Average Error: 3.9 → 3.6
Time: 7.2s
Precision: binary64
\[z > 0.5\]
\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
↓
\[\begin{array}{l} t_0 := \sqrt{0.9999999999998099 + \left(\frac{\frac{\mathsf{fma}\left(z, -582.6188486005177, 676.5203681218851\right)}{z}}{z + 1} + \frac{771.3234287776531}{2 + z}\right)}\\ \frac{\left(\sqrt{\pi \cdot 2} \cdot {\left(z + 6.5\right)}^{\left(z + -0.5\right)}\right) \cdot \left(t_0 \cdot t_0 + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)}{e^{z} \cdot e^{6.5}} \end{array} \]
(FPCore (z)
 :precision binary64
 (*
  (*
   (* (sqrt (* PI 2.0)) (pow (+ (+ (- z 1.0) 7.0) 0.5) (+ (- z 1.0) 0.5)))
   (exp (- (+ (+ (- z 1.0) 7.0) 0.5))))
  (+
   (+
    (+
     (+
      (+
       (+
        (+
         (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- z 1.0) 1.0)))
         (/ -1259.1392167224028 (+ (- z 1.0) 2.0)))
        (/ 771.3234287776531 (+ (- z 1.0) 3.0)))
       (/ -176.6150291621406 (+ (- z 1.0) 4.0)))
      (/ 12.507343278686905 (+ (- z 1.0) 5.0)))
     (/ -0.13857109526572012 (+ (- z 1.0) 6.0)))
    (/ 9.984369578019572e-6 (+ (- z 1.0) 7.0)))
   (/ 1.5056327351493116e-7 (+ (- z 1.0) 8.0)))))
↓
(FPCore (z)
 :precision binary64
 (let* ((t_0
         (sqrt
          (+
           0.9999999999998099
           (+
            (/ (/ (fma z -582.6188486005177 676.5203681218851) z) (+ z 1.0))
            (/ 771.3234287776531 (+ 2.0 z)))))))
   (/
    (*
     (* (sqrt (* PI 2.0)) (pow (+ z 6.5) (+ z -0.5)))
     (+
      (* t_0 t_0)
      (+
       (/ -176.6150291621406 (+ z 3.0))
       (+
        (/ 12.507343278686905 (+ z 4.0))
        (+
         (/ -0.13857109526572012 (+ z 5.0))
         (+
          (/ 9.984369578019572e-6 (+ z 6.0))
          (/ 1.5056327351493116e-7 (+ z 7.0))))))))
    (* (exp z) (exp 6.5)))))
double code(double z) {
	return ((sqrt((((double) M_PI) * 2.0)) * pow((((z - 1.0) + 7.0) + 0.5), ((z - 1.0) + 0.5))) * exp(-(((z - 1.0) + 7.0) + 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / ((z - 1.0) + 1.0))) + (-1259.1392167224028 / ((z - 1.0) + 2.0))) + (771.3234287776531 / ((z - 1.0) + 3.0))) + (-176.6150291621406 / ((z - 1.0) + 4.0))) + (12.507343278686905 / ((z - 1.0) + 5.0))) + (-0.13857109526572012 / ((z - 1.0) + 6.0))) + (9.984369578019572e-6 / ((z - 1.0) + 7.0))) + (1.5056327351493116e-7 / ((z - 1.0) + 8.0)));
}
↓
double code(double z) {
	double t_0 = sqrt((0.9999999999998099 + (((fma(z, -582.6188486005177, 676.5203681218851) / z) / (z + 1.0)) + (771.3234287776531 / (2.0 + z)))));
	return ((sqrt((((double) M_PI) * 2.0)) * pow((z + 6.5), (z + -0.5))) * ((t_0 * t_0) + ((-176.6150291621406 / (z + 3.0)) + ((12.507343278686905 / (z + 4.0)) + ((-0.13857109526572012 / (z + 5.0)) + ((9.984369578019572e-6 / (z + 6.0)) + (1.5056327351493116e-7 / (z + 7.0)))))))) / (exp(z) * exp(6.5));
}

function code(z)
	return Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (Float64(Float64(Float64(z - 1.0) + 7.0) + 0.5) ^ Float64(Float64(z - 1.0) + 0.5))) * exp(Float64(-Float64(Float64(Float64(z - 1.0) + 7.0) + 0.5)))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(Float64(z - 1.0) + 1.0))) + Float64(-1259.1392167224028 / Float64(Float64(z - 1.0) + 2.0))) + Float64(771.3234287776531 / Float64(Float64(z - 1.0) + 3.0))) + Float64(-176.6150291621406 / Float64(Float64(z - 1.0) + 4.0))) + Float64(12.507343278686905 / Float64(Float64(z - 1.0) + 5.0))) + Float64(-0.13857109526572012 / Float64(Float64(z - 1.0) + 6.0))) + Float64(9.984369578019572e-6 / Float64(Float64(z - 1.0) + 7.0))) + Float64(1.5056327351493116e-7 / Float64(Float64(z - 1.0) + 8.0))))
end
↓
function code(z)
	t_0 = sqrt(Float64(0.9999999999998099 + Float64(Float64(Float64(fma(z, -582.6188486005177, 676.5203681218851) / z) / Float64(z + 1.0)) + Float64(771.3234287776531 / Float64(2.0 + z)))))
	return Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (Float64(z + 6.5) ^ Float64(z + -0.5))) * Float64(Float64(t_0 * t_0) + Float64(Float64(-176.6150291621406 / Float64(z + 3.0)) + Float64(Float64(12.507343278686905 / Float64(z + 4.0)) + Float64(Float64(-0.13857109526572012 / Float64(z + 5.0)) + Float64(Float64(9.984369578019572e-6 / Float64(z + 6.0)) + Float64(1.5056327351493116e-7 / Float64(z + 7.0)))))))) / Float64(exp(z) * exp(6.5)))
end

code[z_] := N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[(N[(z - 1.0), $MachinePrecision] + 7.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(z - 1.0), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[(N[(z - 1.0), $MachinePrecision] + 7.0), $MachinePrecision] + 0.5), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(N[(z - 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(N[(z - 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(N[(z - 1.0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(z - 1.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(N[(z - 1.0), $MachinePrecision] + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(z - 1.0), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(N[(z - 1.0), $MachinePrecision] + 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(z - 1.0), $MachinePrecision] + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[z_] := Block[{t$95$0 = N[Sqrt[N[(0.9999999999998099 + N[(N[(N[(N[(z * -582.6188486005177 + 676.5203681218851), $MachinePrecision] / z), $MachinePrecision] / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(2.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(z + 6.5), $MachinePrecision], N[(z + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[(-176.6150291621406 / N[(z + 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(z + 4.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(z + 5.0), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(z + 6.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(z + 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Exp[z], $MachinePrecision] * N[Exp[6.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)
↓
\begin{array}{l}
t_0 := \sqrt{0.9999999999998099 + \left(\frac{\frac{\mathsf{fma}\left(z, -582.6188486005177, 676.5203681218851\right)}{z}}{z + 1} + \frac{771.3234287776531}{2 + z}\right)}\\
\frac{\left(\sqrt{\pi \cdot 2} \cdot {\left(z + 6.5\right)}^{\left(z + -0.5\right)}\right) \cdot \left(t_0 \cdot t_0 + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)}{e^{z} \cdot e^{6.5}}
\end{array}

Error

Derivation

  1. Initial program 3.9

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

  2. Simplified3.8

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

  3. Applied egg-rr3.9

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

  4. Taylor expanded in z around 0 3.6

    \[\leadsto \frac{\left(\sqrt{\pi \cdot 2} \cdot {\left(z + 6.5\right)}^{\left(z + -0.5\right)}\right) \cdot \left(\left(\left(0.9999999999998099 + \frac{\color{blue}{676.5203681218851 + -582.6188486005177 \cdot z}}{z \cdot \left(z + 1\right)}\right) + \frac{771.3234287776531}{2 + z}\right) + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)}{e^{z + 6.5}} \]

  5. Simplified3.6

    \[\leadsto \frac{\left(\sqrt{\pi \cdot 2} \cdot {\left(z + 6.5\right)}^{\left(z + -0.5\right)}\right) \cdot \left(\left(\left(0.9999999999998099 + \frac{\color{blue}{\mathsf{fma}\left(z, -582.6188486005177, 676.5203681218851\right)}}{z \cdot \left(z + 1\right)}\right) + \frac{771.3234287776531}{2 + z}\right) + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)}{e^{z + 6.5}} \]

  6. Applied egg-rr3.6

    \[\leadsto \frac{\left(\sqrt{\pi \cdot 2} \cdot {\left(z + 6.5\right)}^{\left(z + -0.5\right)}\right) \cdot \left(\color{blue}{\sqrt{0.9999999999998099 + \left(\frac{\frac{\mathsf{fma}\left(z, -582.6188486005177, 676.5203681218851\right)}{z}}{z + 1} + \frac{771.3234287776531}{2 + z}\right)} \cdot \sqrt{0.9999999999998099 + \left(\frac{\frac{\mathsf{fma}\left(z, -582.6188486005177, 676.5203681218851\right)}{z}}{z + 1} + \frac{771.3234287776531}{2 + z}\right)}} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)}{e^{z + 6.5}} \]

  7. Applied egg-rr3.6

    \[\leadsto \frac{\left(\sqrt{\pi \cdot 2} \cdot {\left(z + 6.5\right)}^{\left(z + -0.5\right)}\right) \cdot \left(\sqrt{0.9999999999998099 + \left(\frac{\frac{\mathsf{fma}\left(z, -582.6188486005177, 676.5203681218851\right)}{z}}{z + 1} + \frac{771.3234287776531}{2 + z}\right)} \cdot \sqrt{0.9999999999998099 + \left(\frac{\frac{\mathsf{fma}\left(z, -582.6188486005177, 676.5203681218851\right)}{z}}{z + 1} + \frac{771.3234287776531}{2 + z}\right)} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)}{\color{blue}{e^{z} \cdot e^{6.5}}} \]

  8. Final simplification3.6

    \[\leadsto \frac{\left(\sqrt{\pi \cdot 2} \cdot {\left(z + 6.5\right)}^{\left(z + -0.5\right)}\right) \cdot \left(\sqrt{0.9999999999998099 + \left(\frac{\frac{\mathsf{fma}\left(z, -582.6188486005177, 676.5203681218851\right)}{z}}{z + 1} + \frac{771.3234287776531}{2 + z}\right)} \cdot \sqrt{0.9999999999998099 + \left(\frac{\frac{\mathsf{fma}\left(z, -582.6188486005177, 676.5203681218851\right)}{z}}{z + 1} + \frac{771.3234287776531}{2 + z}\right)} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)}{e^{z} \cdot e^{6.5}} \]

Reproduce

herbie shell --seed 2022190 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z greater than 0.5"
  :precision binary64
  :pre (> z 0.5)
  (* (* (* (sqrt (* PI 2.0)) (pow (+ (+ (- z 1.0) 7.0) 0.5) (+ (- z 1.0) 0.5))) (exp (- (+ (+ (- z 1.0) 7.0) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- z 1.0) 1.0))) (/ -1259.1392167224028 (+ (- z 1.0) 2.0))) (/ 771.3234287776531 (+ (- z 1.0) 3.0))) (/ -176.6150291621406 (+ (- z 1.0) 4.0))) (/ 12.507343278686905 (+ (- z 1.0) 5.0))) (/ -0.13857109526572012 (+ (- z 1.0) 6.0))) (/ 9.984369578019572e-6 (+ (- z 1.0) 7.0))) (/ 1.5056327351493116e-7 (+ (- z 1.0) 8.0)))))