Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.6% → 99.2%

Time: 1.9min

Alternatives: 14

Speedup: 1.5×

• could not determine a ground truth

  1. z = -6.0355560928800776e+32

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: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2 \cdot \pi}\\ t_1 := \frac{\pi}{\sin \left(z \cdot \pi\right)}\\ \mathbf{if}\;z \leq -140:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \left(0.9999999999998099 \cdot e^{z + \left(\left(0.5 - z\right) \cdot \log \left(7.5 - z\right) - 7.5\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\left(\left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right) + \frac{12.507343278686905}{5 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\right)\right) \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sqrt (* 2.0 PI))) (t_1 (/ PI (sin (* z PI)))))
   (if (<= z -140.0)
     (*
      t_0
      (*
       t_1
       (*
        0.9999999999998099
        (exp (+ z (- (* (- 0.5 z) (log (- 7.5 z))) 7.5))))))
     (*
      t_0
      (*
       t_1
       (*
        (+
         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))
            (+
             (/ 9.984369578019572e-6 (- 7.0 z))
             (/ -0.13857109526572012 (- 6.0 z)))))))
        (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5)))))))))

C

double code(double z) {
	double t_0 = sqrt((2.0 * ((double) M_PI)));
	double t_1 = ((double) M_PI) / sin((z * ((double) M_PI)));
	double tmp;
	if (z <= -140.0) {
		tmp = t_0 * (t_1 * (0.9999999999998099 * exp((z + (((0.5 - z) * log((7.5 - z))) - 7.5)))));
	} else {
		tmp = t_0 * (t_1 * ((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)) + ((9.984369578019572e-6 / (7.0 - z)) + (-0.13857109526572012 / (6.0 - z))))))) * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5)))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.sqrt((2.0 * Math.PI));
	double t_1 = Math.PI / Math.sin((z * Math.PI));
	double tmp;
	if (z <= -140.0) {
		tmp = t_0 * (t_1 * (0.9999999999998099 * Math.exp((z + (((0.5 - z) * Math.log((7.5 - z))) - 7.5)))));
	} else {
		tmp = t_0 * (t_1 * ((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)) + ((9.984369578019572e-6 / (7.0 - z)) + (-0.13857109526572012 / (6.0 - z))))))) * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5)))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.sqrt((2.0 * math.pi))
	t_1 = math.pi / math.sin((z * math.pi))
	tmp = 0
	if z <= -140.0:
		tmp = t_0 * (t_1 * (0.9999999999998099 * math.exp((z + (((0.5 - z) * math.log((7.5 - z))) - 7.5)))))
	else:
		tmp = t_0 * (t_1 * ((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)) + ((9.984369578019572e-6 / (7.0 - z)) + (-0.13857109526572012 / (6.0 - z))))))) * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5)))))
	return tmp

Julia

function code(z)
	t_0 = sqrt(Float64(2.0 * pi))
	t_1 = Float64(pi / sin(Float64(z * pi)))
	tmp = 0.0
	if (z <= -140.0)
		tmp = Float64(t_0 * Float64(t_1 * Float64(0.9999999999998099 * exp(Float64(z + Float64(Float64(Float64(0.5 - z) * log(Float64(7.5 - z))) - 7.5))))));
	else
		tmp = Float64(t_0 * Float64(t_1 * Float64(Float64(0.9999999999998099 + Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(-1259.1392167224028 / Float64(2.0 - z))) + Float64(Float64(Float64(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(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(-0.13857109526572012 / Float64(6.0 - z))))))) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = sqrt((2.0 * pi));
	t_1 = pi / sin((z * pi));
	tmp = 0.0;
	if (z <= -140.0)
		tmp = t_0 * (t_1 * (0.9999999999998099 * exp((z + (((0.5 - z) * log((7.5 - z))) - 7.5)))));
	else
		tmp = t_0 * (t_1 * ((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)) + ((9.984369578019572e-6 / (7.0 - z)) + (-0.13857109526572012 / (6.0 - z))))))) * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(Pi / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -140.0], N[(t$95$0 * N[(t$95$1 * N[(0.9999999999998099 * N[Exp[N[(z + N[(N[(N[(0.5 - z), $MachinePrecision] * N[Log[N[(7.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - 7.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(t$95$1 * N[(N[(0.9999999999998099 + N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(771.3234287776531 / N[(3.0 - z), $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[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < -140

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

      1. expm1-log1p-u 0.0%

         \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

      2. expm1-udef 0.0%

         \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)} - 1\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   6. Applied egg-rr 0.0%

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\left(e^{z + -7.5} \cdot \left(0.9999999999998099 + \left(\left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right)\right)\right)\right) \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)}\right)} - 1\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. Simplified 0.0%

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

      1. add-exp-log 0.0%

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

      2. sub-neg 0.0%

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

      3. +-commutative 0.0%

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

      4. log-prod 0.0%

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

      5. log-pow 1.6%

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

      6. +-commutative 1.6%

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

      7. sub-neg 1.6%

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

      8. add-log-exp 100.0%

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

   10. Applied egg-rr 100.0%

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

   11. Taylor expanded in z around inf 100.0%

       \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\left(0.9999999999998099 \cdot e^{\left(z + \log \left(7.5 - z\right) \cdot \left(0.5 - z\right)\right) - 7.5}\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   12. Step-by-step derivation

       1. *-commutative 100.0%

          \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\left(e^{\left(z + \log \left(7.5 - z\right) \cdot \left(0.5 - z\right)\right) - 7.5} \cdot 0.9999999999998099\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

       2. associate--l+ 100.0%

          \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(e^{\color{blue}{z + \left(\log \left(7.5 - z\right) \cdot \left(0.5 - z\right) - 7.5\right)}} \cdot 0.9999999999998099\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

       3. *-commutative 100.0%

          \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(e^{z + \left(\color{blue}{\left(0.5 - z\right) \cdot \log \left(7.5 - z\right)} - 7.5\right)} \cdot 0.9999999999998099\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   13. Simplified 100.0%

       \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\left(e^{z + \left(\left(0.5 - z\right) \cdot \log \left(7.5 - z\right) - 7.5\right)} \cdot 0.9999999999998099\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   if -140 < z

   1. Initial program 97.3%

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

   3. Simplified 98.2%

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

      1. expm1-log1p-u 98.2%

         \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

      2. expm1-udef 98.2%

         \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)} - 1\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   6. Applied egg-rr 98.2%

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\left(e^{z + -7.5} \cdot \left(0.9999999999998099 + \left(\left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right)\right)\right)\right) \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)}\right)} - 1\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. Simplified 99.2%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -140:\\ \;\;\;\;\sqrt{2 \cdot \pi} \cdot \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(0.9999999999998099 \cdot e^{z + \left(\left(0.5 - z\right) \cdot \log \left(7.5 - z\right) - 7.5\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \pi} \cdot \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\left(\left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right) + \frac{12.507343278686905}{5 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\right)\right) \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 97.1%

   \[\leadsto \color{blue}{\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right)} \]
4. Add Preprocessing

6. Applied egg-rr 98.3%

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

7. Final simplification 98.3%

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

Alternative 3: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(z)
	return Float64(sqrt(Float64(2.0 * pi)) * Float64(Float64(pi / sin(Float64(z * pi))) * Float64(Float64(0.9999999999998099 + Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(-1259.1392167224028 / Float64(2.0 - z))) + Float64(Float64(Float64(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(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(-0.13857109526572012 / Float64(6.0 - z))))))) * exp(Float64(Float64(Float64(0.5 - z) * log(Float64(7.5 - z))) + Float64(z + -7.5))))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 96.3%

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

   1. expm1-log1p-u 96.3%

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   2. expm1-udef 96.3%

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)} - 1\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

6. Applied egg-rr 96.3%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\left(e^{z + -7.5} \cdot \left(0.9999999999998099 + \left(\left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right)\right)\right)\right) \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)}\right)} - 1\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

8. Simplified 97.3%

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

   1. add-exp-log 96.3%

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

   2. sub-neg 96.3%

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

   3. +-commutative 96.3%

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

   4. log-prod 96.3%

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

   5. log-pow 96.4%

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

   6. +-commutative 96.4%

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

   7. sub-neg 96.4%

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

   8. add-log-exp 98.3%

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

10. Applied egg-rr 98.3%

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

11. Final simplification 98.3%

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

Alternative 4: 97.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2 \cdot \pi}\\ \mathbf{if}\;z \leq -13.5:\\ \;\;\;\;t\_0 \cdot \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(0.9999999999998099 \cdot e^{z + \left(\left(0.5 - z\right) \cdot \log \left(7.5 - z\right) - 7.5\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(212.9540523020159 + z \cdot 74.66416387488323\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right)\right) \cdot \frac{1}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sqrt (* 2.0 PI))))
   (if (<= z -13.5)
     (*
      t_0
      (*
       (/ PI (sin (* z PI)))
       (*
        0.9999999999998099
        (exp (+ z (- (* (- 0.5 z) (log (- 7.5 z))) 7.5))))))
     (*
      t_0
      (*
       (*
        (pow (- 7.5 z) (- 0.5 z))
        (*
         (exp (+ z -7.5))
         (+
          (+
           0.9999999999998099
           (+
            (+
             (/ 676.5203681218851 (- 1.0 z))
             (/ -1259.1392167224028 (- 2.0 z)))
            (+ 212.9540523020159 (* z 74.66416387488323))))
          (+ 2.4783749183520145 (* z 0.49644474017195733)))))
       (/ 1.0 z))))))

C

double code(double z) {
	double t_0 = sqrt((2.0 * ((double) M_PI)));
	double tmp;
	if (z <= -13.5) {
		tmp = t_0 * ((((double) M_PI) / sin((z * ((double) M_PI)))) * (0.9999999999998099 * exp((z + (((0.5 - z) * log((7.5 - z))) - 7.5)))));
	} else {
		tmp = t_0 * ((pow((7.5 - z), (0.5 - z)) * (exp((z + -7.5)) * ((0.9999999999998099 + (((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z))) + (212.9540523020159 + (z * 74.66416387488323)))) + (2.4783749183520145 + (z * 0.49644474017195733))))) * (1.0 / z));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.sqrt((2.0 * Math.PI));
	double tmp;
	if (z <= -13.5) {
		tmp = t_0 * ((Math.PI / Math.sin((z * Math.PI))) * (0.9999999999998099 * Math.exp((z + (((0.5 - z) * Math.log((7.5 - z))) - 7.5)))));
	} else {
		tmp = t_0 * ((Math.pow((7.5 - z), (0.5 - z)) * (Math.exp((z + -7.5)) * ((0.9999999999998099 + (((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z))) + (212.9540523020159 + (z * 74.66416387488323)))) + (2.4783749183520145 + (z * 0.49644474017195733))))) * (1.0 / z));
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.sqrt((2.0 * math.pi))
	tmp = 0
	if z <= -13.5:
		tmp = t_0 * ((math.pi / math.sin((z * math.pi))) * (0.9999999999998099 * math.exp((z + (((0.5 - z) * math.log((7.5 - z))) - 7.5)))))
	else:
		tmp = t_0 * ((math.pow((7.5 - z), (0.5 - z)) * (math.exp((z + -7.5)) * ((0.9999999999998099 + (((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z))) + (212.9540523020159 + (z * 74.66416387488323)))) + (2.4783749183520145 + (z * 0.49644474017195733))))) * (1.0 / z))
	return tmp

Julia

function code(z)
	t_0 = sqrt(Float64(2.0 * pi))
	tmp = 0.0
	if (z <= -13.5)
		tmp = Float64(t_0 * Float64(Float64(pi / sin(Float64(z * pi))) * Float64(0.9999999999998099 * exp(Float64(z + Float64(Float64(Float64(0.5 - z) * log(Float64(7.5 - z))) - 7.5))))));
	else
		tmp = Float64(t_0 * Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * Float64(exp(Float64(z + -7.5)) * Float64(Float64(0.9999999999998099 + Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(-1259.1392167224028 / Float64(2.0 - z))) + Float64(212.9540523020159 + Float64(z * 74.66416387488323)))) + Float64(2.4783749183520145 + Float64(z * 0.49644474017195733))))) * Float64(1.0 / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = sqrt((2.0 * pi));
	tmp = 0.0;
	if (z <= -13.5)
		tmp = t_0 * ((pi / sin((z * pi))) * (0.9999999999998099 * exp((z + (((0.5 - z) * log((7.5 - z))) - 7.5)))));
	else
		tmp = t_0 * ((((7.5 - z) ^ (0.5 - z)) * (exp((z + -7.5)) * ((0.9999999999998099 + (((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z))) + (212.9540523020159 + (z * 74.66416387488323)))) + (2.4783749183520145 + (z * 0.49644474017195733))))) * (1.0 / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -13.5], N[(t$95$0 * N[(N[(Pi / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.9999999999998099 * N[Exp[N[(z + N[(N[(N[(0.5 - z), $MachinePrecision] * N[Log[N[(7.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - 7.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(N[(0.9999999999998099 + N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(212.9540523020159 + N[(z * 74.66416387488323), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.4783749183520145 + N[(z * 0.49644474017195733), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -13.5

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

      1. expm1-log1p-u 0.0%

         \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

      2. expm1-udef 0.0%

         \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)} - 1\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   6. Applied egg-rr 0.0%

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\left(e^{z + -7.5} \cdot \left(0.9999999999998099 + \left(\left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right)\right)\right)\right) \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)}\right)} - 1\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. Simplified 0.0%

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

      1. add-exp-log 0.0%

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

      2. sub-neg 0.0%

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

      3. +-commutative 0.0%

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

      4. log-prod 0.0%

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

      5. log-pow 1.6%

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

      6. +-commutative 1.6%

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

      7. sub-neg 1.6%

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

      8. add-log-exp 100.0%

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

   10. Applied egg-rr 100.0%

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

   11. Taylor expanded in z around inf 100.0%

       \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\left(0.9999999999998099 \cdot e^{\left(z + \log \left(7.5 - z\right) \cdot \left(0.5 - z\right)\right) - 7.5}\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   12. Step-by-step derivation

       1. *-commutative 100.0%

          \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\left(e^{\left(z + \log \left(7.5 - z\right) \cdot \left(0.5 - z\right)\right) - 7.5} \cdot 0.9999999999998099\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

       2. associate--l+ 100.0%

          \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(e^{\color{blue}{z + \left(\log \left(7.5 - z\right) \cdot \left(0.5 - z\right) - 7.5\right)}} \cdot 0.9999999999998099\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

       3. *-commutative 100.0%

          \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(e^{z + \left(\color{blue}{\left(0.5 - z\right) \cdot \log \left(7.5 - z\right)} - 7.5\right)} \cdot 0.9999999999998099\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   13. Simplified 100.0%

       \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\left(e^{z + \left(\left(0.5 - z\right) \cdot \log \left(7.5 - z\right) - 7.5\right)} \cdot 0.9999999999998099\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   if -13.5 < z

   1. Initial program 97.3%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in z around 0 98.2%

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \color{blue}{\left(2.4783749183520145 + 0.49644474017195733 \cdot z\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative 98.2%

         \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \left(2.4783749183520145 + \color{blue}{z \cdot 0.49644474017195733}\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   7. Simplified 98.2%

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \color{blue}{\left(2.4783749183520145 + z \cdot 0.49644474017195733\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. Taylor expanded in z around 0 98.0%

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right)\right) \cdot \color{blue}{\frac{1}{z}}\right) \]

   9. Taylor expanded in z around 0 99.2%

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \color{blue}{\left(212.9540523020159 + 74.66416387488323 \cdot z\right)}\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right)\right) \cdot \frac{1}{z}\right) \]
   10. Step-by-step derivation

       1. *-commutative 99.2%

          \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(212.9540523020159 + \color{blue}{z \cdot 74.66416387488323}\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right)\right) \cdot \frac{1}{z}\right) \]

   11. Simplified 99.2%

       \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \color{blue}{\left(212.9540523020159 + z \cdot 74.66416387488323\right)}\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right)\right) \cdot \frac{1}{z}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -13.5:\\ \;\;\;\;\sqrt{2 \cdot \pi} \cdot \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(0.9999999999998099 \cdot e^{z + \left(\left(0.5 - z\right) \cdot \log \left(7.5 - z\right) - 7.5\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \pi} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(212.9540523020159 + z \cdot 74.66416387488323\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right)\right) \cdot \frac{1}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot \pi} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(212.9540523020159 + z \cdot 74.66416387488323\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right)\right) \cdot \frac{1}{z}\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (sqrt (* 2.0 PI))
  (*
   (*
    (pow (- 7.5 z) (- 0.5 z))
    (*
     (exp (+ z -7.5))
     (+
      (+
       0.9999999999998099
       (+
        (+ (/ 676.5203681218851 (- 1.0 z)) (/ -1259.1392167224028 (- 2.0 z)))
        (+ 212.9540523020159 (* z 74.66416387488323))))
      (+ 2.4783749183520145 (* z 0.49644474017195733)))))
   (/ 1.0 z))))

C

double code(double z) {
	return sqrt((2.0 * ((double) M_PI))) * ((pow((7.5 - z), (0.5 - z)) * (exp((z + -7.5)) * ((0.9999999999998099 + (((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z))) + (212.9540523020159 + (z * 74.66416387488323)))) + (2.4783749183520145 + (z * 0.49644474017195733))))) * (1.0 / z));
}

Java

public static double code(double z) {
	return Math.sqrt((2.0 * Math.PI)) * ((Math.pow((7.5 - z), (0.5 - z)) * (Math.exp((z + -7.5)) * ((0.9999999999998099 + (((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z))) + (212.9540523020159 + (z * 74.66416387488323)))) + (2.4783749183520145 + (z * 0.49644474017195733))))) * (1.0 / z));
}

Python

def code(z):
	return math.sqrt((2.0 * math.pi)) * ((math.pow((7.5 - z), (0.5 - z)) * (math.exp((z + -7.5)) * ((0.9999999999998099 + (((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z))) + (212.9540523020159 + (z * 74.66416387488323)))) + (2.4783749183520145 + (z * 0.49644474017195733))))) * (1.0 / z))

Julia

function code(z)
	return Float64(sqrt(Float64(2.0 * pi)) * Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * Float64(exp(Float64(z + -7.5)) * Float64(Float64(0.9999999999998099 + Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(-1259.1392167224028 / Float64(2.0 - z))) + Float64(212.9540523020159 + Float64(z * 74.66416387488323)))) + Float64(2.4783749183520145 + Float64(z * 0.49644474017195733))))) * Float64(1.0 / z)))
end

MATLAB

function tmp = code(z)
	tmp = sqrt((2.0 * pi)) * ((((7.5 - z) ^ (0.5 - z)) * (exp((z + -7.5)) * ((0.9999999999998099 + (((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z))) + (212.9540523020159 + (z * 74.66416387488323)))) + (2.4783749183520145 + (z * 0.49644474017195733))))) * (1.0 / z));
end

Wolfram

code[z_] := N[(N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(N[(0.9999999999998099 + N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(212.9540523020159 + N[(z * 74.66416387488323), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.4783749183520145 + N[(z * 0.49644474017195733), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.4%

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

3. Simplified 96.3%

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

5. Taylor expanded in z around 0 96.3%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \color{blue}{\left(2.4783749183520145 + 0.49644474017195733 \cdot z\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
6. Step-by-step derivation

   1. *-commutative 96.3%

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \left(2.4783749183520145 + \color{blue}{z \cdot 0.49644474017195733}\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

7. Simplified 96.3%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \color{blue}{\left(2.4783749183520145 + z \cdot 0.49644474017195733\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

8. Taylor expanded in z around 0 96.1%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right)\right) \cdot \color{blue}{\frac{1}{z}}\right) \]

9. Taylor expanded in z around 0 97.3%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \color{blue}{\left(212.9540523020159 + 74.66416387488323 \cdot z\right)}\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right)\right) \cdot \frac{1}{z}\right) \]
10. Step-by-step derivation

    1. *-commutative 97.3%

       \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(212.9540523020159 + \color{blue}{z \cdot 74.66416387488323}\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right)\right) \cdot \frac{1}{z}\right) \]

11. Simplified 97.3%

    \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \color{blue}{\left(212.9540523020159 + z \cdot 74.66416387488323\right)}\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right)\right) \cdot \frac{1}{z}\right) \]

12. Final simplification 97.3%

    \[\leadsto \sqrt{2 \cdot \pi} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(212.9540523020159 + z \cdot 74.66416387488323\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right)\right) \cdot \frac{1}{z}\right) \]
13. Add Preprocessing

Alternative 6: 96.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 96.3%

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

5. Taylor expanded in z around 0 96.5%

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

   1. associate-*r* 96.5%

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

   2. *-commutative 96.5%

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

7. Simplified 96.5%

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

8. Taylor expanded in z around 0 96.4%

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

   1. associate-*l/ 96.2%

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

   2. *-commutative 96.2%

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

   3. associate-*r* 97.0%

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

   4. *-commutative 97.0%

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

10. Simplified 97.0%

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

    1. expm1-log1p-u 44.4%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\sqrt{\pi} \cdot e^{-7.5}\right) \cdot \left(\sqrt{7.5} \cdot \sqrt{2}\right)}{z}\right)\right)} \]

    2. expm1-udef 44.4%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\left(\sqrt{\pi} \cdot e^{-7.5}\right) \cdot \left(\sqrt{7.5} \cdot \sqrt{2}\right)}{z}\right)} - 1\right)} \]

    3. associate-/l* 44.4%

       \[\leadsto 263.3831869810514 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{\pi} \cdot e^{-7.5}}{\frac{z}{\sqrt{7.5} \cdot \sqrt{2}}}}\right)} - 1\right) \]

    4. *-commutative 44.4%

       \[\leadsto 263.3831869810514 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{e^{-7.5} \cdot \sqrt{\pi}}}{\frac{z}{\sqrt{7.5} \cdot \sqrt{2}}}\right)} - 1\right) \]

    5. sqrt-unprod 44.4%

       \[\leadsto 263.3831869810514 \cdot \left(e^{\mathsf{log1p}\left(\frac{e^{-7.5} \cdot \sqrt{\pi}}{\frac{z}{\color{blue}{\sqrt{7.5 \cdot 2}}}}\right)} - 1\right) \]

    6. metadata-eval 44.4%

       \[\leadsto 263.3831869810514 \cdot \left(e^{\mathsf{log1p}\left(\frac{e^{-7.5} \cdot \sqrt{\pi}}{\frac{z}{\sqrt{\color{blue}{15}}}}\right)} - 1\right) \]

12. Applied egg-rr 44.4%

    \[\leadsto 263.3831869810514 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{e^{-7.5} \cdot \sqrt{\pi}}{\frac{z}{\sqrt{15}}}\right)} - 1\right)} \]
13. Step-by-step derivation

    1. expm1-def 44.4%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{-7.5} \cdot \sqrt{\pi}}{\frac{z}{\sqrt{15}}}\right)\right)} \]

    2. expm1-log1p 96.7%

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

    3. associate-/r/ 96.7%

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

    4. *-commutative 96.7%

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

    5. *-commutative 96.7%

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

    6. associate-/l* 96.5%

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

14. Simplified 96.5%

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

    1. *-un-lft-identity 96.5%

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

    2. div-inv 96.2%

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

    3. times-frac 96.8%

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

    4. rec-exp 96.8%

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

    5. metadata-eval 96.8%

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

16. Applied egg-rr 96.8%

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

17. Final simplification 96.8%

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

Alternative 7: 96.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ {\left(0.0037967495627271876 \cdot \frac{z}{e^{-7.5} \cdot \sqrt{\pi \cdot 15}}\right)}^{-1} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (pow (* 0.0037967495627271876 (/ z (* (exp -7.5) (sqrt (* PI 15.0))))) -1.0))

C

double code(double z) {
	return pow((0.0037967495627271876 * (z / (exp(-7.5) * sqrt((((double) M_PI) * 15.0))))), -1.0);
}

Java

public static double code(double z) {
	return Math.pow((0.0037967495627271876 * (z / (Math.exp(-7.5) * Math.sqrt((Math.PI * 15.0))))), -1.0);
}

Python

def code(z):
	return math.pow((0.0037967495627271876 * (z / (math.exp(-7.5) * math.sqrt((math.pi * 15.0))))), -1.0)

Julia

function code(z)
	return Float64(0.0037967495627271876 * Float64(z / Float64(exp(-7.5) * sqrt(Float64(pi * 15.0))))) ^ -1.0
end

MATLAB

function tmp = code(z)
	tmp = (0.0037967495627271876 * (z / (exp(-7.5) * sqrt((pi * 15.0))))) ^ -1.0;
end

Wolfram

code[z_] := N[Power[N[(0.0037967495627271876 * N[(z / N[(N[Exp[-7.5], $MachinePrecision] * N[Sqrt[N[(Pi * 15.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 95.4%

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

3. Simplified 96.3%

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

5. Taylor expanded in z around 0 96.5%

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

   1. associate-*r* 96.5%

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

   2. *-commutative 96.5%

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

7. Simplified 96.5%

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

8. Taylor expanded in z around 0 96.4%

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

   1. associate-*l/ 96.2%

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

   2. *-commutative 96.2%

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

   3. associate-*r* 97.0%

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

   4. *-commutative 97.0%

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

10. Simplified 97.0%

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

    1. associate-*r/ 96.8%

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

    2. associate-*l* 95.9%

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

    3. sqrt-unprod 95.9%

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

    4. metadata-eval 95.9%

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

12. Applied egg-rr 95.9%

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

    1. clear-num 95.9%

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

    2. inv-pow 95.9%

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

14. Applied egg-rr 96.8%

    \[\leadsto \color{blue}{{\left(0.0037967495627271876 \cdot \frac{z}{e^{-7.5} \cdot \sqrt{15 \cdot \pi}}\right)}^{-1}} \]

15. Final simplification 96.8%

    \[\leadsto {\left(0.0037967495627271876 \cdot \frac{z}{e^{-7.5} \cdot \sqrt{\pi \cdot 15}}\right)}^{-1} \]
16. Add Preprocessing

Alternative 8: 96.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 96.3%

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

5. Taylor expanded in z around 0 96.5%

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

   1. associate-*r* 96.5%

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

   2. *-commutative 96.5%

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

7. Simplified 96.5%

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

8. Taylor expanded in z around 0 96.4%

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

   1. associate-*l/ 96.2%

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

   2. *-commutative 96.2%

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

   3. associate-*r* 97.0%

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

   4. *-commutative 97.0%

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

10. Simplified 97.0%

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

    1. expm1-log1p-u 44.4%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\sqrt{\pi} \cdot e^{-7.5}\right) \cdot \left(\sqrt{7.5} \cdot \sqrt{2}\right)}{z}\right)\right)} \]

    2. expm1-udef 44.4%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\left(\sqrt{\pi} \cdot e^{-7.5}\right) \cdot \left(\sqrt{7.5} \cdot \sqrt{2}\right)}{z}\right)} - 1\right)} \]

    3. associate-/l* 44.4%

       \[\leadsto 263.3831869810514 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{\pi} \cdot e^{-7.5}}{\frac{z}{\sqrt{7.5} \cdot \sqrt{2}}}}\right)} - 1\right) \]

    4. *-commutative 44.4%

       \[\leadsto 263.3831869810514 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{e^{-7.5} \cdot \sqrt{\pi}}}{\frac{z}{\sqrt{7.5} \cdot \sqrt{2}}}\right)} - 1\right) \]

    5. sqrt-unprod 44.4%

       \[\leadsto 263.3831869810514 \cdot \left(e^{\mathsf{log1p}\left(\frac{e^{-7.5} \cdot \sqrt{\pi}}{\frac{z}{\color{blue}{\sqrt{7.5 \cdot 2}}}}\right)} - 1\right) \]

    6. metadata-eval 44.4%

       \[\leadsto 263.3831869810514 \cdot \left(e^{\mathsf{log1p}\left(\frac{e^{-7.5} \cdot \sqrt{\pi}}{\frac{z}{\sqrt{\color{blue}{15}}}}\right)} - 1\right) \]

12. Applied egg-rr 44.4%

    \[\leadsto 263.3831869810514 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{e^{-7.5} \cdot \sqrt{\pi}}{\frac{z}{\sqrt{15}}}\right)} - 1\right)} \]
13. Step-by-step derivation

    1. expm1-def 44.4%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{-7.5} \cdot \sqrt{\pi}}{\frac{z}{\sqrt{15}}}\right)\right)} \]

    2. expm1-log1p 96.7%

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

    3. associate-/r/ 96.7%

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

    4. *-commutative 96.7%

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

    5. *-commutative 96.7%

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

    6. associate-/l* 96.5%

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

14. Simplified 96.5%

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

    1. associate-/r/ 96.6%

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

16. Applied egg-rr 96.6%

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

17. Final simplification 96.6%

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

Alternative 9: 96.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 96.3%

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

5. Taylor expanded in z around 0 96.5%

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

   1. associate-*r* 96.5%

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

   2. *-commutative 96.5%

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

7. Simplified 96.5%

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

8. Taylor expanded in z around 0 96.4%

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

   1. associate-*l/ 96.2%

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

   2. *-commutative 96.2%

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

   3. associate-*r* 97.0%

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

   4. *-commutative 97.0%

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

10. Simplified 97.0%

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

    1. expm1-log1p-u 44.4%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\sqrt{\pi} \cdot e^{-7.5}\right) \cdot \left(\sqrt{7.5} \cdot \sqrt{2}\right)}{z}\right)\right)} \]

    2. expm1-udef 44.4%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\left(\sqrt{\pi} \cdot e^{-7.5}\right) \cdot \left(\sqrt{7.5} \cdot \sqrt{2}\right)}{z}\right)} - 1\right)} \]

    3. associate-/l* 44.4%

       \[\leadsto 263.3831869810514 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{\pi} \cdot e^{-7.5}}{\frac{z}{\sqrt{7.5} \cdot \sqrt{2}}}}\right)} - 1\right) \]

    4. *-commutative 44.4%

       \[\leadsto 263.3831869810514 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{e^{-7.5} \cdot \sqrt{\pi}}}{\frac{z}{\sqrt{7.5} \cdot \sqrt{2}}}\right)} - 1\right) \]

    5. sqrt-unprod 44.4%

       \[\leadsto 263.3831869810514 \cdot \left(e^{\mathsf{log1p}\left(\frac{e^{-7.5} \cdot \sqrt{\pi}}{\frac{z}{\color{blue}{\sqrt{7.5 \cdot 2}}}}\right)} - 1\right) \]

    6. metadata-eval 44.4%

       \[\leadsto 263.3831869810514 \cdot \left(e^{\mathsf{log1p}\left(\frac{e^{-7.5} \cdot \sqrt{\pi}}{\frac{z}{\sqrt{\color{blue}{15}}}}\right)} - 1\right) \]

12. Applied egg-rr 44.4%

    \[\leadsto 263.3831869810514 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{e^{-7.5} \cdot \sqrt{\pi}}{\frac{z}{\sqrt{15}}}\right)} - 1\right)} \]
13. Step-by-step derivation

    1. expm1-def 44.4%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{-7.5} \cdot \sqrt{\pi}}{\frac{z}{\sqrt{15}}}\right)\right)} \]

    2. expm1-log1p 96.7%

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

    3. associate-/r/ 96.7%

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

    4. *-commutative 96.7%

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

    5. *-commutative 96.7%

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

    6. associate-/l* 96.5%

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

14. Simplified 96.5%

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

    1. *-un-lft-identity 96.5%

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

    2. div-inv 96.2%

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

    3. times-frac 96.8%

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

    4. rec-exp 96.8%

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

    5. metadata-eval 96.8%

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

16. Applied egg-rr 96.8%

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

    1. associate-*l/ 96.7%

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

    2. *-lft-identity 96.7%

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

18. Simplified 96.7%

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

19. Final simplification 96.7%

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

Alternative 10: 95.6% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 96.3%

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

5. Taylor expanded in z around 0 96.5%

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

   1. associate-*r* 96.5%

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

   2. *-commutative 96.5%

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

7. Simplified 96.5%

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

8. Taylor expanded in z around 0 96.4%

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

   1. associate-*l/ 96.2%

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

   2. *-commutative 96.2%

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

   3. associate-*r* 97.0%

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

   4. *-commutative 97.0%

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

10. Simplified 97.0%

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

    1. associate-*r/ 96.8%

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

    2. associate-*l* 95.9%

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

    3. sqrt-unprod 95.9%

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

    4. metadata-eval 95.9%

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

12. Applied egg-rr 95.9%

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

    1. div-inv 95.9%

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

    2. associate-*l* 96.2%

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

    3. *-commutative 96.2%

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

    4. associate-*l* 96.2%

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

    5. add-sqr-sqrt 97.0%

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

    6. sqrt-unprod 96.2%

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

    7. sqrt-unprod 97.0%

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

    8. pow1/2 97.0%

       \[\leadsto 263.3831869810514 \cdot \left(\left(e^{-7.5} \cdot \sqrt{\left(\color{blue}{{15}^{0.5}} \cdot \sqrt{15}\right) \cdot \pi}\right) \cdot \frac{1}{z}\right) \]

    9. pow1/2 97.0%

       \[\leadsto 263.3831869810514 \cdot \left(\left(e^{-7.5} \cdot \sqrt{\left({15}^{0.5} \cdot \color{blue}{{15}^{0.5}}\right) \cdot \pi}\right) \cdot \frac{1}{z}\right) \]

    10. pow-prod-up 96.2%

        \[\leadsto 263.3831869810514 \cdot \left(\left(e^{-7.5} \cdot \sqrt{\color{blue}{{15}^{\left(0.5 + 0.5\right)}} \cdot \pi}\right) \cdot \frac{1}{z}\right) \]

    11. metadata-eval 96.2%

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

    12. metadata-eval 96.2%

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

14. Applied egg-rr 96.2%

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

15. Final simplification 96.2%

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

Alternative 11: 95.7% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 96.3%

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

5. Taylor expanded in z around 0 96.5%

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

   1. associate-*r* 96.5%

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

   2. *-commutative 96.5%

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

7. Simplified 96.5%

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

8. Taylor expanded in z around 0 96.4%

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

   1. associate-*l/ 96.2%

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

   2. *-commutative 96.2%

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

   3. associate-*r* 97.0%

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

   4. *-commutative 97.0%

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

10. Simplified 97.0%

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

    1. associate-*r/ 96.8%

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

    2. associate-*l* 95.9%

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

    3. sqrt-unprod 95.9%

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

    4. metadata-eval 95.9%

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

12. Applied egg-rr 95.9%

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

    1. associate-/l* 96.2%

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

    2. div-inv 96.2%

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

    3. *-commutative 96.2%

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

    4. associate-*l* 96.2%

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

    5. add-sqr-sqrt 96.8%

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

    6. sqrt-unprod 96.2%

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

    7. sqrt-unprod 96.8%

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

    8. pow1/2 96.8%

       \[\leadsto 263.3831869810514 \cdot \frac{1}{\frac{z}{e^{-7.5} \cdot \sqrt{\left(\color{blue}{{15}^{0.5}} \cdot \sqrt{15}\right) \cdot \pi}}} \]

    9. pow1/2 96.8%

       \[\leadsto 263.3831869810514 \cdot \frac{1}{\frac{z}{e^{-7.5} \cdot \sqrt{\left({15}^{0.5} \cdot \color{blue}{{15}^{0.5}}\right) \cdot \pi}}} \]

    10. pow-prod-up 96.2%

        \[\leadsto 263.3831869810514 \cdot \frac{1}{\frac{z}{e^{-7.5} \cdot \sqrt{\color{blue}{{15}^{\left(0.5 + 0.5\right)}} \cdot \pi}}} \]

    11. metadata-eval 96.2%

        \[\leadsto 263.3831869810514 \cdot \frac{1}{\frac{z}{e^{-7.5} \cdot \sqrt{{15}^{\color{blue}{1}} \cdot \pi}}} \]

    12. metadata-eval 96.2%

        \[\leadsto 263.3831869810514 \cdot \frac{1}{\frac{z}{e^{-7.5} \cdot \sqrt{\color{blue}{15} \cdot \pi}}} \]

14. Applied egg-rr 96.2%

    \[\leadsto \color{blue}{263.3831869810514 \cdot \frac{1}{\frac{z}{e^{-7.5} \cdot \sqrt{15 \cdot \pi}}}} \]

15. Final simplification 96.2%

    \[\leadsto 263.3831869810514 \cdot \frac{1}{\frac{z}{e^{-7.5} \cdot \sqrt{\pi \cdot 15}}} \]
16. Add Preprocessing

Alternative 12: 95.6% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 96.3%

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

5. Taylor expanded in z around 0 96.5%

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

   1. associate-*r* 96.5%

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

   2. *-commutative 96.5%

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

7. Simplified 96.5%

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

8. Taylor expanded in z around 0 96.4%

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

   1. associate-*l/ 96.2%

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

   2. *-commutative 96.2%

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

   3. associate-*r* 97.0%

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

   4. *-commutative 97.0%

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

10. Simplified 97.0%

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

    1. expm1-log1p-u 44.4%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\sqrt{\pi} \cdot e^{-7.5}\right) \cdot \left(\sqrt{7.5} \cdot \sqrt{2}\right)}{z}\right)\right)} \]

    2. expm1-udef 44.4%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\left(\sqrt{\pi} \cdot e^{-7.5}\right) \cdot \left(\sqrt{7.5} \cdot \sqrt{2}\right)}{z}\right)} - 1\right)} \]

    3. associate-/l* 44.4%

       \[\leadsto 263.3831869810514 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{\pi} \cdot e^{-7.5}}{\frac{z}{\sqrt{7.5} \cdot \sqrt{2}}}}\right)} - 1\right) \]

    4. *-commutative 44.4%

       \[\leadsto 263.3831869810514 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{e^{-7.5} \cdot \sqrt{\pi}}}{\frac{z}{\sqrt{7.5} \cdot \sqrt{2}}}\right)} - 1\right) \]

    5. sqrt-unprod 44.4%

       \[\leadsto 263.3831869810514 \cdot \left(e^{\mathsf{log1p}\left(\frac{e^{-7.5} \cdot \sqrt{\pi}}{\frac{z}{\color{blue}{\sqrt{7.5 \cdot 2}}}}\right)} - 1\right) \]

    6. metadata-eval 44.4%

       \[\leadsto 263.3831869810514 \cdot \left(e^{\mathsf{log1p}\left(\frac{e^{-7.5} \cdot \sqrt{\pi}}{\frac{z}{\sqrt{\color{blue}{15}}}}\right)} - 1\right) \]

12. Applied egg-rr 44.4%

    \[\leadsto 263.3831869810514 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{e^{-7.5} \cdot \sqrt{\pi}}{\frac{z}{\sqrt{15}}}\right)} - 1\right)} \]
13. Step-by-step derivation

    1. expm1-def 44.4%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{-7.5} \cdot \sqrt{\pi}}{\frac{z}{\sqrt{15}}}\right)\right)} \]

    2. expm1-log1p 96.7%

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

    3. associate-/r/ 96.7%

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

    4. *-commutative 96.7%

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

    5. *-commutative 96.7%

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

    6. associate-/l* 96.5%

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

14. Simplified 96.5%

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

16. Applied egg-rr 44.4%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(263.3831869810514 \cdot \left(\sqrt{15 \cdot \pi} \cdot \frac{e^{-7.5}}{z}\right)\right)} - 1} \]
17. Step-by-step derivation

    1. expm1-def 44.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(263.3831869810514 \cdot \left(\sqrt{15 \cdot \pi} \cdot \frac{e^{-7.5}}{z}\right)\right)\right)} \]

    2. expm1-log1p 96.1%

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

    3. *-commutative 96.1%

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

18. Simplified 96.1%

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

19. Final simplification 96.1%

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

Alternative 13: 95.7% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 96.3%

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

5. Taylor expanded in z around 0 96.5%

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

   1. associate-*r* 96.5%

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

   2. *-commutative 96.5%

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

7. Simplified 96.5%

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

8. Taylor expanded in z around 0 96.4%

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

   1. associate-*l/ 96.2%

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

   2. *-commutative 96.2%

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

   3. associate-*r* 97.0%

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

   4. *-commutative 97.0%

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

10. Simplified 97.0%

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

    1. associate-*r/ 96.8%

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

    2. associate-*l* 95.9%

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

    3. sqrt-unprod 95.9%

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

    4. metadata-eval 95.9%

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

12. Applied egg-rr 95.9%

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

    1. *-un-lft-identity 95.9%

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

    2. times-frac 96.2%

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

    3. metadata-eval 96.2%

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

    4. *-commutative 96.2%

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

    5. associate-*l* 96.2%

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

    6. add-sqr-sqrt 97.0%

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

    7. sqrt-unprod 96.2%

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

    8. sqrt-unprod 97.0%

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

    9. pow1/2 97.0%

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

    10. pow1/2 97.0%

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

    11. pow-prod-up 96.2%

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

    12. metadata-eval 96.2%

        \[\leadsto 263.3831869810514 \cdot \frac{e^{-7.5} \cdot \sqrt{{15}^{\color{blue}{1}} \cdot \pi}}{z} \]

    13. metadata-eval 96.2%

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

14. Applied egg-rr 96.2%

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

15. Final simplification 96.2%

    \[\leadsto 263.3831869810514 \cdot \frac{e^{-7.5} \cdot \sqrt{\pi \cdot 15}}{z} \]
16. Add Preprocessing

Alternative 14: 95.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{\pi \cdot \left(e^{-15} \cdot 1040560.5477644323\right)}}{z} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (/ (sqrt (* PI (* (exp -15.0) 1040560.5477644323))) z))

C

double code(double z) {
	return sqrt((((double) M_PI) * (exp(-15.0) * 1040560.5477644323))) / z;
}

Java

public static double code(double z) {
	return Math.sqrt((Math.PI * (Math.exp(-15.0) * 1040560.5477644323))) / z;
}

Python

def code(z):
	return math.sqrt((math.pi * (math.exp(-15.0) * 1040560.5477644323))) / z

Julia

function code(z)
	return Float64(sqrt(Float64(pi * Float64(exp(-15.0) * 1040560.5477644323))) / z)
end

MATLAB

function tmp = code(z)
	tmp = sqrt((pi * (exp(-15.0) * 1040560.5477644323))) / z;
end

Wolfram

code[z_] := N[(N[Sqrt[N[(Pi * N[(N[Exp[-15.0], $MachinePrecision] * 1040560.5477644323), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / z), $MachinePrecision]

Derivation

1. Initial program 95.4%

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

3. Simplified 96.3%

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

5. Taylor expanded in z around 0 96.5%

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

   1. associate-*r* 96.5%

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

   2. *-commutative 96.5%

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

7. Simplified 96.5%

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

8. Taylor expanded in z around 0 96.4%

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

   1. associate-*l/ 96.2%

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

   2. *-commutative 96.2%

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

   3. associate-*r* 97.0%

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

   4. *-commutative 97.0%

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

10. Simplified 97.0%

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

    1. associate-*r/ 96.8%

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

    2. associate-*l* 95.9%

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

    3. sqrt-unprod 95.9%

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

    4. metadata-eval 95.9%

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

12. Applied egg-rr 95.9%

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

    1. add-sqr-sqrt 95.9%

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

    2. sqrt-unprod 95.9%

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

    3. *-commutative 95.9%

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

    4. *-commutative 95.9%

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

    5. swap-sqr 95.9%

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

14. Applied egg-rr 95.9%

    \[\leadsto \frac{\color{blue}{\sqrt{\left(\pi \cdot \left(15 \cdot e^{-15}\right)\right) \cdot 69370.70318429549}}}{z} \]
15. Step-by-step derivation

    1. associate-*l* 95.9%

       \[\leadsto \frac{\sqrt{\color{blue}{\pi \cdot \left(\left(15 \cdot e^{-15}\right) \cdot 69370.70318429549\right)}}}{z} \]

    2. *-commutative 95.9%

       \[\leadsto \frac{\sqrt{\pi \cdot \left(\color{blue}{\left(e^{-15} \cdot 15\right)} \cdot 69370.70318429549\right)}}{z} \]

    3. associate-*l* 96.8%

       \[\leadsto \frac{\sqrt{\pi \cdot \color{blue}{\left(e^{-15} \cdot \left(15 \cdot 69370.70318429549\right)\right)}}}{z} \]

    4. metadata-eval 95.9%

       \[\leadsto \frac{\sqrt{\pi \cdot \left(e^{-15} \cdot \color{blue}{1040560.5477644323}\right)}}{z} \]

16. Simplified 95.9%

    \[\leadsto \frac{\color{blue}{\sqrt{\pi \cdot \left(e^{-15} \cdot 1040560.5477644323\right)}}}{z} \]

17. Final simplification 95.9%

    \[\leadsto \frac{\sqrt{\pi \cdot \left(e^{-15} \cdot 1040560.5477644323\right)}}{z} \]
18. Add Preprocessing

Reproduce

herbie shell --seed 2024030 
(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))))))