Average Error: 1.8 → 0.4
Time: 3.8m
Precision: 64
Internal Precision: 576
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\frac{\left(\sqrt{\left(\frac{\pi}{e} \cdot e^{-6}\right) \cdot e^{z - 0.5}} \cdot \sqrt{\left(\frac{\pi}{e} \cdot e^{-6}\right) \cdot e^{z - 0.5}}\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot {\left(\left(1 - z\right) - \left(-6 - 0.5\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right)}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(\left(\left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-07}}{1 - \left(z - 7\right)}\right)\right)\right)\]

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 simplify0.6

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

  3. Taylor expanded around 0 0.6

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

  4. Applied simplify0.8

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

  6. Applied add-sqr-sqrt0.4

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

  7. Applied simplify0.4

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

  8. Applied simplify0.4

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

Runtime

Time bar (total: 3.8m)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' 
(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))))))