Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.5% → 98.2%

Time: 1.4min

Alternatives: 11

Speedup: 1.1×

• could not determine a ground truth

  1. z = -8.418160024556002e+247

Specification

\[z \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ 0 - \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{2 \cdot \pi}}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\frac{-9.984369578019572 \cdot 10^{-6}}{7 - z} - \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) - \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + {\left({\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \frac{\frac{\frac{594939.8317813153}{3 - z}}{3 - z} - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{\frac{771.3234287776531}{3 - z} - \frac{-1259.1392167224028}{2 - z}}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (-
  0.0
  (*
   (*
    (exp (+ z -7.5))
    (/ (* (* PI (pow (- 7.5 z) (- 0.5 z))) (sqrt (* 2.0 PI))) (sin (* z PI))))
   (-
    (-
     (/ (- 9.984369578019572e-6) (- 7.0 z))
     (/ 1.5056327351493116e-7 (- 8.0 z)))
    (+
     (+ (/ -176.6150291621406 (- 4.0 z)) (/ 12.507343278686905 (- 5.0 z)))
     (pow
      (pow
       (+
        (/ -0.13857109526572012 (- 6.0 z))
        (+
         (/ 676.5203681218851 (- 1.0 z))
         (+
          0.9999999999998099
          (/
           (-
            (/ (/ 594939.8317813153 (- 3.0 z)) (- 3.0 z))
            (/ (/ 1585431.567088306 (- 2.0 z)) (- 2.0 z)))
           (-
            (/ 771.3234287776531 (- 3.0 z))
            (/ -1259.1392167224028 (- 2.0 z)))))))
       3.0)
      0.3333333333333333))))))

C

double code(double z) {
	return 0.0 - ((exp((z + -7.5)) * (((((double) M_PI) * pow((7.5 - z), (0.5 - z))) * sqrt((2.0 * ((double) M_PI)))) / sin((z * ((double) M_PI))))) * (((-9.984369578019572e-6 / (7.0 - z)) - (1.5056327351493116e-7 / (8.0 - z))) - (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + pow(pow(((-0.13857109526572012 / (6.0 - z)) + ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + ((((594939.8317813153 / (3.0 - z)) / (3.0 - z)) - ((1585431.567088306 / (2.0 - z)) / (2.0 - z))) / ((771.3234287776531 / (3.0 - z)) - (-1259.1392167224028 / (2.0 - z))))))), 3.0), 0.3333333333333333))));
}

Java

public static double code(double z) {
	return 0.0 - ((Math.exp((z + -7.5)) * (((Math.PI * Math.pow((7.5 - z), (0.5 - z))) * Math.sqrt((2.0 * Math.PI))) / Math.sin((z * Math.PI)))) * (((-9.984369578019572e-6 / (7.0 - z)) - (1.5056327351493116e-7 / (8.0 - z))) - (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + Math.pow(Math.pow(((-0.13857109526572012 / (6.0 - z)) + ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + ((((594939.8317813153 / (3.0 - z)) / (3.0 - z)) - ((1585431.567088306 / (2.0 - z)) / (2.0 - z))) / ((771.3234287776531 / (3.0 - z)) - (-1259.1392167224028 / (2.0 - z))))))), 3.0), 0.3333333333333333))));
}

Python

def code(z):
	return 0.0 - ((math.exp((z + -7.5)) * (((math.pi * math.pow((7.5 - z), (0.5 - z))) * math.sqrt((2.0 * math.pi))) / math.sin((z * math.pi)))) * (((-9.984369578019572e-6 / (7.0 - z)) - (1.5056327351493116e-7 / (8.0 - z))) - (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + math.pow(math.pow(((-0.13857109526572012 / (6.0 - z)) + ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + ((((594939.8317813153 / (3.0 - z)) / (3.0 - z)) - ((1585431.567088306 / (2.0 - z)) / (2.0 - z))) / ((771.3234287776531 / (3.0 - z)) - (-1259.1392167224028 / (2.0 - z))))))), 3.0), 0.3333333333333333))))

Julia

function code(z)
	return Float64(0.0 - Float64(Float64(exp(Float64(z + -7.5)) * Float64(Float64(Float64(pi * (Float64(7.5 - z) ^ Float64(0.5 - z))) * sqrt(Float64(2.0 * pi))) / sin(Float64(z * pi)))) * Float64(Float64(Float64(Float64(-9.984369578019572e-6) / Float64(7.0 - z)) - Float64(1.5056327351493116e-7 / Float64(8.0 - z))) - 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(676.5203681218851 / Float64(1.0 - z)) + Float64(0.9999999999998099 + Float64(Float64(Float64(Float64(594939.8317813153 / Float64(3.0 - z)) / Float64(3.0 - z)) - Float64(Float64(1585431.567088306 / Float64(2.0 - z)) / Float64(2.0 - z))) / Float64(Float64(771.3234287776531 / Float64(3.0 - z)) - Float64(-1259.1392167224028 / Float64(2.0 - z))))))) ^ 3.0) ^ 0.3333333333333333)))))
end

MATLAB

function tmp = code(z)
	tmp = 0.0 - ((exp((z + -7.5)) * (((pi * ((7.5 - z) ^ (0.5 - z))) * sqrt((2.0 * pi))) / sin((z * pi)))) * (((-9.984369578019572e-6 / (7.0 - z)) - (1.5056327351493116e-7 / (8.0 - z))) - (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + ((((-0.13857109526572012 / (6.0 - z)) + ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + ((((594939.8317813153 / (3.0 - z)) / (3.0 - z)) - ((1585431.567088306 / (2.0 - z)) / (2.0 - z))) / ((771.3234287776531 / (3.0 - z)) - (-1259.1392167224028 / (2.0 - z))))))) ^ 3.0) ^ 0.3333333333333333))));
end

Wolfram

code[z_] := N[(0.0 - N[(N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(Pi * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[N[(z * Pi), $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[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Power[N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(N[(N[(N[(594939.8317813153 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision] / N[(3.0 - z), $MachinePrecision]), $MachinePrecision] - N[(N[(1585431.567088306 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision] - N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

3. Simplified 95.5%

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

   1. add-cbrt-cube 95.5%

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

   2. pow1/3 97.3%

      \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \color{blue}{{\left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right) + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right) + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right) + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}^{0.3333333333333333}}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

5. Applied egg-rr 98.0%

   \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \color{blue}{{\left({\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)}^{3}\right)}^{0.3333333333333333}}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]
6. Step-by-step derivation

   1. flip-+ 98.0%

      \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + {\left({\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \color{blue}{\frac{\frac{771.3234287776531}{3 - z} \cdot \frac{771.3234287776531}{3 - z} - \frac{-1259.1392167224028}{2 - z} \cdot \frac{-1259.1392167224028}{2 - z}}{\frac{771.3234287776531}{3 - z} - \frac{-1259.1392167224028}{2 - z}}}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

7. Applied egg-rr 98.0%

   \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + {\left({\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \color{blue}{\frac{\frac{771.3234287776531}{3 - z} \cdot \frac{771.3234287776531}{3 - z} - \frac{-1259.1392167224028}{2 - z} \cdot \frac{-1259.1392167224028}{2 - z}}{\frac{771.3234287776531}{3 - z} - \frac{-1259.1392167224028}{2 - z}}}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]
8. Step-by-step derivation

   1. associate-*l/ 98.0%

      \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + {\left({\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \frac{\color{blue}{\frac{771.3234287776531 \cdot \frac{771.3234287776531}{3 - z}}{3 - z}} - \frac{-1259.1392167224028}{2 - z} \cdot \frac{-1259.1392167224028}{2 - z}}{\frac{771.3234287776531}{3 - z} - \frac{-1259.1392167224028}{2 - z}}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

   2. associate-*r/ 98.0%

      \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + {\left({\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \frac{\frac{\color{blue}{\frac{771.3234287776531 \cdot 771.3234287776531}{3 - z}}}{3 - z} - \frac{-1259.1392167224028}{2 - z} \cdot \frac{-1259.1392167224028}{2 - z}}{\frac{771.3234287776531}{3 - z} - \frac{-1259.1392167224028}{2 - z}}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

   3. metadata-eval 98.0%

      \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + {\left({\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \frac{\frac{\frac{\color{blue}{594939.8317813153}}{3 - z}}{3 - z} - \frac{-1259.1392167224028}{2 - z} \cdot \frac{-1259.1392167224028}{2 - z}}{\frac{771.3234287776531}{3 - z} - \frac{-1259.1392167224028}{2 - z}}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

   4. associate-*l/ 98.0%

      \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + {\left({\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \frac{\frac{\frac{594939.8317813153}{3 - z}}{3 - z} - \color{blue}{\frac{-1259.1392167224028 \cdot \frac{-1259.1392167224028}{2 - z}}{2 - z}}}{\frac{771.3234287776531}{3 - z} - \frac{-1259.1392167224028}{2 - z}}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

   5. associate-*r/ 98.0%

      \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + {\left({\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \frac{\frac{\frac{594939.8317813153}{3 - z}}{3 - z} - \frac{\color{blue}{\frac{-1259.1392167224028 \cdot -1259.1392167224028}{2 - z}}}{2 - z}}{\frac{771.3234287776531}{3 - z} - \frac{-1259.1392167224028}{2 - z}}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

   6. metadata-eval 98.0%

      \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + {\left({\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \frac{\frac{\frac{594939.8317813153}{3 - z}}{3 - z} - \frac{\frac{\color{blue}{1585431.567088306}}{2 - z}}{2 - z}}{\frac{771.3234287776531}{3 - z} - \frac{-1259.1392167224028}{2 - z}}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

9. Simplified 98.0%

   \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + {\left({\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \color{blue}{\frac{\frac{\frac{594939.8317813153}{3 - z}}{3 - z} - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{\frac{771.3234287776531}{3 - z} - \frac{-1259.1392167224028}{2 - z}}}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

10. Final simplification 98.0%

    \[\leadsto 0 - \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{2 \cdot \pi}}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\frac{-9.984369578019572 \cdot 10^{-6}}{7 - z} - \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) - \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + {\left({\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \frac{\frac{\frac{594939.8317813153}{3 - z}}{3 - z} - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{\frac{771.3234287776531}{3 - z} - \frac{-1259.1392167224028}{2 - z}}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\right) \]

Alternative 2: 98.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{2 \cdot \pi}}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + {\left({\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (exp (+ z -7.5))
   (/ (* (* PI (pow (- 7.5 z) (- 0.5 z))) (sqrt (* 2.0 PI))) (sin (* z PI))))
  (+
   (+ (/ 9.984369578019572e-6 (- 7.0 z)) (/ 1.5056327351493116e-7 (- 8.0 z)))
   (+
    (+ (/ -176.6150291621406 (- 4.0 z)) (/ 12.507343278686905 (- 5.0 z)))
    (pow
     (pow
      (+
       (/ -0.13857109526572012 (- 6.0 z))
       (+
        (/ 676.5203681218851 (- 1.0 z))
        (+
         0.9999999999998099
         (+
          (/ 771.3234287776531 (- 3.0 z))
          (/ -1259.1392167224028 (- 2.0 z))))))
      3.0)
     0.3333333333333333)))))

C

double code(double z) {
	return (exp((z + -7.5)) * (((((double) M_PI) * pow((7.5 - z), (0.5 - z))) * sqrt((2.0 * ((double) M_PI)))) / sin((z * ((double) M_PI))))) * (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + pow(pow(((-0.13857109526572012 / (6.0 - z)) + ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + ((771.3234287776531 / (3.0 - z)) + (-1259.1392167224028 / (2.0 - z)))))), 3.0), 0.3333333333333333)));
}

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((2.0 * Math.PI))) / Math.sin((z * Math.PI)))) * (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + Math.pow(Math.pow(((-0.13857109526572012 / (6.0 - z)) + ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + ((771.3234287776531 / (3.0 - z)) + (-1259.1392167224028 / (2.0 - z)))))), 3.0), 0.3333333333333333)));
}

Python

def code(z):
	return (math.exp((z + -7.5)) * (((math.pi * math.pow((7.5 - z), (0.5 - z))) * math.sqrt((2.0 * math.pi))) / math.sin((z * math.pi)))) * (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + math.pow(math.pow(((-0.13857109526572012 / (6.0 - z)) + ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + ((771.3234287776531 / (3.0 - z)) + (-1259.1392167224028 / (2.0 - z)))))), 3.0), 0.3333333333333333)))

Julia

function code(z)
	return Float64(Float64(exp(Float64(z + -7.5)) * Float64(Float64(Float64(pi * (Float64(7.5 - z) ^ Float64(0.5 - z))) * sqrt(Float64(2.0 * pi))) / sin(Float64(z * pi)))) * Float64(Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))) + 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(676.5203681218851 / Float64(1.0 - z)) + Float64(0.9999999999998099 + Float64(Float64(771.3234287776531 / Float64(3.0 - z)) + Float64(-1259.1392167224028 / Float64(2.0 - z)))))) ^ 3.0) ^ 0.3333333333333333))))
end

MATLAB

function tmp = code(z)
	tmp = (exp((z + -7.5)) * (((pi * ((7.5 - z) ^ (0.5 - z))) * sqrt((2.0 * pi))) / sin((z * pi)))) * (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + ((((-0.13857109526572012 / (6.0 - z)) + ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + ((771.3234287776531 / (3.0 - z)) + (-1259.1392167224028 / (2.0 - z)))))) ^ 3.0) ^ 0.3333333333333333)));
end

Wolfram

code[z_] := N[(N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(Pi * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[N[(z * Pi), $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[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Power[N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

3. Simplified 95.5%

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

   1. add-cbrt-cube 95.5%

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

   2. pow1/3 97.3%

      \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \color{blue}{{\left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right) + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right) + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{676.5203681218851}{1 - z}\right) + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}^{0.3333333333333333}}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

5. Applied egg-rr 98.0%

   \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \color{blue}{{\left({\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)}^{3}\right)}^{0.3333333333333333}}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{\pi \cdot 2}}{\sin \left(\pi \cdot z\right)}\right) \]

6. Final simplification 98.0%

   \[\leadsto \left(e^{z + -7.5} \cdot \frac{\left(\pi \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \sqrt{2 \cdot \pi}}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + {\left({\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\right) \]

Alternative 3: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[z_] := N[(N[(N[(Pi / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 + N[(N[(1.0 - z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(N[(z + -1.0), $MachinePrecision] - -1.0), $MachinePrecision] - 7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(-1259.1392167224028 / N[(N[(1.0 - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + 0.9999999999998099), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(N[(1.0 - z), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(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 96.2%

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

3. Simplified 97.9%

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

4. Final simplification 97.9%

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

Alternative 4: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

3. Simplified 97.9%

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

   1. metadata-eval 97.9%

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

   2. associate-+l- 97.9%

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

   3. metadata-eval 97.9%

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

   4. associate-+l- 97.9%

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

   5. expm1-log1p-u 97.5%

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

   6. expm1-udef 97.5%

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

5. Applied egg-rr 97.5%

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

   1. expm1-def 97.5%

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

   2. expm1-log1p 97.9%

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

7. Simplified 97.9%

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

8. Final simplification 97.9%

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

Alternative 5: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

3. Simplified 95.5%

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

   1. expm1-log1p-u 42.3%

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

   2. expm1-udef 42.3%

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

5. Applied egg-rr 42.3%

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

   1. expm1-def 42.3%

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

   2. expm1-log1p 95.7%

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

   3. associate-/l* 95.8%

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

   4. *-commutative 95.8%

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

   5. fma-udef 95.8%

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

   6. neg-mul-1 95.8%

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

   7. +-commutative 95.8%

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

   8. sub-neg 95.8%

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

7. Simplified 95.8%

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

   1. expm1-log1p-u 95.8%

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

   2. expm1-udef 95.8%

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

   3. associate-+l+ 95.8%

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

   4. associate-+l+ 95.8%

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

9. Applied egg-rr 95.8%

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

    1. expm1-def 95.8%

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

    2. expm1-log1p 96.6%

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

    3. +-commutative 96.6%

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

    4. associate-+r+ 97.7%

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

    5. +-commutative 97.7%

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

    6. associate-+l+ 97.7%

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

    7. +-commutative 97.7%

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

11. Simplified 97.7%

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

12. Final simplification 97.7%

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

Alternative 6: 96.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(e^{z + -7.5} \cdot \frac{\pi}{\frac{\frac{\sin \left(z \cdot \pi\right)}{\sqrt{2 \cdot \pi}}}{{\left(7.5 - z\right)}^{\left(0.5 - z\right)}}}\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(z \cdot -10.53814559148631 - 41.65228863479777\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (exp (+ z -7.5))
   (/ PI (/ (/ (sin (* z PI)) (sqrt (* 2.0 PI))) (pow (- 7.5 z) (- 0.5 z)))))
  (+
   (+ (/ 9.984369578019572e-6 (- 7.0 z)) (/ 1.5056327351493116e-7 (- 8.0 z)))
   (+
    (- (* z -10.53814559148631) 41.65228863479777)
    (+
     (/ -0.13857109526572012 (- 6.0 z))
     (+
      (/ 676.5203681218851 (- 1.0 z))
      (+
       0.9999999999998099
       (+
        (/ 771.3234287776531 (- 3.0 z))
        (/ -1259.1392167224028 (- 2.0 z))))))))))

C

double code(double z) {
	return (exp((z + -7.5)) * (((double) M_PI) / ((sin((z * ((double) M_PI))) / sqrt((2.0 * ((double) M_PI)))) / pow((7.5 - z), (0.5 - z))))) * (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((z * -10.53814559148631) - 41.65228863479777) + ((-0.13857109526572012 / (6.0 - z)) + ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + ((771.3234287776531 / (3.0 - z)) + (-1259.1392167224028 / (2.0 - z))))))));
}

Java

public static double code(double z) {
	return (Math.exp((z + -7.5)) * (Math.PI / ((Math.sin((z * Math.PI)) / Math.sqrt((2.0 * Math.PI))) / Math.pow((7.5 - z), (0.5 - z))))) * (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((z * -10.53814559148631) - 41.65228863479777) + ((-0.13857109526572012 / (6.0 - z)) + ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + ((771.3234287776531 / (3.0 - z)) + (-1259.1392167224028 / (2.0 - z))))))));
}

Python

def code(z):
	return (math.exp((z + -7.5)) * (math.pi / ((math.sin((z * math.pi)) / math.sqrt((2.0 * math.pi))) / math.pow((7.5 - z), (0.5 - z))))) * (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((z * -10.53814559148631) - 41.65228863479777) + ((-0.13857109526572012 / (6.0 - z)) + ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + ((771.3234287776531 / (3.0 - z)) + (-1259.1392167224028 / (2.0 - z))))))))

Julia

function code(z)
	return Float64(Float64(exp(Float64(z + -7.5)) * Float64(pi / Float64(Float64(sin(Float64(z * pi)) / sqrt(Float64(2.0 * pi))) / (Float64(7.5 - z) ^ Float64(0.5 - z))))) * Float64(Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))) + Float64(Float64(Float64(z * -10.53814559148631) - 41.65228863479777) + Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(0.9999999999998099 + Float64(Float64(771.3234287776531 / Float64(3.0 - z)) + Float64(-1259.1392167224028 / Float64(2.0 - z)))))))))
end

MATLAB

function tmp = code(z)
	tmp = (exp((z + -7.5)) * (pi / ((sin((z * pi)) / sqrt((2.0 * pi))) / ((7.5 - z) ^ (0.5 - z))))) * (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((z * -10.53814559148631) - 41.65228863479777) + ((-0.13857109526572012 / (6.0 - z)) + ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + ((771.3234287776531 / (3.0 - z)) + (-1259.1392167224028 / (2.0 - z))))))));
end

Wolfram

code[z_] := N[(N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(Pi / N[(N[(N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $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[(N[(z * -10.53814559148631), $MachinePrecision] - 41.65228863479777), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

3. Simplified 95.5%

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

   1. expm1-log1p-u 42.3%

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

   2. expm1-udef 42.3%

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

5. Applied egg-rr 42.3%

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

   1. expm1-def 42.3%

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

   2. expm1-log1p 95.7%

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

   3. associate-/l* 95.8%

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

   4. *-commutative 95.8%

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

   5. fma-udef 95.8%

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

   6. neg-mul-1 95.8%

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

   7. +-commutative 95.8%

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

   8. sub-neg 95.8%

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

7. Simplified 95.8%

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

   1. expm1-log1p-u 95.8%

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

   2. expm1-udef 95.8%

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

   3. associate-+l+ 95.8%

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

   4. associate-+l+ 95.8%

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

9. Applied egg-rr 95.8%

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

    1. expm1-def 95.8%

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

    2. expm1-log1p 96.6%

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

    3. +-commutative 96.6%

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

11. Simplified 96.6%

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

12. Taylor expanded in z around 0 97.0%

    \[\leadsto \left(\left(\color{blue}{\left(-10.53814559148631 \cdot z - 41.65228863479777\right)} + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\pi}{\frac{\frac{\sin \left(z \cdot \pi\right)}{\sqrt{2 \cdot \pi}}}{{\left(7.5 - z\right)}^{\left(0.5 - z\right)}}}\right) \]

13. Final simplification 97.0%

    \[\leadsto \left(e^{z + -7.5} \cdot \frac{\pi}{\frac{\frac{\sin \left(z \cdot \pi\right)}{\sqrt{2 \cdot \pi}}}{{\left(7.5 - z\right)}^{\left(0.5 - z\right)}}}\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(z \cdot -10.53814559148631 - 41.65228863479777\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)\right)\right) \]

Alternative 7: 96.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(e^{z + -7.5} \cdot \frac{\pi}{\frac{\frac{\sin \left(z \cdot \pi\right)}{\sqrt{2 \cdot \pi}}}{{\left(7.5 - z\right)}^{\left(0.5 - z\right)}}}\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(305.03547417069024 + z \cdot 447.4343179417107\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (exp (+ z -7.5))
   (/ PI (/ (/ (sin (* z PI)) (sqrt (* 2.0 PI))) (pow (- 7.5 z) (- 0.5 z)))))
  (+
   (+ (/ 9.984369578019572e-6 (- 7.0 z)) (/ 1.5056327351493116e-7 (- 8.0 z)))
   (+
    (+ (/ -176.6150291621406 (- 4.0 z)) (/ 12.507343278686905 (- 5.0 z)))
    (+ 305.03547417069024 (* z 447.4343179417107))))))

C

double code(double z) {
	return (exp((z + -7.5)) * (((double) M_PI) / ((sin((z * ((double) M_PI))) / sqrt((2.0 * ((double) M_PI)))) / pow((7.5 - z), (0.5 - z))))) * (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + (305.03547417069024 + (z * 447.4343179417107))));
}

Java

public static double code(double z) {
	return (Math.exp((z + -7.5)) * (Math.PI / ((Math.sin((z * Math.PI)) / Math.sqrt((2.0 * Math.PI))) / Math.pow((7.5 - z), (0.5 - z))))) * (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + (305.03547417069024 + (z * 447.4343179417107))));
}

Python

def code(z):
	return (math.exp((z + -7.5)) * (math.pi / ((math.sin((z * math.pi)) / math.sqrt((2.0 * math.pi))) / math.pow((7.5 - z), (0.5 - z))))) * (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + (305.03547417069024 + (z * 447.4343179417107))))

Julia

function code(z)
	return Float64(Float64(exp(Float64(z + -7.5)) * Float64(pi / Float64(Float64(sin(Float64(z * pi)) / sqrt(Float64(2.0 * pi))) / (Float64(7.5 - z) ^ Float64(0.5 - z))))) * Float64(Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))) + Float64(Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(12.507343278686905 / Float64(5.0 - z))) + Float64(305.03547417069024 + Float64(z * 447.4343179417107)))))
end

MATLAB

function tmp = code(z)
	tmp = (exp((z + -7.5)) * (pi / ((sin((z * pi)) / sqrt((2.0 * pi))) / ((7.5 - z) ^ (0.5 - z))))) * (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + (305.03547417069024 + (z * 447.4343179417107))));
end

Wolfram

code[z_] := N[(N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(Pi / N[(N[(N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $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[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(305.03547417069024 + N[(z * 447.4343179417107), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

3. Simplified 95.5%

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

   1. expm1-log1p-u 42.3%

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

   2. expm1-udef 42.3%

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

5. Applied egg-rr 42.3%

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

   1. expm1-def 42.3%

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

   2. expm1-log1p 95.7%

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

   3. associate-/l* 95.8%

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

   4. *-commutative 95.8%

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

   5. fma-udef 95.8%

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

   6. neg-mul-1 95.8%

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

   7. +-commutative 95.8%

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

   8. sub-neg 95.8%

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

7. Simplified 95.8%

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

8. Taylor expanded in z around 0 96.6%

   \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \color{blue}{\left(305.03547417069024 + 447.4343179417107 \cdot z\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\pi}{\frac{\frac{\sin \left(z \cdot \pi\right)}{\sqrt{2 \cdot \pi}}}{{\left(7.5 - z\right)}^{\left(0.5 - z\right)}}}\right) \]
9. Step-by-step derivation

   1. *-commutative 96.6%

      \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(305.03547417069024 + \color{blue}{z \cdot 447.4343179417107}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\pi}{\frac{\frac{\sin \left(z \cdot \pi\right)}{\sqrt{2 \cdot \pi}}}{{\left(7.5 - z\right)}^{\left(0.5 - z\right)}}}\right) \]

10. Simplified 96.6%

    \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \color{blue}{\left(305.03547417069024 + z \cdot 447.4343179417107\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\pi}{\frac{\frac{\sin \left(z \cdot \pi\right)}{\sqrt{2 \cdot \pi}}}{{\left(7.5 - z\right)}^{\left(0.5 - z\right)}}}\right) \]

11. Final simplification 96.6%

    \[\leadsto \left(e^{z + -7.5} \cdot \frac{\pi}{\frac{\frac{\sin \left(z \cdot \pi\right)}{\sqrt{2 \cdot \pi}}}{{\left(7.5 - z\right)}^{\left(0.5 - z\right)}}}\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(305.03547417069024 + z \cdot 447.4343179417107\right)\right)\right) \]

Alternative 8: 95.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + 305.03547417069024\right)\right) \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi}\right)\right) \cdot \frac{e^{-7.5}}{\frac{z}{\sqrt{2} \cdot \sqrt{7.5}}}\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (+
   (+ (/ 9.984369578019572e-6 (- 7.0 z)) (/ 1.5056327351493116e-7 (- 8.0 z)))
   (+
    (+ (/ -176.6150291621406 (- 4.0 z)) (/ 12.507343278686905 (- 5.0 z)))
    305.03547417069024))
  (*
   (expm1 (log1p (sqrt PI)))
   (/ (exp -7.5) (/ z (* (sqrt 2.0) (sqrt 7.5)))))))

C

double code(double z) {
	return (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + 305.03547417069024)) * (expm1(log1p(sqrt(((double) M_PI)))) * (exp(-7.5) / (z / (sqrt(2.0) * sqrt(7.5)))));
}

Java

public static double code(double z) {
	return (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + 305.03547417069024)) * (Math.expm1(Math.log1p(Math.sqrt(Math.PI))) * (Math.exp(-7.5) / (z / (Math.sqrt(2.0) * Math.sqrt(7.5)))));
}

Python

def code(z):
	return (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + 305.03547417069024)) * (math.expm1(math.log1p(math.sqrt(math.pi))) * (math.exp(-7.5) / (z / (math.sqrt(2.0) * math.sqrt(7.5)))))

Julia

function code(z)
	return Float64(Float64(Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))) + Float64(Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(12.507343278686905 / Float64(5.0 - z))) + 305.03547417069024)) * Float64(expm1(log1p(sqrt(pi))) * Float64(exp(-7.5) / Float64(z / Float64(sqrt(2.0) * sqrt(7.5))))))
end

Wolfram

code[z_] := 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[(N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 305.03547417069024), $MachinePrecision]), $MachinePrecision] * N[(N[(Exp[N[Log[1 + N[Sqrt[Pi], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] / N[(z / N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

3. Simplified 95.5%

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

4. Taylor expanded in z around 0 93.6%

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

   1. *-commutative 93.6%

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

   2. associate-/l* 93.5%

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

6. Simplified 93.5%

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

7. Taylor expanded in z around 0 95.5%

   \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \color{blue}{305.03547417069024}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(\sqrt{\pi} \cdot \frac{e^{-7.5}}{\frac{z}{\sqrt{2} \cdot \sqrt{7.5}}}\right) \]
8. Step-by-step derivation

   1. expm1-log1p-u 96.0%

      \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + 305.03547417069024\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi}\right)\right)} \cdot \frac{e^{-7.5}}{\frac{z}{\sqrt{2} \cdot \sqrt{7.5}}}\right) \]

9. Applied egg-rr 96.0%

   \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + 305.03547417069024\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi}\right)\right)} \cdot \frac{e^{-7.5}}{\frac{z}{\sqrt{2} \cdot \sqrt{7.5}}}\right) \]

10. Final simplification 96.0%

    \[\leadsto \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + 305.03547417069024\right)\right) \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi}\right)\right) \cdot \frac{e^{-7.5}}{\frac{z}{\sqrt{2} \cdot \sqrt{7.5}}}\right) \]

Alternative 9: 95.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(-44.15375584537623 + \left(\frac{12.507343278686905}{5 - z} + 305.03547417069024\right)\right) \cdot \left(\sqrt{\pi} \cdot \frac{e^{-7.5}}{\frac{z}{\sqrt{15}}}\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (+
   -44.15375584537623
   (+ (/ 12.507343278686905 (- 5.0 z)) 305.03547417069024))
  (* (sqrt PI) (/ (exp -7.5) (/ z (sqrt 15.0))))))

C

double code(double z) {
	return (-44.15375584537623 + ((12.507343278686905 / (5.0 - z)) + 305.03547417069024)) * (sqrt(((double) M_PI)) * (exp(-7.5) / (z / sqrt(15.0))));
}

Java

public static double code(double z) {
	return (-44.15375584537623 + ((12.507343278686905 / (5.0 - z)) + 305.03547417069024)) * (Math.sqrt(Math.PI) * (Math.exp(-7.5) / (z / Math.sqrt(15.0))));
}

Python

def code(z):
	return (-44.15375584537623 + ((12.507343278686905 / (5.0 - z)) + 305.03547417069024)) * (math.sqrt(math.pi) * (math.exp(-7.5) / (z / math.sqrt(15.0))))

Julia

function code(z)
	return Float64(Float64(-44.15375584537623 + Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + 305.03547417069024)) * Float64(sqrt(pi) * Float64(exp(-7.5) / Float64(z / sqrt(15.0)))))
end

MATLAB

function tmp = code(z)
	tmp = (-44.15375584537623 + ((12.507343278686905 / (5.0 - z)) + 305.03547417069024)) * (sqrt(pi) * (exp(-7.5) / (z / sqrt(15.0))));
end

Wolfram

code[z_] := N[(N[(-44.15375584537623 + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + 305.03547417069024), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] / N[(z / N[Sqrt[15.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

3. Simplified 95.5%

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

4. Taylor expanded in z around 0 93.6%

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

   1. *-commutative 93.6%

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

   2. associate-/l* 93.5%

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

6. Simplified 93.5%

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

7. Taylor expanded in z around 0 95.5%

   \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \color{blue}{305.03547417069024}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(\sqrt{\pi} \cdot \frac{e^{-7.5}}{\frac{z}{\sqrt{2} \cdot \sqrt{7.5}}}\right) \]
8. Step-by-step derivation

   1. expm1-log1p-u 40.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + 305.03547417069024\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(\sqrt{\pi} \cdot \frac{e^{-7.5}}{\frac{z}{\sqrt{2} \cdot \sqrt{7.5}}}\right)\right)\right)} \]

   2. expm1-udef 40.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + 305.03547417069024\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(\sqrt{\pi} \cdot \frac{e^{-7.5}}{\frac{z}{\sqrt{2} \cdot \sqrt{7.5}}}\right)\right)} - 1} \]

9. Applied egg-rr 40.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(\frac{-176.6150291621406}{4 - z} + \left(\frac{12.507343278686905}{5 - z} + 305.03547417069024\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(\sqrt{\pi} \cdot \left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right)\right)\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 40.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(\frac{-176.6150291621406}{4 - z} + \left(\frac{12.507343278686905}{5 - z} + 305.03547417069024\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(\sqrt{\pi} \cdot \left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right)\right)\right)\right)} \]

    2. expm1-log1p 95.9%

       \[\leadsto \color{blue}{\left(\left(\frac{-176.6150291621406}{4 - z} + \left(\frac{12.507343278686905}{5 - z} + 305.03547417069024\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(\sqrt{\pi} \cdot \left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right)\right)} \]

    3. +-commutative 95.9%

       \[\leadsto \color{blue}{\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{12.507343278686905}{5 - z} + 305.03547417069024\right)\right)\right)} \cdot \left(\sqrt{\pi} \cdot \left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right)\right) \]

    4. associate-+r+ 95.9%

       \[\leadsto \color{blue}{\left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{12.507343278686905}{5 - z} + 305.03547417069024\right)\right)} \cdot \left(\sqrt{\pi} \cdot \left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right)\right) \]

    5. +-commutative 95.9%

       \[\leadsto \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \color{blue}{\left(305.03547417069024 + \frac{12.507343278686905}{5 - z}\right)}\right) \cdot \left(\sqrt{\pi} \cdot \left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right)\right) \]

    6. associate-*l/ 95.7%

       \[\leadsto \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(305.03547417069024 + \frac{12.507343278686905}{5 - z}\right)\right) \cdot \left(\sqrt{\pi} \cdot \color{blue}{\frac{e^{-7.5} \cdot \sqrt{15}}{z}}\right) \]

    7. associate-/l* 95.9%

       \[\leadsto \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(305.03547417069024 + \frac{12.507343278686905}{5 - z}\right)\right) \cdot \left(\sqrt{\pi} \cdot \color{blue}{\frac{e^{-7.5}}{\frac{z}{\sqrt{15}}}}\right) \]

11. Simplified 95.9%

    \[\leadsto \color{blue}{\left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(305.03547417069024 + \frac{12.507343278686905}{5 - z}\right)\right) \cdot \left(\sqrt{\pi} \cdot \frac{e^{-7.5}}{\frac{z}{\sqrt{15}}}\right)} \]

12. Taylor expanded in z around 0 95.9%

    \[\leadsto \left(\color{blue}{-44.15375584537623} + \left(305.03547417069024 + \frac{12.507343278686905}{5 - z}\right)\right) \cdot \left(\sqrt{\pi} \cdot \frac{e^{-7.5}}{\frac{z}{\sqrt{15}}}\right) \]

13. Final simplification 95.9%

    \[\leadsto \left(-44.15375584537623 + \left(\frac{12.507343278686905}{5 - z} + 305.03547417069024\right)\right) \cdot \left(\sqrt{\pi} \cdot \frac{e^{-7.5}}{\frac{z}{\sqrt{15}}}\right) \]

Alternative 10: 95.7% accurate, 2.0× speedup

Math

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

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(263.3831869810514 * Float64(sqrt(pi) * Float64(exp(-7.5) / Float64(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[(263.3831869810514 * N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] / N[(z / N[Sqrt[15.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

3. Simplified 95.5%

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

4. Taylor expanded in z around 0 93.6%

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

   1. *-commutative 93.6%

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

   2. associate-/l* 93.5%

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

6. Simplified 93.5%

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

7. Taylor expanded in z around 0 95.5%

   \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \color{blue}{305.03547417069024}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(\sqrt{\pi} \cdot \frac{e^{-7.5}}{\frac{z}{\sqrt{2} \cdot \sqrt{7.5}}}\right) \]
8. Step-by-step derivation

   1. expm1-log1p-u 40.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + 305.03547417069024\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(\sqrt{\pi} \cdot \frac{e^{-7.5}}{\frac{z}{\sqrt{2} \cdot \sqrt{7.5}}}\right)\right)\right)} \]

   2. expm1-udef 40.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + 305.03547417069024\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(\sqrt{\pi} \cdot \frac{e^{-7.5}}{\frac{z}{\sqrt{2} \cdot \sqrt{7.5}}}\right)\right)} - 1} \]

9. Applied egg-rr 40.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(\frac{-176.6150291621406}{4 - z} + \left(\frac{12.507343278686905}{5 - z} + 305.03547417069024\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(\sqrt{\pi} \cdot \left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right)\right)\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 40.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(\frac{-176.6150291621406}{4 - z} + \left(\frac{12.507343278686905}{5 - z} + 305.03547417069024\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(\sqrt{\pi} \cdot \left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right)\right)\right)\right)} \]

    2. expm1-log1p 95.9%

       \[\leadsto \color{blue}{\left(\left(\frac{-176.6150291621406}{4 - z} + \left(\frac{12.507343278686905}{5 - z} + 305.03547417069024\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(\sqrt{\pi} \cdot \left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right)\right)} \]

    3. +-commutative 95.9%

       \[\leadsto \color{blue}{\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{12.507343278686905}{5 - z} + 305.03547417069024\right)\right)\right)} \cdot \left(\sqrt{\pi} \cdot \left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right)\right) \]

    4. associate-+r+ 95.9%

       \[\leadsto \color{blue}{\left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{12.507343278686905}{5 - z} + 305.03547417069024\right)\right)} \cdot \left(\sqrt{\pi} \cdot \left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right)\right) \]

    5. +-commutative 95.9%

       \[\leadsto \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \color{blue}{\left(305.03547417069024 + \frac{12.507343278686905}{5 - z}\right)}\right) \cdot \left(\sqrt{\pi} \cdot \left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right)\right) \]

    6. associate-*l/ 95.7%

       \[\leadsto \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(305.03547417069024 + \frac{12.507343278686905}{5 - z}\right)\right) \cdot \left(\sqrt{\pi} \cdot \color{blue}{\frac{e^{-7.5} \cdot \sqrt{15}}{z}}\right) \]

    7. associate-/l* 95.9%

       \[\leadsto \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(305.03547417069024 + \frac{12.507343278686905}{5 - z}\right)\right) \cdot \left(\sqrt{\pi} \cdot \color{blue}{\frac{e^{-7.5}}{\frac{z}{\sqrt{15}}}}\right) \]

11. Simplified 95.9%

    \[\leadsto \color{blue}{\left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(305.03547417069024 + \frac{12.507343278686905}{5 - z}\right)\right) \cdot \left(\sqrt{\pi} \cdot \frac{e^{-7.5}}{\frac{z}{\sqrt{15}}}\right)} \]

12. Taylor expanded in z around 0 95.9%

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

13. Final simplification 95.9%

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

Alternative 11: 95.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

3. Simplified 95.5%

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

4. Taylor expanded in z around 0 93.6%

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

   1. *-commutative 93.6%

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

   2. associate-/l* 93.5%

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

6. Simplified 93.5%

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

7. Taylor expanded in z around 0 95.5%

   \[\leadsto \left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \color{blue}{305.03547417069024}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(\sqrt{\pi} \cdot \frac{e^{-7.5}}{\frac{z}{\sqrt{2} \cdot \sqrt{7.5}}}\right) \]
8. Step-by-step derivation

   1. pow1 95.5%

      \[\leadsto \color{blue}{{\left(\left(\left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + 305.03547417069024\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(\sqrt{\pi} \cdot \frac{e^{-7.5}}{\frac{z}{\sqrt{2} \cdot \sqrt{7.5}}}\right)\right)}^{1}} \]

   2. associate-+l+ 95.9%

      \[\leadsto {\left(\left(\color{blue}{\left(\frac{-176.6150291621406}{4 - z} + \left(\frac{12.507343278686905}{5 - z} + 305.03547417069024\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(\sqrt{\pi} \cdot \frac{e^{-7.5}}{\frac{z}{\sqrt{2} \cdot \sqrt{7.5}}}\right)\right)}^{1} \]

   3. associate-/r/ 95.9%

      \[\leadsto {\left(\left(\left(\frac{-176.6150291621406}{4 - z} + \left(\frac{12.507343278686905}{5 - z} + 305.03547417069024\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(\sqrt{\pi} \cdot \color{blue}{\left(\frac{e^{-7.5}}{z} \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)\right)}\right)\right)}^{1} \]

   4. sqrt-unprod 95.9%

      \[\leadsto {\left(\left(\left(\frac{-176.6150291621406}{4 - z} + \left(\frac{12.507343278686905}{5 - z} + 305.03547417069024\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(\sqrt{\pi} \cdot \left(\frac{e^{-7.5}}{z} \cdot \color{blue}{\sqrt{2 \cdot 7.5}}\right)\right)\right)}^{1} \]

   5. metadata-eval 95.9%

      \[\leadsto {\left(\left(\left(\frac{-176.6150291621406}{4 - z} + \left(\frac{12.507343278686905}{5 - z} + 305.03547417069024\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(\sqrt{\pi} \cdot \left(\frac{e^{-7.5}}{z} \cdot \sqrt{\color{blue}{15}}\right)\right)\right)}^{1} \]

9. Applied egg-rr 95.9%

   \[\leadsto \color{blue}{{\left(\left(\left(\frac{-176.6150291621406}{4 - z} + \left(\frac{12.507343278686905}{5 - z} + 305.03547417069024\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \cdot \left(\sqrt{\pi} \cdot \left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right)\right)\right)}^{1}} \]

10. Taylor expanded in z around 0 95.7%

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

    1. associate-*r* 95.7%

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

    2. associate-/l* 95.9%

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

12. Simplified 95.9%

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

13. Final simplification 95.9%

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

Reproduce

herbie shell --seed 2023293 
(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))))))