Average Error: 1.8 → 1.0
Time: 3.1m
Precision: 64
Internal Precision: 576
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\left(\left(\frac{{\left(\left(0.5 + 7\right) + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}}{e^{\left(0.5 + 7\right) + \left(-z\right)}} \cdot \sqrt[3]{\left(2 \cdot \pi\right) \cdot \sqrt{2 \cdot \pi}}\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\left((z \cdot 0.49644453405676175 + \left((\left(z \cdot z\right) \cdot 0.09941721338104284 + 2.478373473193094)_*\right))_* + \frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(-z\right)}\right) + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{\left(-z\right) + 3}\right)\right) + \left(\left(\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) + \frac{-176.6150291621406}{\left(-z\right) + 4}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{8 - z}\right)\right)\]

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. Applied simplify1.1

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

  4. Applied add-cbrt-cube1.1

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

  5. Applied simplify0.6

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

  6. Taylor expanded around 0 1.1

    \[\leadsto \left(\left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\sqrt[3]{\sqrt{\pi \cdot 2} \cdot \left(\pi \cdot 2\right)} \cdot {\left(\left(0.5 + \left(1 - z\right)\right) - \left(1 - 7\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right)\right) \cdot e^{-\left(\left(0.5 + \left(1 - z\right)\right) - \left(1 - 7\right)\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \left(\frac{771.3234287776531}{\left(3 + 1\right) - \left(1 + z\right)} + \frac{-1259.1392167224028}{\left(2 + 1\right) - \left(1 + z\right)}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) - \left(1 - 4\right)}\right) + \left(\color{blue}{\left(0.49644453405676175 \cdot z + \left(0.09941721338104284 \cdot {z}^{2} + 2.478373473193094\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - \left(1 - 7\right)} + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(8 + 1\right) - \left(1 + z\right)}\right)\right)\right)\]

  7. Applied simplify1.1

    \[\leadsto \color{blue}{\left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{\left(8 + 1\right) - \left(1 + z\right)} + \frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - \left(1 - 7\right)}\right) + (z \cdot 0.49644453405676175 + \left((0.09941721338104284 \cdot \left(z \cdot z\right) + 2.478373473193094)_*\right))_*\right) + \left(\left(\frac{-1259.1392167224028}{\left(1 + 2\right) - \left(1 + z\right)} + \frac{771.3234287776531}{\left(3 + 1\right) - \left(1 + z\right)}\right) + \left(\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) + \frac{-176.6150291621406}{\left(1 - z\right) - \left(1 - 4\right)}\right)\right)\right) \cdot \left(\left(\left(\sqrt[3]{\sqrt{2 \cdot \pi} \cdot \left(2 \cdot \pi\right)} \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \cdot {\left(\left(\left(1 - z\right) + 0.5\right) - \left(1 - 7\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + 0.5\right) - \left(1 - 7\right)\right)}\right)}\]

  8. Taylor expanded around 0 1.1

    \[\leadsto \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{\left(8 + 1\right) - \left(1 + z\right)} + \frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - \left(1 - 7\right)}\right) + (z \cdot 0.49644453405676175 + \left((0.09941721338104284 \cdot \left(z \cdot z\right) + 2.478373473193094)_*\right))_*\right) + \left(\left(\frac{-1259.1392167224028}{\left(1 + 2\right) - \left(1 + z\right)} + \frac{771.3234287776531}{\left(3 + 1\right) - \left(1 + z\right)}\right) + \left(\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) + \frac{-176.6150291621406}{\left(1 - z\right) - \left(1 - 4\right)}\right)\right)\right) \cdot \left(\left(\left(\color{blue}{\sqrt[3]{\sqrt{2 \cdot \pi} \cdot \left(2 \cdot \pi\right)}} \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \cdot {\left(\left(\left(1 - z\right) + 0.5\right) - \left(1 - 7\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + 0.5\right) - \left(1 - 7\right)\right)}\right)\]

  9. Applied simplify1.0

    \[\leadsto \color{blue}{\left(\left(\frac{{\left(\left(7 + 0.5\right) + \left(0 - z\right)\right)}^{\left(0.5 + \left(0 - z\right)\right)}}{e^{\left(7 + 0.5\right) + \left(0 - z\right)}} \cdot \sqrt[3]{\left(2 \cdot \pi\right) \cdot \sqrt{2 \cdot \pi}}\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\left(\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) + \frac{-176.6150291621406}{\left(0 - z\right) + 4}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(8 + 0\right) - z}\right) + \left(\left(\frac{-1259.1392167224028}{\left(2 + 0\right) - z} + \frac{771.3234287776531}{3 + \left(0 - z\right)}\right) + \left((z \cdot 0.49644453405676175 + \left((\left(z \cdot z\right) \cdot 0.09941721338104284 + 2.478373473193094)_*\right))_* + \frac{9.984369578019572 \cdot 10^{-06}}{\left(0 - z\right) + 7}\right)\right)\right)}\]

  10. Applied simplify1.0

    \[\leadsto \color{blue}{\left(\left(\frac{{\left(\left(0.5 + 7\right) + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}}{e^{\left(0.5 + 7\right) + \left(-z\right)}} \cdot \sqrt[3]{\left(2 \cdot \pi\right) \cdot \sqrt{2 \cdot \pi}}\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right)} \cdot \left(\left(\left(\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) + \frac{-176.6150291621406}{\left(0 - z\right) + 4}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(8 + 0\right) - z}\right) + \left(\left(\frac{-1259.1392167224028}{\left(2 + 0\right) - z} + \frac{771.3234287776531}{3 + \left(0 - z\right)}\right) + \left((z \cdot 0.49644453405676175 + \left((\left(z \cdot z\right) \cdot 0.09941721338104284 + 2.478373473193094)_*\right))_* + \frac{9.984369578019572 \cdot 10^{-06}}{\left(0 - z\right) + 7}\right)\right)\right)\]

  11. Applied simplify1.0

    \[\leadsto \left(\left(\frac{{\left(\left(0.5 + 7\right) + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)}}{e^{\left(0.5 + 7\right) + \left(-z\right)}} \cdot \sqrt[3]{\left(2 \cdot \pi\right) \cdot \sqrt{2 \cdot \pi}}\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \cdot \color{blue}{\left(\left(\left((z \cdot 0.49644453405676175 + \left((\left(z \cdot z\right) \cdot 0.09941721338104284 + 2.478373473193094)_*\right))_* + \frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(-z\right)}\right) + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{\left(-z\right) + 3}\right)\right) + \left(\left(\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) + \frac{-176.6150291621406}{\left(-z\right) + 4}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{8 - z}\right)\right)}\]

Runtime

Time bar (total: 3.1m)Debug logProfile

herbie shell --seed '#(1071979731 1496239409 439705970 2863295848 982327776 189749553)' +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))))))