Octave 3.8, jcobi/1

Average Error: 16.0 → 3.2

Time: 3.9min

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \le -0.999999999987615351:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\frac{4 - \frac{8}{\alpha}}{\alpha}}{\alpha} - \frac{2}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} \cdot \log \left(e^{\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}}\right)\right) \cdot \log \left(e^{\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}}\right) - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \end{array}\]

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 (((((double) (beta - alpha)) / ((double) (((double) (alpha + beta)) + 2.0))) <= -0.9999999999876154)) {
		VAR = (((double) ((beta / ((double) (((double) (alpha + beta)) + 2.0))) - ((double) (((((double) (4.0 - (8.0 / alpha))) / alpha) / alpha) - (2.0 / alpha))))) / 2.0);
	} else {
		VAR = (((double) (((double) (((double) (((double) cbrt((beta / ((double) (((double) (alpha + beta)) + 2.0))))) * ((double) log(((double) exp(((double) cbrt((beta / ((double) (((double) (alpha + beta)) + 2.0))))))))))) * ((double) log(((double) exp(((double) cbrt((beta / ((double) (((double) (alpha + beta)) + 2.0))))))))))) - ((double) ((alpha / ((double) (((double) (alpha + beta)) + 2.0))) - 1.0)))) / 2.0);
	}
	return VAR;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if (/ (- beta alpha) (+ (+ alpha beta) 2.0)) < -0.999999999987615351

   1. Initial program 60.2

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

   3. Applied div-sub 60.2

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

   4. Applied associate-+l- 58.3

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

   5. Taylor expanded around inf 11.4

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

   6. Simplified 11.4

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

   if -0.999999999987615351 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

   1. Initial program 0.3

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

   3. Applied div-sub 0.3

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

   4. Applied associate-+l- 0.3

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

   6. Applied add-cube-cbrt 0.3

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

   8. Applied add-log-exp 0.3

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

   10. Applied add-log-exp 0.3

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

4. Final simplification 3.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \le -0.999999999987615351:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\frac{4 - \frac{8}{\alpha}}{\alpha}}{\alpha} - \frac{2}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} \cdot \log \left(e^{\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}}\right)\right) \cdot \log \left(e^{\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}}\right) - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(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))