Average Error: 1.8 → 0.7
Time: 4.2m
Precision: 64
Internal Precision: 576
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\left(\left(\sqrt{2 \cdot \pi} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \cdot \sqrt[3]{{\left({\left(\left(6 + 0.5\right) + \left(1 - z\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right)}^{3} \cdot e^{(3 \cdot \left(z - 7\right) + \left(0.5 \cdot -3\right))_*}}\right) \cdot \left(\left(\left(\frac{-0.13857109526572012}{5 + \left(1 - z\right)} + \frac{12.507343278686905}{6 - \left(1 + z\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-07}}{9 - \left(1 + z\right)}\right)\right) + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right)\right) + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{5 - \left(1 + z\right)}\right)\right)\right)\]

Error

Bits error versus z

Derivation

  1. Initial program 1.8

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

  2. Initial simplification0.7

    \[\leadsto \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \left({\left(\left(1 - z\right) - \left(-6 - 0.5\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)} \cdot \frac{1}{e^{\left(1 - z\right) - \left(-6 - 0.5\right)}}\right)\right) \cdot \left(\left(\left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{5 - \left(z + 1\right)}\right) + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right)\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{12.507343278686905}{6 - \left(z + 1\right)}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{9 - \left(z + 1\right)} + \frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - -6}\right)\right)\right)\]
  3. Using strategy rm

  4. Applied add-cbrt-cube0.7

    \[\leadsto \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \left({\left(\left(1 - z\right) - \left(-6 - 0.5\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)} \cdot \color{blue}{\sqrt[3]{\left(\frac{1}{e^{\left(1 - z\right) - \left(-6 - 0.5\right)}} \cdot \frac{1}{e^{\left(1 - z\right) - \left(-6 - 0.5\right)}}\right) \cdot \frac{1}{e^{\left(1 - z\right) - \left(-6 - 0.5\right)}}}}\right)\right) \cdot \left(\left(\left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{5 - \left(z + 1\right)}\right) + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right)\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{12.507343278686905}{6 - \left(z + 1\right)}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{9 - \left(z + 1\right)} + \frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - -6}\right)\right)\right)\]

  5. Applied add-cbrt-cube0.7

    \[\leadsto \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \left(\color{blue}{\sqrt[3]{\left({\left(\left(1 - z\right) - \left(-6 - 0.5\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)} \cdot {\left(\left(1 - z\right) - \left(-6 - 0.5\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right) \cdot {\left(\left(1 - z\right) - \left(-6 - 0.5\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}}} \cdot \sqrt[3]{\left(\frac{1}{e^{\left(1 - z\right) - \left(-6 - 0.5\right)}} \cdot \frac{1}{e^{\left(1 - z\right) - \left(-6 - 0.5\right)}}\right) \cdot \frac{1}{e^{\left(1 - z\right) - \left(-6 - 0.5\right)}}}\right)\right) \cdot \left(\left(\left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{5 - \left(z + 1\right)}\right) + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right)\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{12.507343278686905}{6 - \left(z + 1\right)}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{9 - \left(z + 1\right)} + \frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - -6}\right)\right)\right)\]

  6. Applied cbrt-unprod0.7

    \[\leadsto \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \color{blue}{\sqrt[3]{\left(\left({\left(\left(1 - z\right) - \left(-6 - 0.5\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)} \cdot {\left(\left(1 - z\right) - \left(-6 - 0.5\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right) \cdot {\left(\left(1 - z\right) - \left(-6 - 0.5\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right) \cdot \left(\left(\frac{1}{e^{\left(1 - z\right) - \left(-6 - 0.5\right)}} \cdot \frac{1}{e^{\left(1 - z\right) - \left(-6 - 0.5\right)}}\right) \cdot \frac{1}{e^{\left(1 - z\right) - \left(-6 - 0.5\right)}}\right)}}\right) \cdot \left(\left(\left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{5 - \left(z + 1\right)}\right) + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right)\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{12.507343278686905}{6 - \left(z + 1\right)}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{9 - \left(z + 1\right)} + \frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - -6}\right)\right)\right)\]

  7. Simplified0.7

    \[\leadsto \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \sqrt[3]{\color{blue}{{\left({\left(\left(1 - z\right) + \left(0.5 + 6\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right)}^{3} \cdot {\left(\frac{\frac{1}{e} \cdot e^{z}}{e^{0.5 + 6}}\right)}^{3}}}\right) \cdot \left(\left(\left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{5 - \left(z + 1\right)}\right) + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right)\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{12.507343278686905}{6 - \left(z + 1\right)}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{9 - \left(z + 1\right)} + \frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - -6}\right)\right)\right)\]
  8. Using strategy rm

  9. Applied add-exp-log0.7

    \[\leadsto \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \sqrt[3]{{\left({\left(\left(1 - z\right) + \left(0.5 + 6\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right)}^{3} \cdot {\left(\frac{\color{blue}{e^{\log \left(\frac{1}{e} \cdot e^{z}\right)}}}{e^{0.5 + 6}}\right)}^{3}}\right) \cdot \left(\left(\left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{5 - \left(z + 1\right)}\right) + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right)\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{12.507343278686905}{6 - \left(z + 1\right)}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{9 - \left(z + 1\right)} + \frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - -6}\right)\right)\right)\]

  10. Applied div-exp0.7

    \[\leadsto \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \sqrt[3]{{\left({\left(\left(1 - z\right) + \left(0.5 + 6\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right)}^{3} \cdot {\color{blue}{\left(e^{\log \left(\frac{1}{e} \cdot e^{z}\right) - \left(0.5 + 6\right)}\right)}}^{3}}\right) \cdot \left(\left(\left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{5 - \left(z + 1\right)}\right) + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right)\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{12.507343278686905}{6 - \left(z + 1\right)}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{9 - \left(z + 1\right)} + \frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - -6}\right)\right)\right)\]

  11. Applied pow-exp0.7

    \[\leadsto \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \sqrt[3]{{\left({\left(\left(1 - z\right) + \left(0.5 + 6\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right)}^{3} \cdot \color{blue}{e^{\left(\log \left(\frac{1}{e} \cdot e^{z}\right) - \left(0.5 + 6\right)\right) \cdot 3}}}\right) \cdot \left(\left(\left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{5 - \left(z + 1\right)}\right) + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right)\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{12.507343278686905}{6 - \left(z + 1\right)}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{9 - \left(z + 1\right)} + \frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - -6}\right)\right)\right)\]

  12. Simplified0.7

    \[\leadsto \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \sqrt[3]{{\left({\left(\left(1 - z\right) + \left(0.5 + 6\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right)}^{3} \cdot e^{\color{blue}{(3 \cdot \left(z - 7\right) + \left(-3 \cdot 0.5\right))_*}}}\right) \cdot \left(\left(\left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{5 - \left(z + 1\right)}\right) + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right)\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{12.507343278686905}{6 - \left(z + 1\right)}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{9 - \left(z + 1\right)} + \frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - -6}\right)\right)\right)\]

  13. Final simplification0.7

    \[\leadsto \left(\left(\sqrt{2 \cdot \pi} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \cdot \sqrt[3]{{\left({\left(\left(6 + 0.5\right) + \left(1 - z\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right)}^{3} \cdot e^{(3 \cdot \left(z - 7\right) + \left(0.5 \cdot -3\right))_*}}\right) \cdot \left(\left(\left(\frac{-0.13857109526572012}{5 + \left(1 - z\right)} + \frac{12.507343278686905}{6 - \left(1 + z\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-07}}{9 - \left(1 + z\right)}\right)\right) + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right)\right) + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{5 - \left(1 + z\right)}\right)\right)\right)\]

Runtime

Time bar (total: 4.2m)Debug logProfile

herbie shell --seed 2018249 +o rules:numerics
(FPCore (z)
  :name "Jmat.Real.gamma, branch z less than 0.5"
  (* (/ PI (sin (* PI z))) (* (* (* (sqrt (* PI 2)) (pow (+ (+ (- (- 1 z) 1) 7) 0.5) (+ (- (- 1 z) 1) 0.5))) (exp (- (+ (+ (- (- 1 z) 1) 7) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- (- 1 z) 1) 1))) (/ -1259.1392167224028 (+ (- (- 1 z) 1) 2))) (/ 771.3234287776531 (+ (- (- 1 z) 1) 3))) (/ -176.6150291621406 (+ (- (- 1 z) 1) 4))) (/ 12.507343278686905 (+ (- (- 1 z) 1) 5))) (/ -0.13857109526572012 (+ (- (- 1 z) 1) 6))) (/ 9.984369578019572e-06 (+ (- (- 1 z) 1) 7))) (/ 1.5056327351493116e-07 (+ (- (- 1 z) 1) 8))))))