Octave 3.8, jcobi/2

Average Error: 23.7 → 12.5

Time: 7.1s

Precision: binary64

\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]

Math

\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 2463897326213938200:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\frac{\frac{2 + 2 \cdot i}{\alpha + \beta} + 1}{\sqrt[3]{\beta - \alpha}} \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} + 1}{2}\\ \mathbf{elif}\;\alpha \le 1.4898782167186468 \cdot 10^{48}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\beta} \cdot \left(\frac{\sqrt[3]{\beta}}{\frac{2 + 2 \cdot i}{\alpha + \beta} + 1} \cdot \frac{\sqrt[3]{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}}{\sqrt{\alpha + \left(\beta + 2 \cdot i\right)}}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{\beta}}}{\sqrt{\alpha + \left(\beta + 2 \cdot i\right)}} - \left(\frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} - 1\right)}{2}\\ \end{array}\]

C

double code(double alpha, double beta, double i) {
	return (((double) (((((double) (((double) (alpha + beta)) * ((double) (beta - alpha)))) / ((double) (((double) (alpha + beta)) + ((double) (2.0 * i))))) / ((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))) + 2.0))) + 1.0)) / 2.0);
}

↓

double code(double alpha, double beta, double i) {
	double VAR;
	if ((alpha <= 2.463897326213938e+18)) {
		VAR = (((double) ((((double) (((double) cbrt(((double) (beta - alpha)))) * ((double) cbrt(((double) (beta - alpha)))))) / ((double) ((((double) ((((double) (2.0 + ((double) (2.0 * i)))) / ((double) (alpha + beta))) + 1.0)) / ((double) cbrt(((double) (beta - alpha))))) * ((double) (beta + ((double) (alpha + ((double) (2.0 * i))))))))) + 1.0)) / 2.0);
	} else {
		double VAR_1;
		if ((alpha <= 1.4898782167186468e+48)) {
			VAR_1 = (((double) ((beta / ((double) (((double) ((((double) (2.0 + ((double) (2.0 * i)))) / ((double) (alpha + beta))) + 1.0)) * ((double) (alpha + ((double) (beta + ((double) (2.0 * i))))))))) - ((double) ((4.0 / ((double) (alpha * alpha))) - ((double) ((2.0 / alpha) + (8.0 / ((double) pow(alpha, 3.0))))))))) / 2.0);
		} else {
			VAR_1 = (((double) (((double) (((double) (((double) cbrt(beta)) * ((double) ((((double) cbrt(beta)) / ((double) ((((double) (2.0 + ((double) (2.0 * i)))) / ((double) (alpha + beta))) + 1.0))) * (((double) cbrt(((double) (((double) cbrt(beta)) * ((double) cbrt(beta)))))) / ((double) sqrt(((double) (alpha + ((double) (beta + ((double) (2.0 * i))))))))))))) * (((double) cbrt(((double) cbrt(beta)))) / ((double) sqrt(((double) (alpha + ((double) (beta + ((double) (2.0 * i))))))))))) - ((double) ((alpha / ((double) (((double) ((((double) (2.0 + ((double) (2.0 * i)))) / ((double) (alpha + beta))) + 1.0)) * ((double) (beta + ((double) (alpha + ((double) (2.0 * i))))))))) - 1.0)))) / 2.0);
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if alpha < 2463897326213938200

   1. Initial program 11.0

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]

   2. Simplified 0.3

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} + 1}{2}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 0.5

      \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}\right) \cdot \sqrt[3]{\beta - \alpha}}}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} + 1}{2}\]

   5. Applied associate-/l* 0.5

      \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\frac{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)}{\sqrt[3]{\beta - \alpha}}}} + 1}{2}\]

   6. Simplified 0.5

      \[\leadsto \frac{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\color{blue}{\frac{\frac{2 + 2 \cdot i}{\beta + \alpha} + 1}{\sqrt[3]{\beta - \alpha}} \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)}} + 1}{2}\]

   if 2463897326213938200 < alpha < 1.4898782167186468e48

   1. Initial program 32.3

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]

   2. Simplified 22.8

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} + 1}{2}}\]
   3. Using strategy rm

   4. Applied div-sub 22.7

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - \frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)}\right)} + 1}{2}\]

   5. Applied associate-+l- 22.5

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - \left(\frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - 1\right)}}{2}\]

   6. Simplified 22.5

      \[\leadsto \frac{\frac{\beta}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - \color{blue}{\left(\frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\beta + \alpha} + 1\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} - 1\right)}}{2}\]

   7. Taylor expanded around inf 38.3

      \[\leadsto \frac{\frac{\beta}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - \color{blue}{\left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2}\]

   8. Simplified 38.3

      \[\leadsto \frac{\frac{\beta}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - \color{blue}{\left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}}{2}\]

   if 1.4898782167186468e48 < alpha

   1. Initial program 54.5

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]

   2. Simplified 40.7

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} + 1}{2}}\]
   3. Using strategy rm

   4. Applied div-sub 40.7

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - \frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)}\right)} + 1}{2}\]

   5. Applied associate-+l- 39.4

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - \left(\frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - 1\right)}}{2}\]

   6. Simplified 39.4

      \[\leadsto \frac{\frac{\beta}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - \color{blue}{\left(\frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\beta + \alpha} + 1\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} - 1\right)}}{2}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 39.5

      \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}\right) \cdot \sqrt[3]{\beta}}}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - \left(\frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\beta + \alpha} + 1\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} - 1\right)}{2}\]

   9. Applied times-frac 39.5

      \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\frac{2 + 2 \cdot i}{\alpha + \beta} + 1} \cdot \frac{\sqrt[3]{\beta}}{\alpha + \left(\beta + 2 \cdot i\right)}} - \left(\frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\beta + \alpha} + 1\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} - 1\right)}{2}\]

   10. Simplified 39.5

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\beta} \cdot \frac{\sqrt[3]{\beta}}{\frac{2 + 2 \cdot i}{\alpha + \beta} + 1}\right)} \cdot \frac{\sqrt[3]{\beta}}{\alpha + \left(\beta + 2 \cdot i\right)} - \left(\frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\beta + \alpha} + 1\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} - 1\right)}{2}\]

   11. Simplified 39.5

       \[\leadsto \frac{\left(\sqrt[3]{\beta} \cdot \frac{\sqrt[3]{\beta}}{\frac{2 + 2 \cdot i}{\alpha + \beta} + 1}\right) \cdot \color{blue}{\frac{\sqrt[3]{\beta}}{\alpha + \left(2 \cdot i + \beta\right)}} - \left(\frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\beta + \alpha} + 1\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} - 1\right)}{2}\]
   12. Using strategy rm

   13. Applied add-sqr-sqrt 39.5

       \[\leadsto \frac{\left(\sqrt[3]{\beta} \cdot \frac{\sqrt[3]{\beta}}{\frac{2 + 2 \cdot i}{\alpha + \beta} + 1}\right) \cdot \frac{\sqrt[3]{\beta}}{\color{blue}{\sqrt{\alpha + \left(2 \cdot i + \beta\right)} \cdot \sqrt{\alpha + \left(2 \cdot i + \beta\right)}}} - \left(\frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\beta + \alpha} + 1\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} - 1\right)}{2}\]

   14. Applied add-cube-cbrt 39.5

       \[\leadsto \frac{\left(\sqrt[3]{\beta} \cdot \frac{\sqrt[3]{\beta}}{\frac{2 + 2 \cdot i}{\alpha + \beta} + 1}\right) \cdot \frac{\sqrt[3]{\color{blue}{\left(\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}\right) \cdot \sqrt[3]{\beta}}}}{\sqrt{\alpha + \left(2 \cdot i + \beta\right)} \cdot \sqrt{\alpha + \left(2 \cdot i + \beta\right)}} - \left(\frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\beta + \alpha} + 1\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} - 1\right)}{2}\]

   15. Applied cbrt-prod 39.5

       \[\leadsto \frac{\left(\sqrt[3]{\beta} \cdot \frac{\sqrt[3]{\beta}}{\frac{2 + 2 \cdot i}{\alpha + \beta} + 1}\right) \cdot \frac{\color{blue}{\sqrt[3]{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}} \cdot \sqrt[3]{\sqrt[3]{\beta}}}}{\sqrt{\alpha + \left(2 \cdot i + \beta\right)} \cdot \sqrt{\alpha + \left(2 \cdot i + \beta\right)}} - \left(\frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\beta + \alpha} + 1\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} - 1\right)}{2}\]

   16. Applied times-frac 39.5

       \[\leadsto \frac{\left(\sqrt[3]{\beta} \cdot \frac{\sqrt[3]{\beta}}{\frac{2 + 2 \cdot i}{\alpha + \beta} + 1}\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}}{\sqrt{\alpha + \left(2 \cdot i + \beta\right)}} \cdot \frac{\sqrt[3]{\sqrt[3]{\beta}}}{\sqrt{\alpha + \left(2 \cdot i + \beta\right)}}\right)} - \left(\frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\beta + \alpha} + 1\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} - 1\right)}{2}\]

   17. Applied associate-*r* 39.5

       \[\leadsto \frac{\color{blue}{\left(\left(\sqrt[3]{\beta} \cdot \frac{\sqrt[3]{\beta}}{\frac{2 + 2 \cdot i}{\alpha + \beta} + 1}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}}{\sqrt{\alpha + \left(2 \cdot i + \beta\right)}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{\beta}}}{\sqrt{\alpha + \left(2 \cdot i + \beta\right)}}} - \left(\frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\beta + \alpha} + 1\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} - 1\right)}{2}\]

   18. Simplified 39.5

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\beta} \cdot \left(\frac{\sqrt[3]{\beta}}{\frac{2 + 2 \cdot i}{\alpha + \beta} + 1} \cdot \frac{\sqrt[3]{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}}{\sqrt{\alpha + \left(2 \cdot i + \beta\right)}}\right)\right)} \cdot \frac{\sqrt[3]{\sqrt[3]{\beta}}}{\sqrt{\alpha + \left(2 \cdot i + \beta\right)}} - \left(\frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\beta + \alpha} + 1\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} - 1\right)}{2}\]
3. Recombined 3 regimes into one program.

4. Final simplification 12.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2463897326213938200:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\frac{\frac{2 + 2 \cdot i}{\alpha + \beta} + 1}{\sqrt[3]{\beta - \alpha}} \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} + 1}{2}\\ \mathbf{elif}\;\alpha \le 1.4898782167186468 \cdot 10^{48}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\beta} \cdot \left(\frac{\sqrt[3]{\beta}}{\frac{2 + 2 \cdot i}{\alpha + \beta} + 1} \cdot \frac{\sqrt[3]{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}}{\sqrt{\alpha + \left(\beta + 2 \cdot i\right)}}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{\beta}}}{\sqrt{\alpha + \left(\beta + 2 \cdot i\right)}} - \left(\frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} - 1\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))