Average Error: 1.8 → 1.0
Time: 1.1min
Precision: binary64
\[z \leq 0.5\]
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(7 - z\right) + 0.5\right)}^{\left(0.5 - z\right)} \cdot \left(700.2793595373912 \cdot \frac{z}{e^{7.5}} + \left(\frac{263.38318698105144}{e^{7.5}} + 1113.6230738897482 \cdot \frac{z \cdot z}{e^{7.5}}\right)\right)\right)\right)\]
\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{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(7 - z\right) + 0.5\right)}^{\left(0.5 - z\right)} \cdot \left(700.2793595373912 \cdot \frac{z}{e^{7.5}} + \left(\frac{263.38318698105144}{e^{7.5}} + 1113.6230738897482 \cdot \frac{z \cdot z}{e^{7.5}}\right)\right)\right)\right)
(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* PI z)))
  (*
   (*
    (*
     (sqrt (* PI 2.0))
     (pow (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5) (+ (- (- 1.0 z) 1.0) 0.5)))
    (exp (- (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5))))
   (+
    (+
     (+
      (+
       (+
        (+
         (+
          (+
           0.9999999999998099
           (/ 676.5203681218851 (+ (- (- 1.0 z) 1.0) 1.0)))
          (/ -1259.1392167224028 (+ (- (- 1.0 z) 1.0) 2.0)))
         (/ 771.3234287776531 (+ (- (- 1.0 z) 1.0) 3.0)))
        (/ -176.6150291621406 (+ (- (- 1.0 z) 1.0) 4.0)))
       (/ 12.507343278686905 (+ (- (- 1.0 z) 1.0) 5.0)))
      (/ -0.13857109526572012 (+ (- (- 1.0 z) 1.0) 6.0)))
     (/ 9.984369578019572e-06 (+ (- (- 1.0 z) 1.0) 7.0)))
    (/ 1.5056327351493116e-07 (+ (- (- 1.0 z) 1.0) 8.0))))))
↓
(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* PI z)))
  (*
   (sqrt (* PI 2.0))
   (*
    (pow (+ (- 7.0 z) 0.5) (- 0.5 z))
    (+
     (* 700.2793595373912 (/ z (exp 7.5)))
     (+
      (/ 263.38318698105144 (exp 7.5))
      (* 1113.6230738897482 (/ (* z z) (exp 7.5)))))))))
double code(double z) {
	return ((double) ((((double) M_PI) / ((double) sin(((double) (((double) M_PI) * z))))) * ((double) (((double) (((double) (((double) sqrt(((double) (((double) M_PI) * 2.0)))) * ((double) pow(((double) (((double) (((double) (((double) (1.0 - z)) - 1.0)) + 7.0)) + 0.5)), ((double) (((double) (((double) (1.0 - z)) - 1.0)) + 0.5)))))) * ((double) exp(((double) -(((double) (((double) (((double) (((double) (1.0 - z)) - 1.0)) + 7.0)) + 0.5)))))))) * ((double) (((double) (((double) (((double) (((double) (((double) (((double) (((double) (0.9999999999998099 + (676.5203681218851 / ((double) (((double) (((double) (1.0 - z)) - 1.0)) + 1.0))))) + (-1259.1392167224028 / ((double) (((double) (((double) (1.0 - z)) - 1.0)) + 2.0))))) + (771.3234287776531 / ((double) (((double) (((double) (1.0 - z)) - 1.0)) + 3.0))))) + (-176.6150291621406 / ((double) (((double) (((double) (1.0 - z)) - 1.0)) + 4.0))))) + (12.507343278686905 / ((double) (((double) (((double) (1.0 - z)) - 1.0)) + 5.0))))) + (-0.13857109526572012 / ((double) (((double) (((double) (1.0 - z)) - 1.0)) + 6.0))))) + (9.984369578019572e-06 / ((double) (((double) (((double) (1.0 - z)) - 1.0)) + 7.0))))) + (1.5056327351493116e-07 / ((double) (((double) (((double) (1.0 - z)) - 1.0)) + 8.0)))))))));
}
↓
double code(double z) {
	return ((double) ((((double) M_PI) / ((double) sin(((double) (((double) M_PI) * z))))) * ((double) (((double) sqrt(((double) (((double) M_PI) * 2.0)))) * ((double) (((double) pow(((double) (((double) (7.0 - z)) + 0.5)), ((double) (0.5 - z)))) * ((double) (((double) (700.2793595373912 * (z / ((double) exp(7.5))))) + ((double) ((263.38318698105144 / ((double) exp(7.5))) + ((double) (1113.6230738897482 * (((double) (z * z)) / ((double) exp(7.5)))))))))))))));
}

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. Simplified1.8

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

  4. Applied flip3-+_binary641.8

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

  5. Applied frac-add_binary641.2

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

  6. Applied frac-add_binary641.2

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

  7. Simplified1.2

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

  8. Simplified1.2

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

  9. Taylor expanded around 0 1.0

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(7 - z\right) + 0.5\right)}^{\left(0.5 - z\right)} \cdot \color{blue}{\left(700.2793595373912 \cdot \frac{z}{e^{7.5}} + \left(263.38318698105144 \cdot \frac{1}{e^{7.5}} + 1113.6230738897482 \cdot \frac{{z}^{2}}{e^{7.5}}\right)\right)}\right)\right)\]

  10. Simplified1.0

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(7 - z\right) + 0.5\right)}^{\left(0.5 - z\right)} \cdot \color{blue}{\left(700.2793595373912 \cdot \frac{z}{e^{7.5}} + \left(\frac{263.38318698105144}{e^{7.5}} + 1113.6230738897482 \cdot \frac{z \cdot z}{e^{7.5}}\right)\right)}\right)\right)\]

  11. Final simplification1.0

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(7 - z\right) + 0.5\right)}^{\left(0.5 - z\right)} \cdot \left(700.2793595373912 \cdot \frac{z}{e^{7.5}} + \left(\frac{263.38318698105144}{e^{7.5}} + 1113.6230738897482 \cdot \frac{z \cdot z}{e^{7.5}}\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2020205 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z less than 0.5"
  :precision binary64
  :pre (<= z 0.5)
  (* (/ PI (sin (* PI z))) (* (* (* (sqrt (* PI 2.0)) (pow (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5) (+ (- (- 1.0 z) 1.0) 0.5))) (exp (- (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- (- 1.0 z) 1.0) 1.0))) (/ -1259.1392167224028 (+ (- (- 1.0 z) 1.0) 2.0))) (/ 771.3234287776531 (+ (- (- 1.0 z) 1.0) 3.0))) (/ -176.6150291621406 (+ (- (- 1.0 z) 1.0) 4.0))) (/ 12.507343278686905 (+ (- (- 1.0 z) 1.0) 5.0))) (/ -0.13857109526572012 (+ (- (- 1.0 z) 1.0) 6.0))) (/ 9.984369578019572e-06 (+ (- (- 1.0 z) 1.0) 7.0))) (/ 1.5056327351493116e-07 (+ (- (- 1.0 z) 1.0) 8.0))))))