Average Error: 1.8 → 0.6
Time: 2.8m
Precision: 64
Internal Precision: 576
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\left(e^{-\left(\left(\left(1 - z\right) + 0.5\right) - \left(1 - 7\right)\right)} \cdot \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) + 0.5\right) - \left(1 - 7\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right)\right)\right) \cdot \left(\left(\left(\frac{12.507343278686905}{\left(1 - z\right) - \left(1 - 5\right)} + \frac{-0.13857109526572012}{\left(6 + 1\right) - \left(1 + z\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - \left(1 - 7\right)} + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(1 + 8\right) - \left(1 + z\right)}\right)\right) + \frac{(\left(-\left(z - 4\right)\right) \cdot \left((\left((\left(3 + \left(-z\right)\right) \cdot -1259.1392167224028 + \left(\left(\left(-z\right) + 2\right) \cdot 771.3234287776531\right))_*\right) \cdot \left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) + \left((\left(\frac{676.5203681218851}{1 - z}\right) \cdot \left(\frac{-676.5203681218851}{1 - z}\right) + \left(0.9999999999998099 \cdot 0.9999999999998099\right))_* \cdot \left(\left(3 + \left(-z\right)\right) \cdot \left(\left(-z\right) + 2\right)\right)\right))_*\right) + \left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(\left(\left(3 + \left(-z\right)\right) \cdot \left(\left(-z\right) + 2\right)\right) \cdot -176.6150291621406\right)\right))_*}{\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(\left(\left(3 - z\right) \cdot \left(\left(-z\right) + 2\right)\right) \cdot \left(\left(-z\right) + 4\right)\right)}\right)\]

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 simplification1.1

    \[\leadsto \left(\left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot {\left(\left(0.5 + \left(1 - z\right)\right) - \left(1 - 7\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right)\right) \cdot e^{-\left(\left(0.5 + \left(1 - z\right)\right) - \left(1 - 7\right)\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \left(\frac{771.3234287776531}{\left(3 + 1\right) - \left(1 + z\right)} + \frac{-1259.1392167224028}{\left(2 + 1\right) - \left(1 + z\right)}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) - \left(1 - 4\right)}\right) + \left(\left(\frac{-0.13857109526572012}{\left(6 + 1\right) - \left(1 + z\right)} + \frac{12.507343278686905}{\left(1 - z\right) - \left(1 - 5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - \left(1 - 7\right)} + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(8 + 1\right) - \left(1 + z\right)}\right)\right)\right)\]
  3. Using strategy rm

  4. Applied sqrt-prod0.6

    \[\leadsto \left(\left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{\pi}\right)} \cdot {\left(\left(0.5 + \left(1 - z\right)\right) - \left(1 - 7\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right)\right) \cdot e^{-\left(\left(0.5 + \left(1 - z\right)\right) - \left(1 - 7\right)\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \left(\frac{771.3234287776531}{\left(3 + 1\right) - \left(1 + z\right)} + \frac{-1259.1392167224028}{\left(2 + 1\right) - \left(1 + z\right)}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) - \left(1 - 4\right)}\right) + \left(\left(\frac{-0.13857109526572012}{\left(6 + 1\right) - \left(1 + z\right)} + \frac{12.507343278686905}{\left(1 - z\right) - \left(1 - 5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - \left(1 - 7\right)} + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(8 + 1\right) - \left(1 + z\right)}\right)\right)\right)\]
  5. Using strategy rm

  6. Applied frac-add0.6

    \[\leadsto \left(\left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\sqrt{2} \cdot \sqrt{\pi}\right) \cdot {\left(\left(0.5 + \left(1 - z\right)\right) - \left(1 - 7\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right)\right) \cdot e^{-\left(\left(0.5 + \left(1 - z\right)\right) - \left(1 - 7\right)\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \color{blue}{\frac{771.3234287776531 \cdot \left(\left(2 + 1\right) - \left(1 + z\right)\right) + \left(\left(3 + 1\right) - \left(1 + z\right)\right) \cdot -1259.1392167224028}{\left(\left(3 + 1\right) - \left(1 + z\right)\right) \cdot \left(\left(2 + 1\right) - \left(1 + z\right)\right)}}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - \left(1 - 4\right)}\right) + \left(\left(\frac{-0.13857109526572012}{\left(6 + 1\right) - \left(1 + z\right)} + \frac{12.507343278686905}{\left(1 - z\right) - \left(1 - 5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - \left(1 - 7\right)} + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(8 + 1\right) - \left(1 + z\right)}\right)\right)\right)\]

  7. Applied flip-+0.6

    \[\leadsto \left(\left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\sqrt{2} \cdot \sqrt{\pi}\right) \cdot {\left(\left(0.5 + \left(1 - z\right)\right) - \left(1 - 7\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right)\right) \cdot e^{-\left(\left(0.5 + \left(1 - z\right)\right) - \left(1 - 7\right)\right)}\right) \cdot \left(\left(\left(\color{blue}{\frac{0.9999999999998099 \cdot 0.9999999999998099 - \frac{676.5203681218851}{\left(1 - z\right) - 0} \cdot \frac{676.5203681218851}{\left(1 - z\right) - 0}}{0.9999999999998099 - \frac{676.5203681218851}{\left(1 - z\right) - 0}}} + \frac{771.3234287776531 \cdot \left(\left(2 + 1\right) - \left(1 + z\right)\right) + \left(\left(3 + 1\right) - \left(1 + z\right)\right) \cdot -1259.1392167224028}{\left(\left(3 + 1\right) - \left(1 + z\right)\right) \cdot \left(\left(2 + 1\right) - \left(1 + z\right)\right)}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - \left(1 - 4\right)}\right) + \left(\left(\frac{-0.13857109526572012}{\left(6 + 1\right) - \left(1 + z\right)} + \frac{12.507343278686905}{\left(1 - z\right) - \left(1 - 5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - \left(1 - 7\right)} + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(8 + 1\right) - \left(1 + z\right)}\right)\right)\right)\]

  8. Applied frac-add0.6

    \[\leadsto \left(\left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\sqrt{2} \cdot \sqrt{\pi}\right) \cdot {\left(\left(0.5 + \left(1 - z\right)\right) - \left(1 - 7\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right)\right) \cdot e^{-\left(\left(0.5 + \left(1 - z\right)\right) - \left(1 - 7\right)\right)}\right) \cdot \left(\left(\color{blue}{\frac{\left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{676.5203681218851}{\left(1 - z\right) - 0} \cdot \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) \cdot \left(\left(\left(3 + 1\right) - \left(1 + z\right)\right) \cdot \left(\left(2 + 1\right) - \left(1 + z\right)\right)\right) + \left(0.9999999999998099 - \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) \cdot \left(771.3234287776531 \cdot \left(\left(2 + 1\right) - \left(1 + z\right)\right) + \left(\left(3 + 1\right) - \left(1 + z\right)\right) \cdot -1259.1392167224028\right)}{\left(0.9999999999998099 - \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) \cdot \left(\left(\left(3 + 1\right) - \left(1 + z\right)\right) \cdot \left(\left(2 + 1\right) - \left(1 + z\right)\right)\right)}} + \frac{-176.6150291621406}{\left(1 - z\right) - \left(1 - 4\right)}\right) + \left(\left(\frac{-0.13857109526572012}{\left(6 + 1\right) - \left(1 + z\right)} + \frac{12.507343278686905}{\left(1 - z\right) - \left(1 - 5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - \left(1 - 7\right)} + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(8 + 1\right) - \left(1 + z\right)}\right)\right)\right)\]

  9. Applied frac-add0.6

    \[\leadsto \left(\left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\sqrt{2} \cdot \sqrt{\pi}\right) \cdot {\left(\left(0.5 + \left(1 - z\right)\right) - \left(1 - 7\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right)\right) \cdot e^{-\left(\left(0.5 + \left(1 - z\right)\right) - \left(1 - 7\right)\right)}\right) \cdot \left(\color{blue}{\frac{\left(\left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{676.5203681218851}{\left(1 - z\right) - 0} \cdot \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) \cdot \left(\left(\left(3 + 1\right) - \left(1 + z\right)\right) \cdot \left(\left(2 + 1\right) - \left(1 + z\right)\right)\right) + \left(0.9999999999998099 - \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) \cdot \left(771.3234287776531 \cdot \left(\left(2 + 1\right) - \left(1 + z\right)\right) + \left(\left(3 + 1\right) - \left(1 + z\right)\right) \cdot -1259.1392167224028\right)\right) \cdot \left(\left(1 - z\right) - \left(1 - 4\right)\right) + \left(\left(0.9999999999998099 - \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) \cdot \left(\left(\left(3 + 1\right) - \left(1 + z\right)\right) \cdot \left(\left(2 + 1\right) - \left(1 + z\right)\right)\right)\right) \cdot -176.6150291621406}{\left(\left(0.9999999999998099 - \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) \cdot \left(\left(\left(3 + 1\right) - \left(1 + z\right)\right) \cdot \left(\left(2 + 1\right) - \left(1 + z\right)\right)\right)\right) \cdot \left(\left(1 - z\right) - \left(1 - 4\right)\right)}} + \left(\left(\frac{-0.13857109526572012}{\left(6 + 1\right) - \left(1 + z\right)} + \frac{12.507343278686905}{\left(1 - z\right) - \left(1 - 5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - \left(1 - 7\right)} + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(8 + 1\right) - \left(1 + z\right)}\right)\right)\right)\]

  10. Simplified0.6

    \[\leadsto \left(\left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\sqrt{2} \cdot \sqrt{\pi}\right) \cdot {\left(\left(0.5 + \left(1 - z\right)\right) - \left(1 - 7\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right)\right) \cdot e^{-\left(\left(0.5 + \left(1 - z\right)\right) - \left(1 - 7\right)\right)}\right) \cdot \left(\frac{\color{blue}{(\left(0 - \left(z - 4\right)\right) \cdot \left((\left((\left(\left(0 - z\right) + 3\right) \cdot -1259.1392167224028 + \left(771.3234287776531 \cdot \left(2 + \left(0 - z\right)\right)\right))_*\right) \cdot \left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) + \left((\left(\frac{676.5203681218851}{1 - z}\right) \cdot \left(\frac{-676.5203681218851}{1 - z}\right) + \left(0.9999999999998099 \cdot 0.9999999999998099\right))_* \cdot \left(\left(\left(0 - z\right) + 3\right) \cdot \left(2 + \left(0 - z\right)\right)\right)\right))_*\right) + \left(\left(\left(\left(\left(0 - z\right) + 3\right) \cdot \left(2 + \left(0 - z\right)\right)\right) \cdot -176.6150291621406\right) \cdot \left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right)\right))_*}}{\left(\left(0.9999999999998099 - \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) \cdot \left(\left(\left(3 + 1\right) - \left(1 + z\right)\right) \cdot \left(\left(2 + 1\right) - \left(1 + z\right)\right)\right)\right) \cdot \left(\left(1 - z\right) - \left(1 - 4\right)\right)} + \left(\left(\frac{-0.13857109526572012}{\left(6 + 1\right) - \left(1 + z\right)} + \frac{12.507343278686905}{\left(1 - z\right) - \left(1 - 5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - \left(1 - 7\right)} + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(8 + 1\right) - \left(1 + z\right)}\right)\right)\right)\]

  11. Simplified0.6

    \[\leadsto \left(\left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\sqrt{2} \cdot \sqrt{\pi}\right) \cdot {\left(\left(0.5 + \left(1 - z\right)\right) - \left(1 - 7\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right)\right) \cdot e^{-\left(\left(0.5 + \left(1 - z\right)\right) - \left(1 - 7\right)\right)}\right) \cdot \left(\frac{(\left(0 - \left(z - 4\right)\right) \cdot \left((\left((\left(\left(0 - z\right) + 3\right) \cdot -1259.1392167224028 + \left(771.3234287776531 \cdot \left(2 + \left(0 - z\right)\right)\right))_*\right) \cdot \left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) + \left((\left(\frac{676.5203681218851}{1 - z}\right) \cdot \left(\frac{-676.5203681218851}{1 - z}\right) + \left(0.9999999999998099 \cdot 0.9999999999998099\right))_* \cdot \left(\left(\left(0 - z\right) + 3\right) \cdot \left(2 + \left(0 - z\right)\right)\right)\right))_*\right) + \left(\left(\left(\left(\left(0 - z\right) + 3\right) \cdot \left(2 + \left(0 - z\right)\right)\right) \cdot -176.6150291621406\right) \cdot \left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right)\right))_*}{\color{blue}{\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(\left(\left(\left(0 - z\right) + 2\right) \cdot \left(\left(3 + 0\right) - z\right)\right) \cdot \left(\left(0 - z\right) + 4\right)\right)}} + \left(\left(\frac{-0.13857109526572012}{\left(6 + 1\right) - \left(1 + z\right)} + \frac{12.507343278686905}{\left(1 - z\right) - \left(1 - 5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - \left(1 - 7\right)} + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(8 + 1\right) - \left(1 + z\right)}\right)\right)\right)\]

  12. Final simplification0.6

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

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed 2018215 +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))))))