Jmat.Real.gamma, branch z less than 0.5

Average Error: 1.8 → 0.4

Time: 1.3min

Precision: binary64

\[z \leq 0.5\]

Math

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

↓

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

FPCore

(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-6 (+ (- (- 1.0 z) 1.0) 7.0)))
    (/ 1.5056327351493116e-7 (+ (- (- 1.0 z) 1.0) 8.0))))))

↓

(FPCore (z)
 :precision binary64
 (/
  (*
   (* (sqrt (* 2.0 PI)) (sqrt (- (+ 0.5 7.0) z)))
   (*
    (+
     0.9999999999998099
     (+
      (+
       (+
        (+ (/ 676.5203681218851 (- 1.0 z)) (/ -1259.1392167224028 (- 2.0 z)))
        (/ 771.3234287776531 (- 3.0 z)))
       (/ -176.6150291621406 (- 4.0 z)))
      (+
       (/ 12.507343278686905 (- 5.0 z))
       (+
        (/ -0.13857109526572012 (- 6.0 z))
        (+
         (/ 9.984369578019572e-6 (- 7.0 z))
         (/ 1.5056327351493116e-7 (- 8.0 z)))))))
    (* PI (exp z))))
  (* (pow (+ 7.0 (- 0.5 z)) z) (* (sin (* PI z)) (exp (+ 0.5 7.0))))))

C

double code(double z) {
	return (((double) M_PI) / sin(((double) M_PI) * z)) * (((sqrt(((double) M_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-6 / (((1.0 - z) - 1.0) + 7.0))) + (1.5056327351493116e-7 / (((1.0 - z) - 1.0) + 8.0))));
}

↓

double code(double z) {
	return ((sqrt(2.0 * ((double) M_PI)) * sqrt((0.5 + 7.0) - z)) * ((0.9999999999998099 + (((((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z))) + (771.3234287776531 / (3.0 - z))) + (-176.6150291621406 / (4.0 - z))) + ((12.507343278686905 / (5.0 - z)) + ((-0.13857109526572012 / (6.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))))) * (((double) M_PI) * exp(z)))) / (pow((7.0 + (0.5 - z)), z) * (sin(((double) M_PI) * z) * exp(0.5 + 7.0)));
}

Error

Bits error versus z

Derivation

1. Initial program 1.8

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

2. Simplified 0.7

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

4. Applied associate-+l+_binary64_9559 0.7

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

5. Simplified 0.7

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

7. Applied exp-diff_binary64_9674 0.6

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

8. Applied frac-times_binary64_9636 0.6

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

9. Applied associate-*r/_binary64_9568 0.6

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

10. Applied pow-sub_binary64_9702 0.6

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

11. Applied frac-times_binary64_9636 0.7

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

12. Applied associate-*r/_binary64_9568 1.0

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

13. Simplified 0.4

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

14. Final simplification 0.4

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

Reproduce

herbie shell --seed 2021204 
(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-6 (+ (- (- 1.0 z) 1.0) 7.0))) (/ 1.5056327351493116e-7 (+ (- (- 1.0 z) 1.0) 8.0))))))