Jmat.Real.gamma, branch z less than 0.5

Average Error: 1.8 → 0.9

Time: 57.1s

Precision: binary64

Cost: 36608

\[z \leq 0.5\]

Math

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

↓

\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \left(\frac{9.984369578019572 \cdot 10^{-06}}{7 - z} + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{771.3234287776531}{3 - z} + \left(\left(47.9507597606835 + z \cdot 361.7355639412844\right) + \left(519.1279660315847 + z \cdot 597.824167076735\right) \cdot \left(z \cdot z\right)\right)\right)\right)\right)\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{\pi} \cdot \sqrt{2}\right)\right)\right)\right)\]

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)))
  (*
   (+
    (/ 1.5056327351493116e-07 (- 8.0 z))
    (+
     (/ 9.984369578019572e-06 (- 7.0 z))
     (+
      (/ -0.13857109526572012 (- 6.0 z))
      (+
       (/ 12.507343278686905 (- 5.0 z))
       (+
        (/ -176.6150291621406 (- 4.0 z))
        (+
         (/ 771.3234287776531 (- 3.0 z))
         (+
          (+ 47.9507597606835 (* z 361.7355639412844))
          (* (+ 519.1279660315847 (* z 597.824167076735)) (* z z)))))))))
   (*
    (exp (+ z -7.5))
    (* (pow (- 7.5 z) (- 0.5 z)) (* (sqrt PI) (sqrt 2.0)))))))

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-06 / (((1.0 - z) - 1.0) + 7.0))) + (1.5056327351493116e-07 / (((1.0 - z) - 1.0) + 8.0))));
}

↓

double code(double z) {
	return (((double) M_PI) / sin(((double) M_PI) * z)) * (((1.5056327351493116e-07 / (8.0 - z)) + ((9.984369578019572e-06 / (7.0 - z)) + ((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) + ((-176.6150291621406 / (4.0 - z)) + ((771.3234287776531 / (3.0 - z)) + ((47.9507597606835 + (z * 361.7355639412844)) + ((519.1279660315847 + (z * 597.824167076735)) * (z * z))))))))) * (exp(z + -7.5) * (pow((7.5 - z), (0.5 - z)) * (sqrt((double) M_PI) * sqrt(2.0)))));
}

Error

Bits error versus z

Alternatives

Alternative 1
Error	1.2
Cost	31744

\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(e^{z + -7.5} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right)\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \left(\frac{9.984369578019572 \cdot 10^{-06}}{7 - z} + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{-1259.1392167224028}{2 - z}}\right)\right)\right)\right)\right)\right)\right)\]

Alternative 2
Error	1.2
Cost	30080

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

Alternative 3
Error	1.7
Cost	29824

\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(e^{z + -7.5} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right)\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \left(\frac{9.984369578019572 \cdot 10^{-06}}{7 - z} + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{771.3234287776531}{3 - z} + \left(47.9507597606835 + z \cdot \left(361.7355639412844 + z \cdot 519.1279660315847\right)\right)\right)\right)\right)\right)\right)\right)\right)\]

Alternative 4
Error	2.3
Cost	29440

\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(e^{z + -7.5} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right)\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \left(\frac{9.984369578019572 \cdot 10^{-06}}{7 - z} + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \left(305.05856935323453 + z \cdot \left(447.4381671388014 + z \cdot 547.6955004307571\right)\right)\right)\right)\right)\right)\right)\right)\]

Alternative 5
Error	2.4
Cost	29312

\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(e^{z + -7.5} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right)\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \left(\frac{9.984369578019572 \cdot 10^{-06}}{7 - z} + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{771.3234287776531}{3 - z} + 47.9507597606835\right)\right)\right)\right)\right)\right)\right)\]

Alternative 6
Error	2.5
Cost	28544

\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(e^{z + -7.5} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right)\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \left(\frac{9.984369578019572 \cdot 10^{-06}}{7 - z} + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + 260.9048120626994\right)\right)\right)\right)\right)\]

Alternative 7
Error	60.6
Cost	385

\[\begin{array}{l} \mathbf{if}\;z \leq 5.523404440305184 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Alternative 8
Error	61.9
Cost	64

\[1\]

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 1.8

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

3. Taylor expanded around 0 1.6

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\left(597.824167076735 \cdot {z}^{3} + \left(519.1279660315847 \cdot {z}^{2} + \left(361.7355639412844 \cdot z + 47.9507597606835\right)\right)\right)} + \frac{771.3234287776531}{3 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \frac{12.507343278686905}{5 - z}\right) + \frac{-0.13857109526572012}{6 - z}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{8 - z}\right)\right)\]

4. Simplified 1.6

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\left(\left(47.9507597606835 + z \cdot 361.7355639412844\right) + \left(z \cdot z\right) \cdot \left(519.1279660315847 + 597.824167076735 \cdot z\right)\right)} + \frac{771.3234287776531}{3 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \frac{12.507343278686905}{5 - z}\right) + \frac{-0.13857109526572012}{6 - z}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{8 - z}\right)\right)\]
5. Using strategy rm

6. Applied sqrt-prod_binary64_1799 0.9

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{2}\right)} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(47.9507597606835 + z \cdot 361.7355639412844\right) + \left(z \cdot z\right) \cdot \left(519.1279660315847 + 597.824167076735 \cdot z\right)\right) + \frac{771.3234287776531}{3 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \frac{12.507343278686905}{5 - z}\right) + \frac{-0.13857109526572012}{6 - z}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{8 - z}\right)\right)\]

7. Simplified 0.9

   \[\leadsto \color{blue}{\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \left(\frac{9.984369578019572 \cdot 10^{-06}}{7 - z} + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{771.3234287776531}{3 - z} + \left(\left(47.9507597606835 + z \cdot 361.7355639412844\right) + \left(519.1279660315847 + z \cdot 597.824167076735\right) \cdot \left(z \cdot z\right)\right)\right)\right)\right)\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{\pi} \cdot \sqrt{2}\right)\right)\right)\right)}\]

8. Final simplification 0.9

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \left(\frac{9.984369578019572 \cdot 10^{-06}}{7 - z} + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{771.3234287776531}{3 - z} + \left(\left(47.9507597606835 + z \cdot 361.7355639412844\right) + \left(519.1279660315847 + z \cdot 597.824167076735\right) \cdot \left(z \cdot z\right)\right)\right)\right)\right)\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{\pi} \cdot \sqrt{2}\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2021044 
(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))))))