Average Error: 1.8 → 0.5
Time: 4.1m
Precision: 64
Internal Precision: 384
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\left(\left(\frac{{\left(0.5 - \left(z - 7\right)\right)}^{0.5}}{\sqrt{e^{0.5 - \left(z - 7\right)}}} \cdot \frac{{\left(\left(0.5 - z\right) + 7\right)}^{\left(-z\right)}}{\sqrt{e^{\left(0.5 - z\right) + 7}}}\right) \cdot \left(\sqrt{\pi + \pi} \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right)\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-06}}{\left(7 + 1\right) - \left(1 + z\right)} + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(1 - z\right) - \left(1 - 8\right)}\right) + \left(\left(\left(\frac{-0.13857109526572012}{\left(6 + 1\right) - \left(1 + z\right)} + \frac{12.507343278686905}{\left(5 + 1\right) - \left(1 + z\right)}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - \left(1 - 4\right)}\right) + \left(\left(\frac{771.3234287776531}{\left(1 + 3\right) - \left(1 + z\right)} + 0.9999999999998099\right) + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(2 + 1\right) - \left(1 + z\right)}\right)\right)\right)\right)\]

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. Applied simplify1.4

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

  4. Applied add-sqr-sqrt1.4

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

  5. Applied associate--l+1.4

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

  6. Applied unpow-prod-up1.4

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

  7. Applied times-frac0.5

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

  8. Applied simplify0.5

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

  9. Applied simplify0.5

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

Runtime

Time bar (total: 4.1m)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' +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))))))