Jmat.Real.gamma, branch z greater than 0.5

?

Percentage Accurate: 0.9% → 99.1%
Time: 1.1min
Precision: binary64
Cost: 61376

?

\[z > 0.5\]
\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
↓
\[\left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{771.3234287776531}{2 + z} + \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(z, 19.623892129126734 + z \cdot -6.541297376375578, \mathsf{fma}\left({z}^{3}, 2.1804324587918593, -58.8716763873802\right)\right)\right)\right)\right)\right)\right) + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right) \]
(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
 (*
  (* (sqrt (* PI 2.0)) (/ (sqrt 0.15384615384615385) (exp 6.5)))
  (+
   0.9999999999998099
   (+
    (+
     (/ 676.5203681218851 z)
     (+
      (/ -1259.1392167224028 (+ z 1.0))
      (+
       (/ 771.3234287776531 (+ 2.0 z))
       (log1p
        (expm1
         (fma
          z
          (+ 19.623892129126734 (* z -6.541297376375578))
          (fma (pow z 3.0) 2.1804324587918593 -58.8716763873802)))))))
    (+
     (/ 12.507343278686905 (+ z 4.0))
     (+
      (/ -0.13857109526572012 (+ z 5.0))
      (+
       (/ 9.984369578019572e-6 (+ z 6.0))
       (/ 1.5056327351493116e-7 (+ z 7.0)))))))))
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) {
	return (sqrt((((double) M_PI) * 2.0)) * (sqrt(0.15384615384615385) / exp(6.5))) * (0.9999999999998099 + (((676.5203681218851 / z) + ((-1259.1392167224028 / (z + 1.0)) + ((771.3234287776531 / (2.0 + z)) + log1p(expm1(fma(z, (19.623892129126734 + (z * -6.541297376375578)), fma(pow(z, 3.0), 2.1804324587918593, -58.8716763873802))))))) + ((12.507343278686905 / (z + 4.0)) + ((-0.13857109526572012 / (z + 5.0)) + ((9.984369578019572e-6 / (z + 6.0)) + (1.5056327351493116e-7 / (z + 7.0)))))));
}

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)
	return Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64(sqrt(0.15384615384615385) / exp(6.5))) * Float64(0.9999999999998099 + Float64(Float64(Float64(676.5203681218851 / z) + Float64(Float64(-1259.1392167224028 / Float64(z + 1.0)) + Float64(Float64(771.3234287776531 / Float64(2.0 + z)) + log1p(expm1(fma(z, Float64(19.623892129126734 + Float64(z * -6.541297376375578)), fma((z ^ 3.0), 2.1804324587918593, -58.8716763873802))))))) + 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))))))))
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_] := N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[0.15384615384615385], $MachinePrecision] / N[Exp[6.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.9999999999998099 + N[(N[(N[(676.5203681218851 / z), $MachinePrecision] + N[(N[(-1259.1392167224028 / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(2.0 + z), $MachinePrecision]), $MachinePrecision] + N[Log[1 + N[(Exp[N[(z * N[(19.623892129126734 + N[(z * -6.541297376375578), $MachinePrecision]), $MachinePrecision] + N[(N[Power[z, 3.0], $MachinePrecision] * 2.1804324587918593 + -58.8716763873802), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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]

\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)
↓
\left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{771.3234287776531}{2 + z} + \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(z, 19.623892129126734 + z \cdot -6.541297376375578, \mathsf{fma}\left({z}^{3}, 2.1804324587918593, -58.8716763873802\right)\right)\right)\right)\right)\right)\right) + \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)

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 9 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Derivation?

  1. Initial program 0.8%

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

  2. Simplified0.8%

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

    [Start]0.8%

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

    *-commutative [=>]0.8%

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

    associate-*r* [=>]0.8%

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

    exp-neg [=>]0.8%

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

  3. Taylor expanded in z around 0 3.2%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \color{blue}{\frac{\sqrt{0.15384615384615385}}{e^{6.5}}}\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{771.3234287776531}{2 + z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right) \]

  4. Taylor expanded in z around 0 22.5%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{771.3234287776531}{2 + z} + \color{blue}{\left(\left(-6.541297376375578 \cdot {z}^{2} + \left(19.623892129126734 \cdot z + 2.1804324587918593 \cdot {z}^{3}\right)\right) - 58.8716763873802\right)}\right)\right)\right) + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right) \]

  5. Simplified22.5%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{771.3234287776531}{2 + z} + \color{blue}{\left(z \cdot \left(19.623892129126734 + z \cdot -6.541297376375578\right) + \mathsf{fma}\left({z}^{3}, 2.1804324587918593, -58.8716763873802\right)\right)}\right)\right)\right) + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right) \]
    Step-by-step derivation

    [Start]22.5%

    \[ \left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{771.3234287776531}{2 + z} + \left(\left(-6.541297376375578 \cdot {z}^{2} + \left(19.623892129126734 \cdot z + 2.1804324587918593 \cdot {z}^{3}\right)\right) - 58.8716763873802\right)\right)\right)\right) + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right) \]

    associate-+r+ [=>]22.5%

    \[ \left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{771.3234287776531}{2 + z} + \left(\color{blue}{\left(\left(-6.541297376375578 \cdot {z}^{2} + 19.623892129126734 \cdot z\right) + 2.1804324587918593 \cdot {z}^{3}\right)} - 58.8716763873802\right)\right)\right)\right) + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right) \]

    associate--l+ [=>]22.5%

    \[ \left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{771.3234287776531}{2 + z} + \color{blue}{\left(\left(-6.541297376375578 \cdot {z}^{2} + 19.623892129126734 \cdot z\right) + \left(2.1804324587918593 \cdot {z}^{3} - 58.8716763873802\right)\right)}\right)\right)\right) + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right) \]

    +-commutative [=>]22.5%

    \[ \left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{771.3234287776531}{2 + z} + \left(\color{blue}{\left(19.623892129126734 \cdot z + -6.541297376375578 \cdot {z}^{2}\right)} + \left(2.1804324587918593 \cdot {z}^{3} - 58.8716763873802\right)\right)\right)\right)\right) + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right) \]

    *-commutative [=>]22.5%

    \[ \left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{771.3234287776531}{2 + z} + \left(\left(\color{blue}{z \cdot 19.623892129126734} + -6.541297376375578 \cdot {z}^{2}\right) + \left(2.1804324587918593 \cdot {z}^{3} - 58.8716763873802\right)\right)\right)\right)\right) + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right) \]

    *-commutative [=>]22.5%

    \[ \left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{771.3234287776531}{2 + z} + \left(\left(z \cdot 19.623892129126734 + \color{blue}{{z}^{2} \cdot -6.541297376375578}\right) + \left(2.1804324587918593 \cdot {z}^{3} - 58.8716763873802\right)\right)\right)\right)\right) + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right) \]

    unpow2 [=>]22.5%

    \[ \left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{771.3234287776531}{2 + z} + \left(\left(z \cdot 19.623892129126734 + \color{blue}{\left(z \cdot z\right)} \cdot -6.541297376375578\right) + \left(2.1804324587918593 \cdot {z}^{3} - 58.8716763873802\right)\right)\right)\right)\right) + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right) \]

    associate-*l* [=>]22.5%

    \[ \left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{771.3234287776531}{2 + z} + \left(\left(z \cdot 19.623892129126734 + \color{blue}{z \cdot \left(z \cdot -6.541297376375578\right)}\right) + \left(2.1804324587918593 \cdot {z}^{3} - 58.8716763873802\right)\right)\right)\right)\right) + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right) \]

    distribute-lft-out [=>]22.5%

    \[ \left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{771.3234287776531}{2 + z} + \left(\color{blue}{z \cdot \left(19.623892129126734 + z \cdot -6.541297376375578\right)} + \left(2.1804324587918593 \cdot {z}^{3} - 58.8716763873802\right)\right)\right)\right)\right) + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right) \]

    *-commutative [=>]22.5%

    \[ \left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{771.3234287776531}{2 + z} + \left(z \cdot \left(19.623892129126734 + z \cdot -6.541297376375578\right) + \left(\color{blue}{{z}^{3} \cdot 2.1804324587918593} - 58.8716763873802\right)\right)\right)\right)\right) + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right) \]

    fma-neg [=>]22.5%

    \[ \left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{771.3234287776531}{2 + z} + \left(z \cdot \left(19.623892129126734 + z \cdot -6.541297376375578\right) + \color{blue}{\mathsf{fma}\left({z}^{3}, 2.1804324587918593, -58.8716763873802\right)}\right)\right)\right)\right) + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right) \]

    metadata-eval [=>]22.5%

    \[ \left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{771.3234287776531}{2 + z} + \left(z \cdot \left(19.623892129126734 + z \cdot -6.541297376375578\right) + \mathsf{fma}\left({z}^{3}, 2.1804324587918593, \color{blue}{-58.8716763873802}\right)\right)\right)\right)\right) + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right) \]

  6. Applied egg-rr98.5%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{771.3234287776531}{2 + z} + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(z, 19.623892129126734 + z \cdot -6.541297376375578, \mathsf{fma}\left({z}^{3}, 2.1804324587918593, -58.8716763873802\right)\right)\right)\right)}\right)\right)\right) + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right) \]
    Step-by-step derivation

    [Start]22.5%

    \[ \left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{771.3234287776531}{2 + z} + \left(z \cdot \left(19.623892129126734 + z \cdot -6.541297376375578\right) + \mathsf{fma}\left({z}^{3}, 2.1804324587918593, -58.8716763873802\right)\right)\right)\right)\right) + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right) \]

    log1p-expm1-u [=>]52.0%

    \[ \left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{771.3234287776531}{2 + z} + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(z \cdot \left(19.623892129126734 + z \cdot -6.541297376375578\right) + \mathsf{fma}\left({z}^{3}, 2.1804324587918593, -58.8716763873802\right)\right)\right)}\right)\right)\right) + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right) \]

    fma-def [=>]98.5%

    \[ \left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{771.3234287776531}{2 + z} + \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(z, 19.623892129126734 + z \cdot -6.541297376375578, \mathsf{fma}\left({z}^{3}, 2.1804324587918593, -58.8716763873802\right)\right)}\right)\right)\right)\right)\right) + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right) \]

  7. Final simplification98.5%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{771.3234287776531}{2 + z} + \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(z, 19.623892129126734 + z \cdot -6.541297376375578, \mathsf{fma}\left({z}^{3}, 2.1804324587918593, -58.8716763873802\right)\right)\right)\right)\right)\right)\right) + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right) \]

Alternatives

Alternative 1
Accuracy99.1%
Cost61376
\[\left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{771.3234287776531}{2 + z} + \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(z, 19.623892129126734 + z \cdot -6.541297376375578, \mathsf{fma}\left({z}^{3}, 2.1804324587918593, -58.8716763873802\right)\right)\right)\right)\right)\right)\right) + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right) \]
Alternative 2
Accuracy68.7%
Cost35392
\[\left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{771.3234287776531}{2 + z} + {z}^{3} \cdot 2.1804324587918593\right)\right)\right)\right)\right) \]
Alternative 3
Accuracy53.0%
Cost34560
\[\left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\frac{12.0895510149948}{z} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{z + 7} + \left(\mathsf{fma}\left(z, -2.7734359938943256 \cdot 10^{-7}, 1.6640615963365953 \cdot 10^{-6}\right) + \left(z \cdot z\right) \cdot 4.622393323157209 \cdot 10^{-8}\right)\right)\right)\right)\right)\right) \]
Alternative 4
Accuracy6.3%
Cost28416
\[\left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) + \left(\left(z \cdot 1085.9322516571162 + 676.5203681218851 \cdot \frac{1}{z}\right) - 932.3491787209565\right)\right)\right) \]
Alternative 5
Accuracy5.3%
Cost28288
\[\left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{771.3234287776531}{2 + z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\frac{12.507343278686905}{z + 4} + \left(z \cdot 0.005542563394309548 - 0.027712533482508614\right)\right)\right)\right) \]
Alternative 6
Accuracy5.3%
Cost27136
\[\left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\frac{12.0895510149948}{z} + \left(\frac{12.507343278686905}{z + 4} + \left(z \cdot 0.005542563394309548 - 0.027712533482508614\right)\right)\right)\right) \]
Alternative 7
Accuracy3.2%
Cost27008
\[\left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\frac{12.0895510149948}{z} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13856096033286858}{z}\right)\right)\right) \]
Alternative 8
Accuracy3.2%
Cost26880
\[\left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(\frac{12.0895510149948}{z} + \left(\frac{12.507343278686905}{z + 4} + -0.027712533482508614\right)\right)\right) \]
Alternative 9
Accuracy3.2%
Cost26112
\[\left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot 0.9722874665173012 \]

Reproduce?

herbie shell --seed 2023271 
(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)))))