Average Error: 1.8 → 0.7
Time: 3.6m
Precision: 64
Internal Precision: 384
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\frac{\left(\sqrt{\pi + \pi} \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \cdot \left({\left(\sqrt[3]{0.5 - \left(z - 7\right)} \cdot \sqrt[3]{0.5 - \left(z - 7\right)}\right)}^{\left(0.5 - z\right)} \cdot {\left(\sqrt[3]{\left(7 - z\right) + 0.5}\right)}^{\left(0.5 - z\right)}\right)}{e^{0.5 + \left(\left(7 + 1\right) - \left(1 + z\right)\right)}} \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-06}}{\left(7 + 1\right) - \left(1 + z\right)} + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(1 - z\right) - \left(1 - 8\right)}\right) + \left(\left(\left(\frac{-0.13857109526572012}{\left(6 + 1\right) - \left(1 + z\right)} + \frac{12.507343278686905}{\left(1 - z\right) - \left(1 - 5\right)}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - \left(1 - 3\right)} + \frac{-176.6150291621406}{\left(4 + 1\right) - \left(1 + z\right)}\right)\right) + \left(\left(0.9999999999998099 + \frac{-1259.1392167224028}{\left(1 - z\right) - \left(1 - 2\right)}\right) + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right)\right)\right)\]

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 simplify0.6

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

  4. Applied add-cube-cbrt0.7

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

  5. Applied unpow-prod-down0.7

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

  6. Applied simplify0.7

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

  7. Applied simplify0.7

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

Runtime

Time bar (total: 3.6m)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit
(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))))))