Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.3% → 98.0%

Time: 1.8min

Alternatives: 10

Speedup: 1.1×

• could not determine a ground truth

  1. z = -5.0189368750157564e+249

Specification

\[z \leq 0.5\]

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)) (t_1 (+ t_0 7.0)) (t_2 (+ t_1 0.5)))
   (*
    (/ PI (sin (* PI z)))
    (*
     (* (* (sqrt (* PI 2.0)) (pow t_2 (+ t_0 0.5))) (exp (- t_2)))
     (+
      (+
       (+
        (+
         (+
          (+
           (+
            (+ 0.9999999999998099 (/ 676.5203681218851 (+ t_0 1.0)))
            (/ -1259.1392167224028 (+ t_0 2.0)))
           (/ 771.3234287776531 (+ t_0 3.0)))
          (/ -176.6150291621406 (+ t_0 4.0)))
         (/ 12.507343278686905 (+ t_0 5.0)))
        (/ -0.13857109526572012 (+ t_0 6.0)))
       (/ 9.984369578019572e-6 t_1))
      (/ 1.5056327351493116e-7 (+ t_0 8.0)))))))

C

double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = t_0 + 7.0;
	double t_2 = t_1 + 0.5;
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt((((double) M_PI) * 2.0)) * pow(t_2, (t_0 + 0.5))) * exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = t_0 + 7.0;
	double t_2 = t_1 + 0.5;
	return (Math.PI / Math.sin((Math.PI * z))) * (((Math.sqrt((Math.PI * 2.0)) * Math.pow(t_2, (t_0 + 0.5))) * Math.exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
}

Python

def code(z):
	t_0 = (1.0 - z) - 1.0
	t_1 = t_0 + 7.0
	t_2 = t_1 + 0.5
	return (math.pi / math.sin((math.pi * z))) * (((math.sqrt((math.pi * 2.0)) * math.pow(t_2, (t_0 + 0.5))) * math.exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - 1.0)
	t_1 = Float64(t_0 + 7.0)
	t_2 = Float64(t_1 + 0.5)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (t_2 ^ Float64(t_0 + 0.5))) * exp(Float64(-t_2))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(t_0 + 1.0))) + Float64(-1259.1392167224028 / Float64(t_0 + 2.0))) + Float64(771.3234287776531 / Float64(t_0 + 3.0))) + Float64(-176.6150291621406 / Float64(t_0 + 4.0))) + Float64(12.507343278686905 / Float64(t_0 + 5.0))) + Float64(-0.13857109526572012 / Float64(t_0 + 6.0))) + Float64(9.984369578019572e-6 / t_1)) + Float64(1.5056327351493116e-7 / Float64(t_0 + 8.0)))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - 1.0;
	t_1 = t_0 + 7.0;
	t_2 = t_1 + 0.5;
	tmp = (pi / sin((pi * z))) * (((sqrt((pi * 2.0)) * (t_2 ^ (t_0 + 0.5))) * exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 7.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + 0.5), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$2, N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$2)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(t$95$0 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(t$95$0 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(t$95$0 + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(t$95$0 + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(t$95$0 + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Initial Program: 96.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)) (t_1 (+ t_0 7.0)) (t_2 (+ t_1 0.5)))
   (*
    (/ PI (sin (* PI z)))
    (*
     (* (* (sqrt (* PI 2.0)) (pow t_2 (+ t_0 0.5))) (exp (- t_2)))
     (+
      (+
       (+
        (+
         (+
          (+
           (+
            (+ 0.9999999999998099 (/ 676.5203681218851 (+ t_0 1.0)))
            (/ -1259.1392167224028 (+ t_0 2.0)))
           (/ 771.3234287776531 (+ t_0 3.0)))
          (/ -176.6150291621406 (+ t_0 4.0)))
         (/ 12.507343278686905 (+ t_0 5.0)))
        (/ -0.13857109526572012 (+ t_0 6.0)))
       (/ 9.984369578019572e-6 t_1))
      (/ 1.5056327351493116e-7 (+ t_0 8.0)))))))

C

double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = t_0 + 7.0;
	double t_2 = t_1 + 0.5;
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt((((double) M_PI) * 2.0)) * pow(t_2, (t_0 + 0.5))) * exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = t_0 + 7.0;
	double t_2 = t_1 + 0.5;
	return (Math.PI / Math.sin((Math.PI * z))) * (((Math.sqrt((Math.PI * 2.0)) * Math.pow(t_2, (t_0 + 0.5))) * Math.exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
}

Python

def code(z):
	t_0 = (1.0 - z) - 1.0
	t_1 = t_0 + 7.0
	t_2 = t_1 + 0.5
	return (math.pi / math.sin((math.pi * z))) * (((math.sqrt((math.pi * 2.0)) * math.pow(t_2, (t_0 + 0.5))) * math.exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - 1.0)
	t_1 = Float64(t_0 + 7.0)
	t_2 = Float64(t_1 + 0.5)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (t_2 ^ Float64(t_0 + 0.5))) * exp(Float64(-t_2))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(t_0 + 1.0))) + Float64(-1259.1392167224028 / Float64(t_0 + 2.0))) + Float64(771.3234287776531 / Float64(t_0 + 3.0))) + Float64(-176.6150291621406 / Float64(t_0 + 4.0))) + Float64(12.507343278686905 / Float64(t_0 + 5.0))) + Float64(-0.13857109526572012 / Float64(t_0 + 6.0))) + Float64(9.984369578019572e-6 / t_1)) + Float64(1.5056327351493116e-7 / Float64(t_0 + 8.0)))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - 1.0;
	t_1 = t_0 + 7.0;
	t_2 = t_1 + 0.5;
	tmp = (pi / sin((pi * z))) * (((sqrt((pi * 2.0)) * (t_2 ^ (t_0 + 0.5))) * exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 7.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + 0.5), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$2, N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$2)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(t$95$0 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(t$95$0 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(t$95$0 + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(t$95$0 + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(t$95$0 + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Alternative 1: 98.0% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (/ PI (sin (* PI z)))
   (* (pow (- 7.5 z) (- 0.5 z)) (* (exp (+ z -7.5)) (sqrt (* PI 2.0)))))
  (+
   (+
    (+
     (+
      0.9999999999998099
      (+ (/ 676.5203681218851 (- 1.0 z)) (/ -1259.1392167224028 (- 2.0 z))))
     (+ (/ -176.6150291621406 (- 4.0 z)) (/ 771.3234287776531 (- 3.0 z))))
    (+
     (/ 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))))))

C

double code(double z) {
	return ((((double) M_PI) / sin((((double) M_PI) * z))) * (pow((7.5 - z), (0.5 - z)) * (exp((z + -7.5)) * sqrt((((double) M_PI) * 2.0))))) * ((((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + ((-176.6150291621406 / (4.0 - z)) + (771.3234287776531 / (3.0 - z)))) + ((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))));
}

Java

public static double code(double z) {
	return ((Math.PI / Math.sin((Math.PI * z))) * (Math.pow((7.5 - z), (0.5 - z)) * (Math.exp((z + -7.5)) * Math.sqrt((Math.PI * 2.0))))) * ((((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + ((-176.6150291621406 / (4.0 - z)) + (771.3234287776531 / (3.0 - z)))) + ((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))));
}

Python

def code(z):
	return ((math.pi / math.sin((math.pi * z))) * (math.pow((7.5 - z), (0.5 - z)) * (math.exp((z + -7.5)) * math.sqrt((math.pi * 2.0))))) * ((((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + ((-176.6150291621406 / (4.0 - z)) + (771.3234287776531 / (3.0 - z)))) + ((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))))

Julia

function code(z)
	return Float64(Float64(Float64(pi / sin(Float64(pi * z))) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * Float64(exp(Float64(z + -7.5)) * sqrt(Float64(pi * 2.0))))) * Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(-1259.1392167224028 / Float64(2.0 - z)))) + Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(771.3234287776531 / Float64(3.0 - z)))) + 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)))))
end

MATLAB

function tmp = code(z)
	tmp = ((pi / sin((pi * z))) * (((7.5 - z) ^ (0.5 - z)) * (exp((z + -7.5)) * sqrt((pi * 2.0))))) * ((((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + ((-176.6150291621406 / (4.0 - z)) + (771.3234287776531 / (3.0 - z)))) + ((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))));
end

Wolfram

code[z_] := N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.9999999999998099 + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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]), $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]

Derivation

1. Initial program 97.0%

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

3. Simplified 98.6%

   \[\leadsto \color{blue}{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right)} \]
4. Step-by-step derivation

   1. expm1-log1p-u 98.6%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right)}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   2. expm1-udef 89.5%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)} - 1\right)}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

5. Applied egg-rr 89.5%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)} \cdot e^{-\mathsf{fma}\left(-1, z, 7.5\right)}\right)\right)} - 1\right)}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
6. Step-by-step derivation

   1. expm1-def 98.6%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)} \cdot e^{-\mathsf{fma}\left(-1, z, 7.5\right)}\right)\right)\right)}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   2. expm1-log1p 98.6%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)} \cdot e^{-\mathsf{fma}\left(-1, z, 7.5\right)}\right)\right)}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   3. *-commutative 98.6%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(\left({\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)} \cdot e^{-\mathsf{fma}\left(-1, z, 7.5\right)}\right) \cdot \sqrt{\pi \cdot 2}\right)}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   4. fma-udef 98.6%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left({\color{blue}{\left(-1 \cdot z + 7.5\right)}}^{\left(0.5 - z\right)} \cdot e^{-\mathsf{fma}\left(-1, z, 7.5\right)}\right) \cdot \sqrt{\pi \cdot 2}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   5. neg-mul-1 98.6%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left({\left(\color{blue}{\left(-z\right)} + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{-\mathsf{fma}\left(-1, z, 7.5\right)}\right) \cdot \sqrt{\pi \cdot 2}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   6. +-commutative 98.6%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left({\color{blue}{\left(7.5 + \left(-z\right)\right)}}^{\left(0.5 - z\right)} \cdot e^{-\mathsf{fma}\left(-1, z, 7.5\right)}\right) \cdot \sqrt{\pi \cdot 2}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   7. sub-neg 98.6%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left({\color{blue}{\left(7.5 - z\right)}}^{\left(0.5 - z\right)} \cdot e^{-\mathsf{fma}\left(-1, z, 7.5\right)}\right) \cdot \sqrt{\pi \cdot 2}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   8. exp-to-pow 98.6%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{e^{\log \left(7.5 - z\right) \cdot \left(0.5 - z\right)}} \cdot e^{-\mathsf{fma}\left(-1, z, 7.5\right)}\right) \cdot \sqrt{\pi \cdot 2}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   9. associate-*l* 98.6%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(e^{\log \left(7.5 - z\right) \cdot \left(0.5 - z\right)} \cdot \left(e^{-\mathsf{fma}\left(-1, z, 7.5\right)} \cdot \sqrt{\pi \cdot 2}\right)\right)}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   10. exp-to-pow 98.6%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\color{blue}{{\left(7.5 - z\right)}^{\left(0.5 - z\right)}} \cdot \left(e^{-\mathsf{fma}\left(-1, z, 7.5\right)} \cdot \sqrt{\pi \cdot 2}\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   11. *-commutative 98.6%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot e^{-\mathsf{fma}\left(-1, z, 7.5\right)}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   12. *-commutative 98.6%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{\color{blue}{2 \cdot \pi}} \cdot e^{-\mathsf{fma}\left(-1, z, 7.5\right)}\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   13. fma-udef 98.6%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{-\color{blue}{\left(-1 \cdot z + 7.5\right)}}\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   14. neg-mul-1 98.6%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{-\left(\color{blue}{\left(-z\right)} + 7.5\right)}\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   15. +-commutative 98.6%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{-\color{blue}{\left(7.5 + \left(-z\right)\right)}}\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

7. Simplified 98.6%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{z + -7.5}\right)\right)}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
8. Step-by-step derivation

   1. *-un-lft-identity 98.6%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{z + -7.5}\right)\right)\right) \cdot \left(\left(\left(\color{blue}{1 \cdot \left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   2. associate-+l+ 98.6%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{z + -7.5}\right)\right)\right) \cdot \left(\left(\left(1 \cdot \color{blue}{\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   3. --rgt-identity 98.6%

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

   4. metadata-eval 98.6%

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

   5. associate-+l- 98.6%

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

   6. +-commutative 98.6%

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

   7. expm1-log1p-u 98.6%

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

   8. log1p-udef 98.6%

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

   9. +-commutative 98.6%

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

   10. associate-+l- 98.6%

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

   11. metadata-eval 98.6%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{z + -7.5}\right)\right)\right) \cdot \left(\left(\left(1 \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \mathsf{expm1}\left(\log \left(\left(1 - z\right) - \color{blue}{0}\right)\right)}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   12. --rgt-identity 98.6%

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

   13. sub-neg 98.6%

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

   14. log1p-udef 98.6%

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

   15. expm1-log1p-u 98.6%

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

   16. sub-neg 98.6%

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

9. Applied egg-rr 98.6%

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

    1. expm1-log1p-u 97.3%

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

    2. expm1-udef 97.3%

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

    3. sub-neg 97.3%

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

    4. metadata-eval 97.3%

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

    5. associate--l- 97.3%

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

11. Applied egg-rr 97.3%

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

    1. expm1-def 97.3%

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

    2. expm1-log1p 98.6%

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

    3. +-commutative 98.6%

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

    4. +-commutative 98.6%

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

    5. associate--r+ 98.6%

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

    6. metadata-eval 98.6%

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

    7. +-commutative 98.6%

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

    8. associate-+r- 98.6%

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

    9. metadata-eval 98.6%

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

13. Simplified 98.6%

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

14. Final simplification 98.6%

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

Alternative 2: 96.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.0%

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

3. Simplified 96.2%

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

5. Applied egg-rr 97.4%

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

   1. expm1-def 98.3%

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

   2. expm1-log1p 96.7%

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

   3. *-commutative 96.7%

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

   4. fma-udef 96.7%

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

   5. neg-mul-1 96.7%

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

   6. +-commutative 96.7%

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

   7. sub-neg 96.7%

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

   8. *-commutative 96.7%

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

7. Simplified 96.7%

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

   1. expm1-log1p-u 96.7%

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

   2. expm1-udef 96.7%

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

9. Applied egg-rr 96.7%

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

    1. expm1-def 96.7%

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

    2. expm1-log1p 97.7%

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

    3. +-commutative 97.7%

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

    4. associate-+l+ 98.8%

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

11. Simplified 97.7%

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

12. Final simplification 97.7%

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

Alternative 3: 96.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.0%

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

3. Simplified 96.2%

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

5. Applied egg-rr 97.4%

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

   1. expm1-def 98.3%

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

   2. expm1-log1p 96.7%

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

   3. *-commutative 96.7%

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

   4. fma-udef 96.7%

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

   5. neg-mul-1 96.7%

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

   6. +-commutative 96.7%

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

   7. sub-neg 96.7%

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

   8. *-commutative 96.7%

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

7. Simplified 96.7%

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

   1. expm1-log1p-u 96.7%

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

   2. expm1-udef 96.7%

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

9. Applied egg-rr 96.7%

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

    1. expm1-def 96.7%

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

    2. expm1-log1p 97.7%

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

    3. associate-+l+ 97.7%

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

11. Simplified 97.7%

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

12. Final simplification 97.7%

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

Alternative 4: 97.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.0%

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

3. Simplified 96.2%

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

   1. add-cube-cbrt 96.9%

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

   2. pow3 98.0%

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

5. Applied egg-rr 97.5%

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

   1. rem-cube-cbrt 97.0%

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

   2. associate-+l+ 98.1%

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

   3. associate-+r+ 96.9%

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

   4. associate-+r+ 96.9%

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

7. Applied egg-rr 96.9%

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

   1. associate-+r+ 96.2%

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

   2. +-commutative 96.2%

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

   3. associate-+l+ 96.9%

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

   4. +-commutative 96.9%

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

   5. associate-+l+ 98.1%

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

9. Simplified 98.1%

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

10. Final simplification 98.1%

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

Alternative 5: 96.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.0%

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

3. Simplified 96.2%

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

4. Taylor expanded in z around 0 97.6%

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

   1. *-commutative 97.6%

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

6. Simplified 97.6%

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

   1. associate-*r/ 97.7%

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

   2. associate-*l* 96.9%

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

   3. neg-mul-1 96.9%

      \[\leadsto \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \frac{e^{z + -7.5} \cdot \left(\pi \cdot \left({\left(\color{blue}{-1 \cdot z} + 7.5\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right)\right)}{\sin \left(\pi \cdot z\right)} \]

   4. fma-def 96.9%

      \[\leadsto \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \frac{e^{z + -7.5} \cdot \left(\pi \cdot \left({\color{blue}{\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right)\right)}{\sin \left(\pi \cdot z\right)} \]

   5. *-commutative 96.9%

      \[\leadsto \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \frac{e^{z + -7.5} \cdot \left(\pi \cdot \left({\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\color{blue}{2 \cdot \pi}}\right)\right)}{\sin \left(\pi \cdot z\right)} \]

8. Applied egg-rr 96.9%

   \[\leadsto \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \color{blue}{\frac{e^{z + -7.5} \cdot \left(\pi \cdot \left({\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)} \cdot \sqrt{2 \cdot \pi}\right)\right)}{\sin \left(\pi \cdot z\right)}} \]
9. Step-by-step derivation

   1. +-commutative 96.9%

      \[\leadsto \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \frac{e^{\color{blue}{-7.5 + z}} \cdot \left(\pi \cdot \left({\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)} \cdot \sqrt{2 \cdot \pi}\right)\right)}{\sin \left(\pi \cdot z\right)} \]

   2. *-commutative 96.9%

      \[\leadsto \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \frac{e^{-7.5 + z} \cdot \left(\pi \cdot \color{blue}{\left(\sqrt{2 \cdot \pi} \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)}\right)}\right)}{\sin \left(\pi \cdot z\right)} \]

   3. *-commutative 96.9%

      \[\leadsto \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \frac{e^{-7.5 + z} \cdot \left(\pi \cdot \left(\sqrt{\color{blue}{\pi \cdot 2}} \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)}\right)\right)}{\sin \left(\pi \cdot z\right)} \]

   4. fma-udef 96.9%

      \[\leadsto \left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \frac{e^{-7.5 + z} \cdot \left(\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot {\color{blue}{\left(-1 \cdot z + 7.5\right)}}^{\left(0.5 - z\right)}\right)\right)}{\sin \left(\pi \cdot z\right)} \]

   5. mul-1-neg 96.9%

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

   6. +-commutative 96.9%

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

   7. sub-neg 96.9%

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

10. Simplified 96.9%

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

11. Final simplification 96.9%

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

Alternative 6: 96.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.0%

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

3. Simplified 96.2%

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

4. Taylor expanded in z around 0 97.6%

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

   1. *-commutative 97.6%

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

6. Simplified 97.6%

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

   1. pow1 97.6%

      \[\leadsto \color{blue}{{\left(\left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right)\right)}^{1}} \]

8. Applied egg-rr 97.3%

   \[\leadsto \color{blue}{{\left(\left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \left(e^{z + -7.5} \cdot \frac{\pi \cdot {\left(7.5 + \left(-z\right)\right)}^{\left(0.5 - z\right)}}{\frac{\sin \left(z \cdot \pi\right)}{\sqrt{2 \cdot \pi}}}\right)\right)}^{1}} \]
9. Step-by-step derivation

   1. unpow1 97.3%

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

   2. *-commutative 97.3%

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

   3. associate-*l/ 97.5%

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

   4. *-commutative 97.5%

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

   5. sub-neg 97.5%

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

10. Simplified 97.5%

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

11. Final simplification 97.5%

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

Alternative 7: 96.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.0%

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

3. Simplified 96.2%

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

4. Taylor expanded in z around 0 97.6%

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

   1. *-commutative 97.6%

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

6. Simplified 97.6%

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

7. Final simplification 97.6%

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

Alternative 8: 95.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{\left(z + -1\right) - 6.5}\right)\right) \cdot \left(436.8961725563396 + \frac{263.3831869810514}{z}\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (pow (+ (- 1.0 z) 6.5) (- 1.0 (+ z 0.5)))
   (* (sqrt (* PI 2.0)) (exp (- (+ z -1.0) 6.5))))
  (+ 436.8961725563396 (/ 263.3831869810514 z))))

C

double code(double z) {
	return (pow(((1.0 - z) + 6.5), (1.0 - (z + 0.5))) * (sqrt((((double) M_PI) * 2.0)) * exp(((z + -1.0) - 6.5)))) * (436.8961725563396 + (263.3831869810514 / z));
}

Java

public static double code(double z) {
	return (Math.pow(((1.0 - z) + 6.5), (1.0 - (z + 0.5))) * (Math.sqrt((Math.PI * 2.0)) * Math.exp(((z + -1.0) - 6.5)))) * (436.8961725563396 + (263.3831869810514 / z));
}

Python

def code(z):
	return (math.pow(((1.0 - z) + 6.5), (1.0 - (z + 0.5))) * (math.sqrt((math.pi * 2.0)) * math.exp(((z + -1.0) - 6.5)))) * (436.8961725563396 + (263.3831869810514 / z))

Julia

function code(z)
	return Float64(Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(1.0 - Float64(z + 0.5))) * Float64(sqrt(Float64(pi * 2.0)) * exp(Float64(Float64(z + -1.0) - 6.5)))) * Float64(436.8961725563396 + Float64(263.3831869810514 / z)))
end

MATLAB

function tmp = code(z)
	tmp = ((((1.0 - z) + 6.5) ^ (1.0 - (z + 0.5))) * (sqrt((pi * 2.0)) * exp(((z + -1.0) - 6.5)))) * (436.8961725563396 + (263.3831869810514 / z));
end

Wolfram

code[z_] := N[(N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(1.0 - N[(z + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(z + -1.0), $MachinePrecision] - 6.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(436.8961725563396 + N[(263.3831869810514 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.0%

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

3. Simplified 98.4%

   \[\leadsto \color{blue}{\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)} \]

4. Taylor expanded in z around 0 97.1%

   \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(260.9048120626994 + 436.3997278161676 \cdot z\right)} + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
5. Step-by-step derivation

   1. *-commutative 97.1%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + \color{blue}{z \cdot 436.3997278161676}\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

6. Simplified 97.1%

   \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(260.9048120626994 + z \cdot 436.3997278161676\right)} + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
7. Step-by-step derivation

   1. expm1-log1p-u 97.1%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right)}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   2. expm1-udef 97.1%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)} - 1\right)}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   3. associate-+l+ 97.1%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{12.507343278686905}{\left(1 - z\right) + 4} + \left(\frac{-0.13857109526572012}{1 - \left(z + -5\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)}\right)} - 1\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   4. associate--r+ 97.1%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(e^{\mathsf{log1p}\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \left(\frac{-0.13857109526572012}{\color{blue}{\left(1 - z\right) - -5}} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right)} - 1\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

8. Applied egg-rr 97.1%

   \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right)} - 1\right)}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
9. Step-by-step derivation

   1. expm1-def 97.1%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right)\right)}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   2. expm1-log1p 97.1%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \color{blue}{\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right)}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   3. associate-+r+ 97.1%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \color{blue}{\left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   4. +-commutative 97.1%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\color{blue}{\left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   5. associate-+l+ 97.1%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \color{blue}{\left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right)}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   6. sub-neg 97.1%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\frac{-0.13857109526572012}{\color{blue}{\left(1 - z\right) + \left(--5\right)}} + \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   7. metadata-eval 97.1%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) + \color{blue}{5}} + \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. +-commutative 97.1%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(\frac{12.507343278686905}{\color{blue}{4 + \left(1 - z\right)}} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   9. associate-+r- 97.1%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(\frac{12.507343278686905}{\color{blue}{\left(4 + 1\right) - z}} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   10. metadata-eval 97.1%

       \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(\frac{12.507343278686905}{\color{blue}{5} - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

10. Simplified 97.1%

    \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \color{blue}{\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

11. Taylor expanded in z around 0 96.9%

    \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\left(436.8961725563396 + 263.3831869810514 \cdot \frac{1}{z}\right)} \]
12. Step-by-step derivation

    1. associate-*r/ 96.9%

       \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(436.8961725563396 + \color{blue}{\frac{263.3831869810514 \cdot 1}{z}}\right) \]

    2. metadata-eval 96.9%

       \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(436.8961725563396 + \frac{\color{blue}{263.3831869810514}}{z}\right) \]

13. Simplified 96.9%

    \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\left(436.8961725563396 + \frac{263.3831869810514}{z}\right)} \]

14. Final simplification 96.9%

    \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{\left(z + -1\right) - 6.5}\right)\right) \cdot \left(436.8961725563396 + \frac{263.3831869810514}{z}\right) \]

Alternative 9: 95.8% accurate, 2.0× speedup

Math

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

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(sqrt(pi) * Float64(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[Sqrt[Pi], $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.0%

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

3. Simplified 96.2%

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

4. Taylor expanded in z around 0 96.8%

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

5. Taylor expanded in z around 0 96.5%

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

   1. *-un-lft-identity 96.5%

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

   2. associate-/l* 96.6%

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

   3. sqrt-unprod 96.6%

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

   4. metadata-eval 96.6%

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

7. Applied egg-rr 96.6%

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

   1. *-lft-identity 96.6%

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

   2. associate-/r/ 96.7%

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

9. Simplified 96.7%

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

10. Taylor expanded in z around 0 96.5%

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

    1. associate-*l/ 96.7%

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

    2. associate-*r* 96.3%

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

    3. *-commutative 96.3%

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

    4. associate-*l* 96.7%

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

12. Simplified 96.7%

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

13. Final simplification 96.7%

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

Alternative 10: 95.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.0%

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

3. Simplified 96.2%

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

4. Taylor expanded in z around 0 96.8%

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

5. Taylor expanded in z around 0 96.5%

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

   1. pow1 96.5%

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

   2. *-commutative 96.5%

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

   3. associate-/l* 96.6%

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

   4. sqrt-unprod 96.6%

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

   5. metadata-eval 96.6%

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

7. Applied egg-rr 96.6%

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

   1. unpow1 96.6%

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

   2. associate-*r* 96.7%

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

   3. associate-/r/ 96.8%

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

9. Simplified 96.8%

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

10. Final simplification 96.8%

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

Reproduce

herbie shell --seed 2023287 
(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))))))