Average Error: 1.8 → 0.5
Time: 1.1min
Precision: binary64
\[z \le 0.5\]
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot \frac{\frac{0.99999999999980993 + \left(\frac{676.520368121885099}{1 - z} + \left(\left(\frac{771.32342877765313}{3 - z} + \frac{-176.615029162140587}{4 - z}\right) + \left(\frac{-1259.13921672240281}{2 - z} + \left(\frac{12.5073432786869052}{5 - z} + \left(\frac{-0.138571095265720118}{6 - z} + \left(\frac{9.98436957801957158 \cdot 10^{-6}}{7 - z} + \frac{1.50563273514931162 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)\right)}{e^{7 + \left(0.5 - z\right)}} \cdot {\left(7 + \left(0.5 - z\right)\right)}^{\left(0.5 - z\right)}}{\sin \left(\pi \cdot z\right)}\right)\]

Error

Bits error versus z

Derivation

  1. Initial program 1.8

    \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

  2. Simplified1.3

    \[\leadsto \color{blue}{\pi \cdot \frac{\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + \left(7 + 0.5\right)\right)}^{\left(\left(-z\right) + 0.5\right)} \cdot \frac{0.99999999999980993 + \left(\left(\frac{676.520368121885099}{1 - z} + \left(\frac{-1259.13921672240281}{2 + \left(-z\right)} + \left(\frac{771.32342877765313}{\left(-z\right) + 3} + \frac{-176.615029162140587}{\left(-z\right) + 4}\right)\right)\right) + \left(\frac{12.5073432786869052}{\left(-z\right) + 5} + \left(\frac{-0.138571095265720118}{\left(-z\right) + 6} + \left(\frac{9.98436957801957158 \cdot 10^{-6}}{\left(-z\right) + 7} + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(-z\right) + 8}\right)\right)\right)\right)}{e^{\left(-z\right) + \left(7 + 0.5\right)}}\right)}{\sin \left(\pi \cdot z\right)}}\]

  3. Simplified0.5

    \[\leadsto \color{blue}{\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot \frac{\frac{0.99999999999980993 + \left(\frac{676.520368121885099}{1 - z} + \left(\left(\frac{771.32342877765313}{3 - z} + \frac{-176.615029162140587}{4 - z}\right) + \left(\frac{-1259.13921672240281}{2 - z} + \left(\frac{12.5073432786869052}{5 - z} + \left(\frac{-0.138571095265720118}{6 - z} + \left(\frac{9.98436957801957158 \cdot 10^{-6}}{7 - z} + \frac{1.50563273514931162 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)\right)}{e^{7 + \left(0.5 - z\right)}} \cdot {\left(7 + \left(0.5 - z\right)\right)}^{\left(0.5 - z\right)}}{\sin \left(\pi \cdot z\right)}\right)}\]

  4. Final simplification0.5

    \[\leadsto \pi \cdot \left(\sqrt{\pi \cdot 2} \cdot \frac{\frac{0.99999999999980993 + \left(\frac{676.520368121885099}{1 - z} + \left(\left(\frac{771.32342877765313}{3 - z} + \frac{-176.615029162140587}{4 - z}\right) + \left(\frac{-1259.13921672240281}{2 - z} + \left(\frac{12.5073432786869052}{5 - z} + \left(\frac{-0.138571095265720118}{6 - z} + \left(\frac{9.98436957801957158 \cdot 10^{-6}}{7 - z} + \frac{1.50563273514931162 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)\right)}{e^{7 + \left(0.5 - z\right)}} \cdot {\left(7 + \left(0.5 - z\right)\right)}^{\left(0.5 - z\right)}}{\sin \left(\pi \cdot z\right)}\right)\]

Reproduce

herbie shell --seed 2020184 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z less than 0.5"
  :precision binary64
  :pre (<= z 0.5)
  (* (/ PI (sin (* PI z))) (* (* (* (sqrt (* PI 2.0)) (pow (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5) (+ (- (- 1.0 z) 1.0) 0.5))) (exp (neg (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- (- 1.0 z) 1.0) 1.0))) (/ -1259.1392167224028 (+ (- (- 1.0 z) 1.0) 2.0))) (/ 771.3234287776531 (+ (- (- 1.0 z) 1.0) 3.0))) (/ -176.6150291621406 (+ (- (- 1.0 z) 1.0) 4.0))) (/ 12.507343278686905 (+ (- (- 1.0 z) 1.0) 5.0))) (/ -0.13857109526572012 (+ (- (- 1.0 z) 1.0) 6.0))) (/ 9.984369578019572e-06 (+ (- (- 1.0 z) 1.0) 7.0))) (/ 1.5056327351493116e-07 (+ (- (- 1.0 z) 1.0) 8.0))))))