Average Error: 1.8 → 0.4
Time: 1.3min
Precision: binary64
\[z \le 0.5\]
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + \left(7 + 0.5\right)\right)}^{\left(\left(-z\right) + 0.5\right)} \cdot \left(e^{z} \cdot \frac{\frac{771.32342877765313}{\left(-z\right) + 3} + \left(\frac{-176.615029162140587}{\left(-z\right) + 4} + \left(\frac{676.520368121885099}{1 - z} + \left(\frac{-1259.13921672240281}{2 + \left(-z\right)} + \left(\frac{12.5073432786869052}{\left(-z\right) + 5} + \left(\frac{-0.138571095265720118}{\left(-z\right) + 6} + \left(\frac{9.98436957801957158 \cdot 10^{-6}}{\left(-z\right) + 7} + \left(\frac{1.50563273514931162 \cdot 10^{-7}}{\left(-z\right) + 8} + 0.99999999999980993\right)\right)\right)\right)\right)\right)\right)}{e^{7 + 0.5}}\right)\right)\right)\]

Error

Bits error versus z

Derivation

  1. Initial program 1.8

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

  2. Simplified1.3

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

  4. Applied exp-sum1.3

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

  5. Applied *-un-lft-identity1.3

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

  6. Applied times-frac1.3

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

  7. Simplified1.3

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

  8. Simplified0.4

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

  10. Applied *-un-lft-identity0.4

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

  11. Applied associate-/r*0.4

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

  12. Simplified0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2020179 
(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))))))