Average Error: 1.8 → 0.9
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\left(\sqrt[3]{\left(\left(\frac{1}{e^{\left(6 + \left(1 - z\right)\right) + 0.5}} \cdot \frac{1}{e^{\left(6 + \left(1 - z\right)\right) + 0.5}}\right) \cdot \frac{1}{e^{\left(6 + \left(1 - z\right)\right) + 0.5}}\right) \cdot \left(\left({\left(\left(6 + \left(1 - z\right)\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)} \cdot {\left(\left(6 + \left(1 - z\right)\right) + 0.5\right)}^{\left(\sqrt[3]{\left(1 - z\right) - 1} \cdot \left(\sqrt[3]{\left(1 - z\right) - 1} \cdot \sqrt[3]{\left(1 - z\right) - 1}\right) + 0.5\right)}\right) \cdot {\left(\left(6 + \left(1 - z\right)\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-06}}{6 + \left(1 - z\right)} + \frac{1.5056327351493116 \cdot 10^{-07}}{9 - \left(1 + z\right)}\right) + \left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \frac{676.5203681218851}{1 - z}\right) + \left(0.9999999999998099 + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \left(\frac{-0.13857109526572012}{7 - \left(1 + z\right)} + \left(\frac{12.507343278686905}{6 - \left(1 + z\right)} + \frac{-176.6150291621406}{3 + \left(1 - z\right)}\right)\right)\right)\right)\]

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. Simplified1.4

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

  4. Applied add-cbrt-cube0.9

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

  5. Applied add-cbrt-cube0.9

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

  6. Applied cbrt-unprod0.9

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

  8. Applied add-cube-cbrt0.9

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

  9. Final simplification0.9

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

Reproduce

herbie shell --seed 2019053 
(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))))))