Average Error: 59.9 → 0.9
Time: 9.4m
Precision: 64
Internal Precision: 1344
\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)\]
↓
\[\frac{\left(\sqrt{2 \cdot \pi} \cdot {\left(\left(z - 1\right) + \left(7 + 0.5\right)\right)}^{\left(\left(0.5 + z\right) - 1\right)}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-07}}{\left(8 + z\right) - 1} + \left(\left(\left(\left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z + 3\right) - 1}\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(z - 1\right)} + \frac{-0.13857109526572012}{6 + \left(z - 1\right)}\right) + \left(\frac{12.507343278686905}{z - \left(1 - 5\right)} + \frac{-176.6150291621406}{4 + \left(z - 1\right)}\right)\right)\right)\right)}{e^{\left(z - 1\right) + \left(7 + 0.5\right)}}\]

Error

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Initial simplification1.1

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

  4. Applied associate-*l/1.1

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

  5. Applied associate-*l/0.9

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

  6. Simplified0.9

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

  7. Final simplification0.9

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

Runtime

Time bar (total: 9.4m)Debug logProfile

herbie shell --seed 2018214 
(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)))))