Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.6% → 98.3%

Time: 16.5s

Alternatives: 7

Speedup: 3.6×

• could not determine a ground truth

  1. z = -114885512352024.28

Specification

\[z \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(0.5 - z, \log \left(7.5 - z\right), z\right) - 7.5} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ (fma (* 0.16666666666666666 (* z z)) (* PI PI) 1.0) z)
  (*
   (* (* (exp (- (fma (- 0.5 z) (log (- 7.5 z)) z) 7.5)) (sqrt 2.0)) (sqrt PI))
   (+
    (fma
     (fma (fma 606.676680916724 z 545.0353078425886) z 436.896172553987)
     z
     263.383186962231)
    (/ 1.5056327351493116e-7 (+ (- (- 1.0 z) 1.0) 8.0))))))

C

double code(double z) {
	return (fma((0.16666666666666666 * (z * z)), (((double) M_PI) * ((double) M_PI)), 1.0) / z) * (((exp((fma((0.5 - z), log((7.5 - z)), z) - 7.5)) * sqrt(2.0)) * sqrt(((double) M_PI))) * (fma(fma(fma(606.676680916724, z, 545.0353078425886), z, 436.896172553987), z, 263.383186962231) + (1.5056327351493116e-7 / (((1.0 - z) - 1.0) + 8.0))));
}

Julia

function code(z)
	return Float64(Float64(fma(Float64(0.16666666666666666 * Float64(z * z)), Float64(pi * pi), 1.0) / z) * Float64(Float64(Float64(exp(Float64(fma(Float64(0.5 - z), log(Float64(7.5 - z)), z) - 7.5)) * sqrt(2.0)) * sqrt(pi)) * Float64(fma(fma(fma(606.676680916724, z, 545.0353078425886), z, 436.896172553987), z, 263.383186962231) + Float64(1.5056327351493116e-7 / Float64(Float64(Float64(1.0 - z) - 1.0) + 8.0)))))
end

Wolfram

code[z_] := N[(N[(N[(N[(0.16666666666666666 * N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision] * N[(N[(N[(N[Exp[N[(N[(N[(0.5 - z), $MachinePrecision] * N[Log[N[(7.5 - z), $MachinePrecision]], $MachinePrecision] + z), $MachinePrecision] - 7.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(606.676680916724 * z + 545.0353078425886), $MachinePrecision] * z + 436.896172553987), $MachinePrecision] * z + 263.383186962231), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(N[(1.0 - z), $MachinePrecision] - 1.0), $MachinePrecision] + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

2. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\frac{1382761731551712743134679}{5250000000000000000000} + z \cdot \left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(z \cdot \left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)\right) + \color{blue}{\frac{1382761731551712743134679}{5250000000000000000000}}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)\right) \cdot z + \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right), \color{blue}{z}, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right) + \frac{16055934341359023345617179}{36750000000000000000000}, z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right) \cdot z + \frac{16055934341359023345617179}{36750000000000000000000}, z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z, z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z + \frac{1261892996482553330703662111}{2315250000000000000000000}, z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lower-fma.f64 96.9

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 96.9%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

5. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1 + \frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1 + \frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{\color{blue}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \frac{\left(\frac{1}{6} \cdot {z}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + 1}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot {z}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot {z}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   10. lift-PI.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   11. lift-PI.f64 96.9

       \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

7. Applied rewrites 96.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

8. Taylor expanded in z around inf

   \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

10. Applied rewrites 98.3%

    \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\color{blue}{\left(\left(e^{\mathsf{fma}\left(\log \left(7.5 - z\right), 0.5 - z, z - 7.5\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
11. Step-by-step derivation

    1. lift--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(\frac{15}{2} - z\right), \frac{1}{2} - z, z - \frac{15}{2}\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

    2. lift--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(\frac{15}{2} - z\right), \frac{1}{2} - z, z - \frac{15}{2}\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

    3. lift-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right) + \left(z - \frac{15}{2}\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

    4. lift--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right) + \left(z - \frac{15}{2}\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

    5. lift-log.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right) + \left(z - \frac{15}{2}\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

    6. associate-+r- N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\left(\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right) + z\right) - \frac{15}{2}} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

    7. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\left(z + \log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)\right) - \frac{15}{2}} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

    8. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\left(z + \log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)\right) - \frac{15}{2}} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

    9. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\left(\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right) + z\right) - \frac{15}{2}} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\left(\left(\frac{1}{2} - z\right) \cdot \log \left(\frac{15}{2} - z\right) + z\right) - \frac{15}{2}} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

    11. lower-fma.f64 N/A

        \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\frac{1}{2} - z, \log \left(\frac{15}{2} - z\right), z\right) - \frac{15}{2}} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

    12. lift--.f64 N/A

        \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\frac{1}{2} - z, \log \left(\frac{15}{2} - z\right), z\right) - \frac{15}{2}} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

    13. lift-log.f64 N/A

        \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\frac{1}{2} - z, \log \left(\frac{15}{2} - z\right), z\right) - \frac{15}{2}} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

    14. lift--.f64 98.3

        \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(0.5 - z, \log \left(7.5 - z\right), z\right) - 7.5} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

12. Applied rewrites 98.3%

    \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(0.5 - z, \log \left(7.5 - z\right), z\right) - 7.5} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
13. Add Preprocessing

Alternative 2: 98.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(7.5 - z\right), 0.5 - z, z - 7.5\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.6766809167608, z, 545.0353078428827\right), z, 436.8961725563396\right), z, 263.3831869810514\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ (fma (* 0.16666666666666666 (* z z)) (* PI PI) 1.0) z)
  (*
   (* (* (exp (fma (log (- 7.5 z)) (- 0.5 z) (- z 7.5))) (sqrt 2.0)) (sqrt PI))
   (fma
    (fma (fma 606.6766809167608 z 545.0353078428827) z 436.8961725563396)
    z
    263.3831869810514))))

C

double code(double z) {
	return (fma((0.16666666666666666 * (z * z)), (((double) M_PI) * ((double) M_PI)), 1.0) / z) * (((exp(fma(log((7.5 - z)), (0.5 - z), (z - 7.5))) * sqrt(2.0)) * sqrt(((double) M_PI))) * fma(fma(fma(606.6766809167608, z, 545.0353078428827), z, 436.8961725563396), z, 263.3831869810514));
}

Julia

function code(z)
	return Float64(Float64(fma(Float64(0.16666666666666666 * Float64(z * z)), Float64(pi * pi), 1.0) / z) * Float64(Float64(Float64(exp(fma(log(Float64(7.5 - z)), Float64(0.5 - z), Float64(z - 7.5))) * sqrt(2.0)) * sqrt(pi)) * fma(fma(fma(606.6766809167608, z, 545.0353078428827), z, 436.8961725563396), z, 263.3831869810514)))
end

Wolfram

code[z_] := N[(N[(N[(N[(0.16666666666666666 * N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision] * N[(N[(N[(N[Exp[N[(N[Log[N[(7.5 - z), $MachinePrecision]], $MachinePrecision] * N[(0.5 - z), $MachinePrecision] + N[(z - 7.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(606.6766809167608 * z + 545.0353078428827), $MachinePrecision] * z + 436.8961725563396), $MachinePrecision] * z + 263.3831869810514), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

2. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\frac{1382761731551712743134679}{5250000000000000000000} + z \cdot \left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(z \cdot \left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)\right) + \color{blue}{\frac{1382761731551712743134679}{5250000000000000000000}}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)\right) \cdot z + \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right), \color{blue}{z}, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right) + \frac{16055934341359023345617179}{36750000000000000000000}, z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right) \cdot z + \frac{16055934341359023345617179}{36750000000000000000000}, z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z, z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z + \frac{1261892996482553330703662111}{2315250000000000000000000}, z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lower-fma.f64 96.9

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 96.9%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

5. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1 + \frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1 + \frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{\color{blue}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \frac{\left(\frac{1}{6} \cdot {z}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + 1}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot {z}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot {z}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   10. lift-PI.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   11. lift-PI.f64 96.9

       \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

7. Applied rewrites 96.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

8. Taylor expanded in z around inf

   \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

10. Applied rewrites 98.3%

    \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\color{blue}{\left(\left(e^{\mathsf{fma}\left(\log \left(7.5 - z\right), 0.5 - z, z - 7.5\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

11. Taylor expanded in z around 0

    \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(\frac{15}{2} - z\right), \frac{1}{2} - z, z - \frac{15}{2}\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)}\right) \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(\frac{15}{2} - z\right), \frac{1}{2} - z, z - \frac{15}{2}\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \left(z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right) + \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000}}\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(\frac{15}{2} - z\right), \frac{1}{2} - z, z - \frac{15}{2}\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \left(\left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right) \cdot z + \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

    3. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(\frac{15}{2} - z\right), \frac{1}{2} - z, z - \frac{15}{2}\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \mathsf{fma}\left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right), \color{blue}{z}, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

    4. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(\frac{15}{2} - z\right), \frac{1}{2} - z, z - \frac{15}{2}\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \mathsf{fma}\left(z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right) + \frac{102757979785251069442117317613}{235200000000000000000000000}, z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

    5. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(\frac{15}{2} - z\right), \frac{1}{2} - z, z - \frac{15}{2}\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \mathsf{fma}\left(\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right) \cdot z + \frac{102757979785251069442117317613}{235200000000000000000000000}, z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

    6. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(\frac{15}{2} - z\right), \frac{1}{2} - z, z - \frac{15}{2}\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z, z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

    7. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(\frac{15}{2} - z\right), \frac{1}{2} - z, z - \frac{15}{2}\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z + \frac{64608921419941589693928044520019}{118540800000000000000000000000}, z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

    8. lower-fma.f64 98.3

       \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(7.5 - z\right), 0.5 - z, z - 7.5\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.6766809167608, z, 545.0353078428827\right), z, 436.8961725563396\right), z, 263.3831869810514\right)\right) \]

13. Applied rewrites 98.3%

    \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(7.5 - z\right), 0.5 - z, z - 7.5\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.6766809167608, z, 545.0353078428827\right), z, 436.8961725563396\right), z, 263.3831869810514\right)}\right) \]
14. Add Preprocessing

Alternative 3: 98.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(7.5 - z\right), 0.5 - z, z - 7.5\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(545.0353078428827, z, 436.8961725563396\right), z, 263.3831869810514\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ (fma (* 0.16666666666666666 (* z z)) (* PI PI) 1.0) z)
  (*
   (* (* (exp (fma (log (- 7.5 z)) (- 0.5 z) (- z 7.5))) (sqrt 2.0)) (sqrt PI))
   (fma (fma 545.0353078428827 z 436.8961725563396) z 263.3831869810514))))

C

double code(double z) {
	return (fma((0.16666666666666666 * (z * z)), (((double) M_PI) * ((double) M_PI)), 1.0) / z) * (((exp(fma(log((7.5 - z)), (0.5 - z), (z - 7.5))) * sqrt(2.0)) * sqrt(((double) M_PI))) * fma(fma(545.0353078428827, z, 436.8961725563396), z, 263.3831869810514));
}

Julia

function code(z)
	return Float64(Float64(fma(Float64(0.16666666666666666 * Float64(z * z)), Float64(pi * pi), 1.0) / z) * Float64(Float64(Float64(exp(fma(log(Float64(7.5 - z)), Float64(0.5 - z), Float64(z - 7.5))) * sqrt(2.0)) * sqrt(pi)) * fma(fma(545.0353078428827, z, 436.8961725563396), z, 263.3831869810514)))
end

Wolfram

code[z_] := N[(N[(N[(N[(0.16666666666666666 * N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision] * N[(N[(N[(N[Exp[N[(N[Log[N[(7.5 - z), $MachinePrecision]], $MachinePrecision] * N[(0.5 - z), $MachinePrecision] + N[(z - 7.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(545.0353078428827 * z + 436.8961725563396), $MachinePrecision] * z + 263.3831869810514), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

2. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\frac{1382761731551712743134679}{5250000000000000000000} + z \cdot \left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(z \cdot \left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)\right) + \color{blue}{\frac{1382761731551712743134679}{5250000000000000000000}}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)\right) \cdot z + \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right), \color{blue}{z}, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right) + \frac{16055934341359023345617179}{36750000000000000000000}, z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right) \cdot z + \frac{16055934341359023345617179}{36750000000000000000000}, z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z, z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z + \frac{1261892996482553330703662111}{2315250000000000000000000}, z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lower-fma.f64 96.9

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 96.9%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

5. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1 + \frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1 + \frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{\color{blue}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \frac{\left(\frac{1}{6} \cdot {z}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + 1}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot {z}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot {z}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   10. lift-PI.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   11. lift-PI.f64 96.9

       \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

7. Applied rewrites 96.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

8. Taylor expanded in z around inf

   \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

10. Applied rewrites 98.3%

    \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\color{blue}{\left(\left(e^{\mathsf{fma}\left(\log \left(7.5 - z\right), 0.5 - z, z - 7.5\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

11. Taylor expanded in z around 0

    \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(\frac{15}{2} - z\right), \frac{1}{2} - z, z - \frac{15}{2}\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)}\right) \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(\frac{15}{2} - z\right), \frac{1}{2} - z, z - \frac{15}{2}\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \left(z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right) + \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000}}\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(\frac{15}{2} - z\right), \frac{1}{2} - z, z - \frac{15}{2}\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \left(\left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right) \cdot z + \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

    3. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(\frac{15}{2} - z\right), \frac{1}{2} - z, z - \frac{15}{2}\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \mathsf{fma}\left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z, \color{blue}{z}, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

    4. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(\frac{15}{2} - z\right), \frac{1}{2} - z, z - \frac{15}{2}\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \mathsf{fma}\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z + \frac{102757979785251069442117317613}{235200000000000000000000000}, z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

    5. lower-fma.f64 98.1

       \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(7.5 - z\right), 0.5 - z, z - 7.5\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(545.0353078428827, z, 436.8961725563396\right), z, 263.3831869810514\right)\right) \]

13. Applied rewrites 98.1%

    \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(7.5 - z\right), 0.5 - z, z - 7.5\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(545.0353078428827, z, 436.8961725563396\right), z, 263.3831869810514\right)}\right) \]
14. Add Preprocessing

Alternative 4: 97.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(7.5 - z\right), 0.5 - z, z - 7.5\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \mathsf{fma}\left(436.8961725563396, z, 263.3831869810514\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ (fma (* 0.16666666666666666 (* z z)) (* PI PI) 1.0) z)
  (*
   (* (* (exp (fma (log (- 7.5 z)) (- 0.5 z) (- z 7.5))) (sqrt 2.0)) (sqrt PI))
   (fma 436.8961725563396 z 263.3831869810514))))

C

double code(double z) {
	return (fma((0.16666666666666666 * (z * z)), (((double) M_PI) * ((double) M_PI)), 1.0) / z) * (((exp(fma(log((7.5 - z)), (0.5 - z), (z - 7.5))) * sqrt(2.0)) * sqrt(((double) M_PI))) * fma(436.8961725563396, z, 263.3831869810514));
}

Julia

function code(z)
	return Float64(Float64(fma(Float64(0.16666666666666666 * Float64(z * z)), Float64(pi * pi), 1.0) / z) * Float64(Float64(Float64(exp(fma(log(Float64(7.5 - z)), Float64(0.5 - z), Float64(z - 7.5))) * sqrt(2.0)) * sqrt(pi)) * fma(436.8961725563396, z, 263.3831869810514)))
end

Wolfram

code[z_] := N[(N[(N[(N[(0.16666666666666666 * N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision] * N[(N[(N[(N[Exp[N[(N[Log[N[(7.5 - z), $MachinePrecision]], $MachinePrecision] * N[(0.5 - z), $MachinePrecision] + N[(z - 7.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(436.8961725563396 * z + 263.3831869810514), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

2. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\frac{1382761731551712743134679}{5250000000000000000000} + z \cdot \left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(z \cdot \left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)\right) + \color{blue}{\frac{1382761731551712743134679}{5250000000000000000000}}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)\right) \cdot z + \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right), \color{blue}{z}, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right) + \frac{16055934341359023345617179}{36750000000000000000000}, z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right) \cdot z + \frac{16055934341359023345617179}{36750000000000000000000}, z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z, z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z + \frac{1261892996482553330703662111}{2315250000000000000000000}, z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lower-fma.f64 96.9

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 96.9%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

5. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1 + \frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1 + \frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{\color{blue}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \frac{\left(\frac{1}{6} \cdot {z}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + 1}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot {z}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot {z}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   10. lift-PI.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   11. lift-PI.f64 96.9

       \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

7. Applied rewrites 96.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

8. Taylor expanded in z around inf

   \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

10. Applied rewrites 98.3%

    \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\color{blue}{\left(\left(e^{\mathsf{fma}\left(\log \left(7.5 - z\right), 0.5 - z, z - 7.5\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

11. Taylor expanded in z around 0

    \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(\frac{15}{2} - z\right), \frac{1}{2} - z, z - \frac{15}{2}\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot z\right)}\right) \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(\frac{15}{2} - z\right), \frac{1}{2} - z, z - \frac{15}{2}\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot z + \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000}}\right)\right) \]

    2. lower-fma.f64 97.8

       \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(7.5 - z\right), 0.5 - z, z - 7.5\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \mathsf{fma}\left(436.8961725563396, \color{blue}{z}, 263.3831869810514\right)\right) \]

13. Applied rewrites 97.8%

    \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(7.5 - z\right), 0.5 - z, z - 7.5\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \color{blue}{\mathsf{fma}\left(436.8961725563396, z, 263.3831869810514\right)}\right) \]
14. Add Preprocessing

Alternative 5: 96.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(7.5 - z\right), 0.5 - z, z - 7.5\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot 263.3831869810514\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ (fma (* 0.16666666666666666 (* z z)) (* PI PI) 1.0) z)
  (*
   (* (* (exp (fma (log (- 7.5 z)) (- 0.5 z) (- z 7.5))) (sqrt 2.0)) (sqrt PI))
   263.3831869810514)))

C

double code(double z) {
	return (fma((0.16666666666666666 * (z * z)), (((double) M_PI) * ((double) M_PI)), 1.0) / z) * (((exp(fma(log((7.5 - z)), (0.5 - z), (z - 7.5))) * sqrt(2.0)) * sqrt(((double) M_PI))) * 263.3831869810514);
}

Julia

function code(z)
	return Float64(Float64(fma(Float64(0.16666666666666666 * Float64(z * z)), Float64(pi * pi), 1.0) / z) * Float64(Float64(Float64(exp(fma(log(Float64(7.5 - z)), Float64(0.5 - z), Float64(z - 7.5))) * sqrt(2.0)) * sqrt(pi)) * 263.3831869810514))
end

Wolfram

code[z_] := N[(N[(N[(N[(0.16666666666666666 * N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision] * N[(N[(N[(N[Exp[N[(N[Log[N[(7.5 - z), $MachinePrecision]], $MachinePrecision] * N[(0.5 - z), $MachinePrecision] + N[(z - 7.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * 263.3831869810514), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

2. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\frac{1382761731551712743134679}{5250000000000000000000} + z \cdot \left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(z \cdot \left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)\right) + \color{blue}{\frac{1382761731551712743134679}{5250000000000000000000}}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)\right) \cdot z + \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right), \color{blue}{z}, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right) + \frac{16055934341359023345617179}{36750000000000000000000}, z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right) \cdot z + \frac{16055934341359023345617179}{36750000000000000000000}, z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z, z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z + \frac{1261892996482553330703662111}{2315250000000000000000000}, z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lower-fma.f64 96.9

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 96.9%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

5. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1 + \frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1 + \frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{\color{blue}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \frac{\left(\frac{1}{6} \cdot {z}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + 1}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot {z}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot {z}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   10. lift-PI.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   11. lift-PI.f64 96.9

       \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

7. Applied rewrites 96.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

8. Taylor expanded in z around inf

   \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

10. Applied rewrites 98.3%

    \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\color{blue}{\left(\left(e^{\mathsf{fma}\left(\log \left(7.5 - z\right), 0.5 - z, z - 7.5\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

11. Taylor expanded in z around 0

    \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(\frac{15}{2} - z\right), \frac{1}{2} - z, z - \frac{15}{2}\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000}}\right) \]
12. Step-by-step derivation

13. Applied rewrites 96.8%

    \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \pi \cdot \pi, 1\right)}{z} \cdot \left(\left(\left(e^{\mathsf{fma}\left(\log \left(7.5 - z\right), 0.5 - z, z - 7.5\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \color{blue}{263.3831869810514}\right) \]
14. Add Preprocessing

Alternative 6: 96.1% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

2. Taylor expanded in z around 0

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

   1. associate-*l/ N/A

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

   2. *-commutative N/A

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

4. Applied rewrites 95.4%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-exp.f64 96.1

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

6. Applied rewrites 96.1%

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

Alternative 7: 95.9% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

2. Taylor expanded in z around 0

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

   1. associate-*l/ N/A

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

   2. *-commutative N/A

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

4. Applied rewrites 95.4%

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

6. Applied rewrites 95.9%

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

Reproduce

herbie shell --seed 2025130 
(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))))))