Octave 3.8, jcobi/1

Average Error: 16.8 → 3.3

Time: 4.6s

Precision: 64

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

Math

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

↓

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

C

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

↓

double code(double alpha, double beta) {
	double VAR;
	if ((((double) (((double) (beta - alpha)) / ((double) (((double) (alpha + beta)) + 2.0)))) <= -1.0)) {
		VAR = ((double) (((double) (((double) (((double) (((double) (((double) cbrt(beta)) * ((double) cbrt(beta)))) / ((double) (((double) cbrt(((double) (((double) (alpha + beta)) + 2.0)))) * ((double) cbrt(((double) (((double) (alpha + beta)) + 2.0)))))))) * ((double) (((double) cbrt(beta)) / ((double) cbrt(((double) (((double) (alpha + beta)) + 2.0)))))))) - ((double) fma(4.0, ((double) (1.0 / ((double) pow(alpha, 2.0)))), ((double) -(((double) fma(2.0, ((double) (1.0 / alpha)), ((double) (8.0 * ((double) (1.0 / ((double) pow(alpha, 3.0)))))))))))))) / 2.0));
	} else {
		VAR = ((double) (((double) (((double) (((double) pow(((double) (((double) (beta - alpha)) / ((double) (((double) (alpha + beta)) + 2.0)))), 3.0)) + ((double) pow(1.0, 3.0)))) / ((double) fma(1.0, ((double) (1.0 - ((double) (((double) (beta - alpha)) / ((double) (((double) (alpha + beta)) + 2.0)))))), ((double) (((double) (((double) (beta - alpha)) / ((double) (((double) (alpha + beta)) + 2.0)))) * ((double) (((double) (beta - alpha)) / ((double) (((double) (alpha + beta)) + 2.0)))))))))) / 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)) < -1.0

   1. Initial program 60.6

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

   3. Applied div-sub 60.6

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

   4. Applied associate-+l- 58.8

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

   6. Applied add-log-exp 58.8

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

   7. Applied add-log-exp 58.8

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

   8. Applied diff-log 58.8

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

   9. Simplified 58.8

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

   11. Applied add-cube-cbrt 58.8

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

   12. Applied add-cube-cbrt 58.8

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

   13. Applied times-frac 58.8

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

   14. Taylor expanded around inf 10.6

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

   15. Simplified 10.6

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

   if -1.0 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

   1. Initial program 0.5

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

   3. Applied flip3-+ 0.5

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

   4. Simplified 0.5

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

4. Final simplification 3.3

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

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))