Jmat.Real.gamma, branch z less than 0.5

Average Error: 1.8 → 0.4

Time: 1.0min

Precision: binary64

• could not determine a ground truth for program body

  1. z = -5.4740488407523195e+26

\[z \leq 0.5\]

Math

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

↓

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

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-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 (- 0.5 z)) (- 0.5 z))
    (*
     (exp z)
     (/
      (+
       (+ (/ 771.3234287776531 (- 3.0 z)) (/ -176.6150291621406 (- 4.0 z)))
       (+
        (+ (/ 676.5203681218851 (- 1.0 z)) (/ -1259.1392167224028 (- 2.0 z)))
        (+
         (/ 12.507343278686905 (- 5.0 z))
         (+
          (/ -0.13857109526572012 (- 6.0 z))
          (+
           (+
            (/ 9.984369578019572e-06 (- 7.0 z))
            (/ 1.5056327351493116e-07 (- 8.0 z)))
           0.9999999999998099)))))
      (exp (+ 7.0 0.5))))))))

C

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) (7.0 + ((double) (0.5 - z)))), ((double) (0.5 - z)))) * ((double) (((double) exp(z)) * (((double) (((double) ((771.3234287776531 / ((double) (3.0 - z))) + (-176.6150291621406 / ((double) (4.0 - z))))) + ((double) (((double) ((676.5203681218851 / ((double) (1.0 - z))) + (-1259.1392167224028 / ((double) (2.0 - z))))) + ((double) ((12.507343278686905 / ((double) (5.0 - z))) + ((double) ((-0.13857109526572012 / ((double) (6.0 - z))) + ((double) (((double) ((9.984369578019572e-06 / ((double) (7.0 - z))) + (1.5056327351493116e-07 / ((double) (8.0 - z))))) + 0.9999999999998099)))))))))) / ((double) exp(((double) (7.0 + 0.5)))))))))))));
}

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. Simplified 1.3

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

4. Applied sub-neg 1.3

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

5. Applied exp-sum 1.2

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

6. Applied associate-*l* 1.2

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

7. Simplified 0.4

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

8. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020198 
(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))))))