Average Error: 60.2 → 1.0
Time: 6.4m
Precision: 64
Internal Precision: 128
\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)\]
↓
\[\frac{\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(0.5 + z\right) + 6\right)}^{\left(-1 + \left(0.5 + z\right)\right)}\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{z - -1} + 0.9999999999998099\right) + \left(\frac{771.3234287776531}{2 + z} + \left(\frac{12.507343278686905}{z + 4} + \frac{676.5203681218851}{z}\right)\right)\right) + \frac{-176.6150291621406}{z + 3}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{7 + z} + \left(\frac{9.984369578019572 \cdot 10^{-06}}{6 + z} + \frac{-0.13857109526572012}{z + 5}\right)\right)\right)}{e^{\left(0.5 + z\right) + 6}}\]

Error

Bits error versus z

Derivation

  1. Initial program 60.2

    \[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)\]

  2. Simplified0.9

    \[\leadsto \color{blue}{\left(\frac{12.507343278686905}{z + 4} + \left(\left(\left(\frac{771.3234287776531}{z + 2} + \left(\frac{676.5203681218851}{z} + \left(0.9999999999998099 + \frac{-1259.1392167224028}{z - -1}\right)\right)\right) + \frac{-176.6150291621406}{z + 3}\right) + \frac{-0.13857109526572012}{z - -5}\right)\right) \cdot \left(\left({\left(\left(7 + \left(z - 1\right)\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot e^{-\left(\left(7 + \left(z - 1\right)\right) + 0.5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(z - 1\right)} + \frac{1.5056327351493116 \cdot 10^{-07}}{z + 7}\right) \cdot \left(\left({\left(\left(7 + \left(z - 1\right)\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot e^{-\left(\left(7 + \left(z - 1\right)\right) + 0.5\right)}\right)}\]

  3. Simplified1.2

    \[\leadsto \color{blue}{\left({\left(\left(z + 0.5\right) + 6\right)}^{\left(0.5 + \left(z + -1\right)\right)} \cdot \frac{\sqrt{\pi \cdot 2}}{e^{\left(z + 0.5\right) + 6}}\right) \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-06}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-07}}{z + 7}\right) + \left(\left(\left(\frac{771.3234287776531}{z + 2} + \frac{12.507343278686905}{z + 4}\right) + \frac{-176.6150291621406}{3 + z}\right) + \left(\frac{676.5203681218851}{z} + \left(0.9999999999998099 + \frac{-1259.1392167224028}{z - -1}\right)\right)\right)\right) + \frac{-0.13857109526572012}{z - -5}\right)}\]

  4. Simplified1.0

    \[\leadsto \color{blue}{\frac{\left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{7 + z} + \left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-06}}{6 + z}\right)\right) + \left(\left(\left(0.9999999999998099 + \frac{-1259.1392167224028}{z - -1}\right) + \left(\frac{771.3234287776531}{z + 2} + \left(\frac{12.507343278686905}{z + 4} + \frac{676.5203681218851}{z}\right)\right)\right) + \frac{-176.6150291621406}{z + 3}\right)\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(6 + \left(0.5 + z\right)\right)}^{\left(-1 + \left(0.5 + z\right)\right)}\right)}{e^{6 + \left(0.5 + z\right)}}}\]

  5. Final simplification1.0

    \[\leadsto \frac{\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(0.5 + z\right) + 6\right)}^{\left(-1 + \left(0.5 + z\right)\right)}\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{z - -1} + 0.9999999999998099\right) + \left(\frac{771.3234287776531}{2 + z} + \left(\frac{12.507343278686905}{z + 4} + \frac{676.5203681218851}{z}\right)\right)\right) + \frac{-176.6150291621406}{z + 3}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{7 + z} + \left(\frac{9.984369578019572 \cdot 10^{-06}}{6 + z} + \frac{-0.13857109526572012}{z + 5}\right)\right)\right)}{e^{\left(0.5 + z\right) + 6}}\]

Reproduce

herbie shell --seed 2019094 +o rules:numerics
(FPCore (z)
  :name "Jmat.Real.gamma, branch z greater than 0.5"
  (* (* (* (sqrt (* PI 2)) (pow (+ (+ (- z 1) 7) 0.5) (+ (- z 1) 0.5))) (exp (- (+ (+ (- z 1) 7) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- z 1) 1))) (/ -1259.1392167224028 (+ (- z 1) 2))) (/ 771.3234287776531 (+ (- z 1) 3))) (/ -176.6150291621406 (+ (- z 1) 4))) (/ 12.507343278686905 (+ (- z 1) 5))) (/ -0.13857109526572012 (+ (- z 1) 6))) (/ 9.984369578019572e-06 (+ (- z 1) 7))) (/ 1.5056327351493116e-07 (+ (- z 1) 8)))))