Octave 3.8, jcobi/1

Average Error: 16.2 → 5.9

Time: 5.3s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 18187930964.3279305:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(1 \cdot 1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)\right) - \left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\log \left(\sqrt{e^{{\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}}\right) + \left(\log \left(\sqrt{e^{{\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}}\right) - {1}^{3}\right)\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(1 \cdot 1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\alpha + \beta\right)} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

C

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

↓

double code(double alpha, double beta) {
	double VAR;
	if ((alpha <= 18187930964.32793)) {
		VAR = ((((double) (((double) (beta * ((double) (((double) (1.0 * 1.0)) + ((double) ((alpha / ((double) (alpha + ((double) (beta + 2.0))))) * ((double) (1.0 + (alpha / ((double) (alpha + ((double) (beta + 2.0))))))))))))) - ((double) (((double) (alpha + ((double) (beta + 2.0)))) * ((double) (((double) log(((double) sqrt(((double) exp(((double) pow((alpha / ((double) (alpha + ((double) (beta + 2.0))))), 3.0)))))))) + ((double) (((double) log(((double) sqrt(((double) exp(((double) pow((alpha / ((double) (alpha + ((double) (beta + 2.0))))), 3.0)))))))) - ((double) pow(1.0, 3.0)))))))))) / ((double) (((double) (alpha + ((double) (beta + 2.0)))) * ((double) (((double) (1.0 * 1.0)) + ((double) ((alpha / ((double) (alpha + ((double) (beta + 2.0))))) * ((double) (1.0 + (alpha / ((double) (alpha + ((double) (beta + 2.0)))))))))))))) / 2.0);
	} else {
		VAR = (((double) ((beta / ((double) (2.0 + ((double) (alpha + beta))))) - ((double) ((4.0 / ((double) (alpha * alpha))) - ((double) ((2.0 / alpha) + (8.0 / ((double) pow(alpha, 3.0))))))))) / 2.0);
	}
	return VAR;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if alpha < 18187930964.3279305

   1. Initial program 0.1

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
   2. Using strategy rm

   3. Applied div-sub 0.1

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

   4. Applied associate-+l- 0.1

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

   5. Simplified 0.1

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

   7. Applied flip3-- 0.2

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

   8. Applied frac-sub 0.2

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

   9. Simplified 0.2

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

   10. Simplified 0.2

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

   12. Applied add-log-exp 0.2

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

   14. Applied add-sqr-sqrt 0.2

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

   15. Applied log-prod 0.2

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

   16. Applied associate--l+ 0.2

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

   if 18187930964.3279305 < alpha

   1. Initial program 50.4

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
   2. Using strategy rm

   3. Applied div-sub 50.4

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

   4. Applied associate-+l- 48.8

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

   5. Simplified 48.8

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

   6. Taylor expanded around inf 18.1

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

   7. Simplified 18.1

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}}{2}\]
3. Recombined 2 regimes into one program.

4. Final simplification 5.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 18187930964.3279305:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(1 \cdot 1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)\right) - \left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\log \left(\sqrt{e^{{\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}}\right) + \left(\log \left(\sqrt{e^{{\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}}}\right) - {1}^{3}\right)\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(1 \cdot 1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\alpha + \beta\right)} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))