Average Error: 1.8 → 0.5
Time: 4.0m
Precision: 64
Internal Precision: 576
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\left(\left(\frac{{\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{e^{7.5 - z}} \cdot \sqrt{2 \cdot \pi}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(\left(\left(\frac{-176.6150291621406}{\left(1 - z\right) - \left(1 - 4\right)} + \frac{771.3234287776531}{\left(1 - z\right) - \left(1 - 3\right)}\right) + \left(\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) + \frac{-1259.1392167224028}{\left(2 + 1\right) - \left(1 + z\right)}\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) - \left(1 - 5\right)} + \frac{-0.13857109526572012}{\left(1 + 6\right) - \left(1 + z\right)}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{\left(1 + 8\right) - \left(z + 1\right)} + \frac{9.984369578019572 \cdot 10^{-06}}{\left(7 + 1\right) - \left(z + 1\right)}\right)\right)\right)\]

Error

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  3. Taylor expanded around inf 1.4

    \[\leadsto \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \color{blue}{\frac{e^{\log \left(7.5 - z\right) \cdot \left(0.5 - z\right)}}{e^{7.5 - z}}}\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(2 + 1\right) - \left(z + 1\right)}\right)\right) + \left(\left(\frac{771.3234287776531}{\left(1 - z\right) - \left(1 - 3\right)} + \frac{-176.6150291621406}{\left(1 - z\right) - \left(1 - 4\right)}\right) + \left(\frac{-0.13857109526572012}{\left(6 + 1\right) - \left(z + 1\right)} + \frac{12.507343278686905}{\left(1 - z\right) - \left(1 - 5\right)}\right)\right)\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{\left(8 + 1\right) - \left(z + 1\right)} + \frac{9.984369578019572 \cdot 10^{-06}}{\left(7 + 1\right) - \left(z + 1\right)}\right)\right)\]

  4. Applied simplify0.5

    \[\leadsto \color{blue}{\left(\left(\frac{{\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{e^{7.5 - z}} \cdot \sqrt{2 \cdot \pi}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(\left(\left(\frac{-176.6150291621406}{\left(1 - z\right) - \left(1 - 4\right)} + \frac{771.3234287776531}{\left(1 - z\right) - \left(1 - 3\right)}\right) + \left(\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) + \frac{-1259.1392167224028}{\left(2 + 1\right) - \left(1 + z\right)}\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) - \left(1 - 5\right)} + \frac{-0.13857109526572012}{\left(1 + 6\right) - \left(1 + z\right)}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{\left(1 + 8\right) - \left(z + 1\right)} + \frac{9.984369578019572 \cdot 10^{-06}}{\left(7 + 1\right) - \left(z + 1\right)}\right)\right)\right)}\]

Runtime

Time bar (total: 4.0m)Debug logProfile

herbie shell --seed 2019053 +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))))))