Average Error: 59.9 → 1.2
Time: 1.6m
Precision: 64
Internal precision: 128
\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)\]
⬇
\[{\left(\frac{\sqrt[3]{{\left(\left(0.5 + 7\right) + \left(z - 1\right)\right)}^{\left(z - \left(1 - 0.5\right)\right)} \cdot \sqrt{\pi + \pi}}}{\sqrt[3]{e^{\left(0.5 + 7\right) + \left(z - 1\right)}}}\right)}^3 \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{\left(8 + z\right) - 1} + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right) + \left(\frac{-0.13857109526572012}{\left(z - 1\right) + 6} + \frac{12.507343278686905}{z - \left(1 - 5\right)}\right)\right) + \left(\left(\left(\frac{676.5203681218851}{z - 0} + 0.9999999999998099\right) + \frac{-176.6150291621406}{z - \left(1 - 4\right)}\right) + \left(\frac{-1259.1392167224028}{\left(z - 1\right) + 2} + \frac{771.3234287776531}{\left(z + 3\right) - 1}\right)\right)\right)\]

Error

Bits error versus z

Derivation

  1. Initial program 59.9

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

  2. Applied simplify 1.2

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

  4. Applied add-cube-cbrt 1.8

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

  5. Applied add-cube-cbrt 1.2

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

  6. Applied cube-undiv 1.2

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

Runtime

Time bar (total: 1.6m) Debug log

Please include this information when filing a bug report:

herbie --seed '#(3647421418 3449660431 1258582581 3090253772 1685670982 2677628283)'
(FPCore (z)
  :name "Jmat.Real.gamma, branch z greater than 0.5"
  (* (* (* (sqrt (* PI 2)) (pow (+ (+ (- z 1) 7) 0.5) (+ (- z 1) 0.5))) (exp (- (+ (+ (- z 1) 7) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- z 1) 1))) (/ -1259.1392167224028 (+ (- z 1) 2))) (/ 771.3234287776531 (+ (- z 1) 3))) (/ -176.6150291621406 (+ (- z 1) 4))) (/ 12.507343278686905 (+ (- z 1) 5))) (/ -0.13857109526572012 (+ (- z 1) 6))) (/ 9.984369578019572e-06 (+ (- z 1) 7))) (/ 1.5056327351493116e-07 (+ (- z 1) 8)))))