000000000000000} \cdot z\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\color{blue}{z}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right)\right)\right) \]

    2. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{158043768442354310053619837}{600000000000000000000000}, \mathsf{*.f64}\left(z, \left(\frac{6291360310315214173820859511}{14400000000000000000000000} + \frac{188364424094192633804566578533}{345600000000000000000000000} \cdot z\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(z, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right)\right)\right) \]

    3. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{158043768442354310053619837}{600000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{6291360310315214173820859511}{14400000000000000000000000}, \left(\frac{188364424094192633804566578533}{345600000000000000000000000} \cdot z\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(z, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right)\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{158043768442354310053619837}{600000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{6291360310315214173820859511}{14400000000000000000000000}, \left(z \cdot \frac{188364424094192633804566578533}{345600000000000000000000000}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(z, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right)\right)\right) \]

    5. *-lowering-*.f64 98.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{158043768442354310053619837}{600000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{6291360310315214173820859511}{14400000000000000000000000}, \mathsf{*.f64}\left(z, \frac{188364424094192633804566578533}{345600000000000000000000000}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(z, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right)\right)\right) \]

12. Simplified 98.8%

    \[\leadsto \frac{\left(\color{blue}{\left(263.4062807372572 + z \cdot \left(436.90002154966766 + z \cdot 545.0359493466223\right)\right)} + \left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right) \cdot \pi}{\frac{\sin \left(z \cdot \pi\right)}{\frac{\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{e^{7.5 - z}}}} \]

13. Final simplification 98.8%

    \[\leadsto \frac{\pi \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(263.4062807372572 + z \cdot \left(436.90002154966766 + z \cdot 545.0359493466223\right)\right)\right)}{\frac{\sin \left(z \cdot \pi\right)}{\frac{\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{e^{7.5 - z}}}} \]
14. Add Preprocessing

Alternative 4: 97.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.0%

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

3. Simplified 97.7%

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

5. Taylor expanded in z around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(z, \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{102757979785251069442117317613}{235200000000000000000000000}, \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{102757979785251069442117317613}{235200000000000000000000000}, \left(z \cdot \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f64 97.6%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{102757979785251069442117317613}{235200000000000000000000000}, \mathsf{*.f64}\left(z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

7. Simplified 97.6%

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

   1. associate-*r* N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right)\right) \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\left(\frac{\frac{\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}}{e^{\frac{15}{2} - z}}}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)}\right)}\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right)\right), \mathsf{PI}\left(\right)\right), \left(\frac{\color{blue}{\frac{\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}}{e^{\frac{15}{2} - z}}}}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right)\right)\right), \mathsf{PI}\left(\right)\right), \left(\frac{\frac{\color{blue}{\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}}}{e^{\frac{15}{2} - z}}}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(z, \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right)\right)\right), \mathsf{PI}\left(\right)\right), \left(\frac{\frac{\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot \color{blue}{{\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}}}{e^{\frac{15}{2} - z}}}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{102757979785251069442117317613}{235200000000000000000000000}, \left(z \cdot \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right)\right)\right)\right), \mathsf{PI}\left(\right)\right), \left(\frac{\frac{\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\frac{15}{2} - z\right)}^{\color{blue}{\left(\frac{1}{2} - z\right)}}}{e^{\frac{15}{2} - z}}}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{102757979785251069442117317613}{235200000000000000000000000}, \mathsf{*.f64}\left(z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right)\right)\right)\right), \mathsf{PI}\left(\right)\right), \left(\frac{\frac{\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - \color{blue}{z}\right)}}{e^{\frac{15}{2} - z}}}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)}\right)\right) \]

   8. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{102757979785251069442117317613}{235200000000000000000000000}, \mathsf{*.f64}\left(z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), \left(\frac{\frac{\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}}{\color{blue}{e^{\frac{15}{2} - z}}}}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)}\right)\right) \]

   9. associate-/l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{102757979785251069442117317613}{235200000000000000000000000}, \mathsf{*.f64}\left(z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}}{\color{blue}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right) \cdot e^{\frac{15}{2} - z}}}\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{102757979785251069442117317613}{235200000000000000000000000}, \mathsf{*.f64}\left(z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right) \cdot e^{\color{blue}{\frac{15}{2}} - z}}\right)\right) \]

9. Applied egg-rr 98.4%

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

10. Final simplification 98.4%

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

Alternative 5: 96.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.0%

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

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z}\right), \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot e^{\frac{-15}{2}}}{z}\right), \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   6. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \frac{e^{\frac{-15}{2}}}{z}\right), \left(\color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right), \left(\frac{e^{\frac{-15}{2}}}{z}\right)\right), \left(\color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{2}\right), \left(\sqrt{\frac{15}{2}}\right)\right), \left(\frac{e^{\frac{-15}{2}}}{z}\right)\right), \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   9. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\sqrt{\frac{15}{2}}\right)\right), \left(\frac{e^{\frac{-15}{2}}}{z}\right)\right), \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \left(\frac{e^{\frac{-15}{2}}}{z}\right)\right), \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{/.f64}\left(\left(e^{\frac{-15}{2}}\right), z\right)\right), \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   12. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right)\right), \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right)\right), \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right) \]

   14. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right)\right), \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right)\right)\right) \]

   15. PI-lowering-PI.f64 97.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right)\right), \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right)\right) \]

5. Simplified 97.6%

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

   1. associate-*l* N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right), \color{blue}{\left(\frac{e^{\frac{-15}{2}}}{z} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)}\right) \]

   3. sqrt-unprod N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{2 \cdot \frac{15}{2}}\right), \left(\color{blue}{\frac{e^{\frac{-15}{2}}}{z}} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot \frac{15}{2}\right)\right), \left(\color{blue}{\frac{e^{\frac{-15}{2}}}{z}} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \left(\frac{\color{blue}{e^{\frac{-15}{2}}}}{z} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{*.f64}\left(\left(\frac{e^{\frac{-15}{2}}}{z}\right), \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{-15}{2}}\right), z\right), \left(\color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   8. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   11. PI-lowering-PI.f64 97.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

7. Applied egg-rr 97.5%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \color{blue}{\frac{e^{\frac{-15}{2}}}{z}}\right)\right) \]

   2. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \left(\frac{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot e^{\frac{-15}{2}}}{\color{blue}{z}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{/.f64}\left(\left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot e^{\frac{-15}{2}}\right), \color{blue}{z}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right), \left(e^{\frac{-15}{2}}\right)\right), z\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(e^{\frac{-15}{2}}\right)\right), z\right)\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right)\right), \left(e^{\frac{-15}{2}}\right)\right), z\right)\right) \]

   7. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \left(e^{\frac{-15}{2}}\right)\right), z\right)\right) \]

   8. exp-lowering-exp.f64 97.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{exp.f64}\left(\frac{-15}{2}\right)\right), z\right)\right) \]

9. Applied egg-rr 97.8%

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

    1. clear-num N/A

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

    2. un-div-inv N/A

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

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{15}\right), \color{blue}{\left(\frac{z}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot e^{\frac{-15}{2}}}\right)}\right) \]

    4. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(15\right), \left(\frac{\color{blue}{z}}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot e^{\frac{-15}{2}}}\right)\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{/.f64}\left(z, \color{blue}{\left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot e^{\frac{-15}{2}}\right)}\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{/.f64}\left(z, \mathsf{*.f64}\left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right), \color{blue}{\left(e^{\frac{-15}{2}}\right)}\right)\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{/.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(e^{\color{blue}{\frac{-15}{2}}}\right)\right)\right)\right) \]

    8. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{/.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right)\right), \left(e^{\frac{-15}{2}}\right)\right)\right)\right) \]

    9. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{/.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \left(e^{\frac{-15}{2}}\right)\right)\right)\right) \]

    10. exp-lowering-exp.f64 97.8%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{/.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{exp.f64}\left(\frac{-15}{2}\right)\right)\right)\right) \]

11. Applied egg-rr 97.8%

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

Alternative 6: 96.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.0%

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

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z}\right), \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot e^{\frac{-15}{2}}}{z}\right), \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   6. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \frac{e^{\frac{-15}{2}}}{z}\right), \left(\color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right), \left(\frac{e^{\frac{-15}{2}}}{z}\right)\right), \left(\color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{2}\right), \left(\sqrt{\frac{15}{2}}\right)\right), \left(\frac{e^{\frac{-15}{2}}}{z}\right)\right), \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   9. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\sqrt{\frac{15}{2}}\right)\right), \left(\frac{e^{\frac{-15}{2}}}{z}\right)\right), \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \left(\frac{e^{\frac{-15}{2}}}{z}\right)\right), \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{/.f64}\left(\left(e^{\frac{-15}{2}}\right), z\right)\right), \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   12. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right)\right), \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right)\right), \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right) \]

   14. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right)\right), \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right)\right)\right) \]

   15. PI-lowering-PI.f64 97.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right)\right), \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right)\right) \]

5. Simplified 97.6%

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

   1. associate-*l* N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right), \color{blue}{\left(\frac{e^{\frac{-15}{2}}}{z} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)}\right) \]

   3. sqrt-unprod N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{2 \cdot \frac{15}{2}}\right), \left(\color{blue}{\frac{e^{\frac{-15}{2}}}{z}} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot \frac{15}{2}\right)\right), \left(\color{blue}{\frac{e^{\frac{-15}{2}}}{z}} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \left(\frac{\color{blue}{e^{\frac{-15}{2}}}}{z} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{*.f64}\left(\left(\frac{e^{\frac{-15}{2}}}{z}\right), \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{-15}{2}}\right), z\right), \left(\color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   8. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   11. PI-lowering-PI.f64 97.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

7. Applied egg-rr 97.5%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \color{blue}{\frac{e^{\frac{-15}{2}}}{z}}\right)\right) \]

   2. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \left(\frac{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot e^{\frac{-15}{2}}}{\color{blue}{z}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{/.f64}\left(\left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot e^{\frac{-15}{2}}\right), \color{blue}{z}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right), \left(e^{\frac{-15}{2}}\right)\right), z\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(e^{\frac{-15}{2}}\right)\right), z\right)\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right)\right), \left(e^{\frac{-15}{2}}\right)\right), z\right)\right) \]

   7. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \left(e^{\frac{-15}{2}}\right)\right), z\right)\right) \]

   8. exp-lowering-exp.f64 97.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{exp.f64}\left(\frac{-15}{2}\right)\right), z\right)\right) \]

9. Applied egg-rr 97.8%

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

Alternative 7: 96.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.0%

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

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z}\right), \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot e^{\frac{-15}{2}}}{z}\right), \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   6. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \frac{e^{\frac{-15}{2}}}{z}\right), \left(\color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right), \left(\frac{e^{\frac{-15}{2}}}{z}\right)\right), \left(\color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{2}\right), \left(\sqrt{\frac{15}{2}}\right)\right), \left(\frac{e^{\frac{-15}{2}}}{z}\right)\right), \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   9. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\sqrt{\frac{15}{2}}\right)\right), \left(\frac{e^{\frac{-15}{2}}}{z}\right)\right), \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \left(\frac{e^{\frac{-15}{2}}}{z}\right)\right), \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{/.f64}\left(\left(e^{\frac{-15}{2}}\right), z\right)\right), \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   12. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right)\right), \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right)\right), \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right) \]

   14. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right)\right), \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right)\right)\right) \]

   15. PI-lowering-PI.f64 97.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right)\right), \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right)\right) \]

5. Simplified 97.6%

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

   1. associate-*l* N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right), \color{blue}{\left(\frac{e^{\frac{-15}{2}}}{z} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)}\right) \]

   3. sqrt-unprod N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{2 \cdot \frac{15}{2}}\right), \left(\color{blue}{\frac{e^{\frac{-15}{2}}}{z}} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot \frac{15}{2}\right)\right), \left(\color{blue}{\frac{e^{\frac{-15}{2}}}{z}} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \left(\frac{\color{blue}{e^{\frac{-15}{2}}}}{z} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{*.f64}\left(\left(\frac{e^{\frac{-15}{2}}}{z}\right), \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{-15}{2}}\right), z\right), \left(\color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   8. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   11. PI-lowering-PI.f64 97.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

7. Applied egg-rr 97.5%

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

8. Final simplification 97.5%

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

Reproduce

herbie shell --seed 2024139 
(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))))))