Octave 3.8, jcobi/1

Average Error: 16.3 → 5.9

Time: 21.3s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 307451316.10967922210693359375:\\ \;\;\;\;\frac{\frac{\beta}{\left(2 + \alpha\right) + \beta} - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\beta}}{{\left(\left(\alpha + \beta\right) + 2\right)}^{\frac{1}{3}}}\right)}^{3} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]

C

double f(double alpha, double beta) {
        double r93315 = beta;
        double r93316 = alpha;
        double r93317 = r93315 - r93316;
        double r93318 = r93316 + r93315;
        double r93319 = 2.0;
        double r93320 = r93318 + r93319;
        double r93321 = r93317 / r93320;
        double r93322 = 1.0;
        double r93323 = r93321 + r93322;
        double r93324 = r93323 / r93319;
        return r93324;
}

↓

double f(double alpha, double beta) {
        double r93325 = alpha;
        double r93326 = 307451316.1096792;
        bool r93327 = r93325 <= r93326;
        double r93328 = beta;
        double r93329 = 2.0;
        double r93330 = r93329 + r93325;
        double r93331 = r93330 + r93328;
        double r93332 = r93328 / r93331;
        double r93333 = r93325 + r93328;
        double r93334 = r93329 + r93333;
        double r93335 = r93325 / r93334;
        double r93336 = 1.0;
        double r93337 = r93335 - r93336;
        double r93338 = r93332 - r93337;
        double r93339 = r93338 / r93329;
        double r93340 = cbrt(r93328);
        double r93341 = r93333 + r93329;
        double r93342 = 0.3333333333333333;
        double r93343 = pow(r93341, r93342);
        double r93344 = r93340 / r93343;
        double r93345 = 3.0;
        double r93346 = pow(r93344, r93345);
        double r93347 = 4.0;
        double r93348 = 2.0;
        double r93349 = pow(r93325, r93348);
        double r93350 = r93347 / r93349;
        double r93351 = 8.0;
        double r93352 = pow(r93325, r93345);
        double r93353 = r93351 / r93352;
        double r93354 = r93350 - r93353;
        double r93355 = r93329 / r93325;
        double r93356 = r93354 - r93355;
        double r93357 = r93346 - r93356;
        double r93358 = r93357 / r93329;
        double r93359 = r93327 ? r93339 : r93358;
        return r93359;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if alpha < 307451316.1096792

   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. Using strategy rm

   6. Applied add-cube-cbrt 0.2

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

   7. Applied add-cube-cbrt 0.4

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

   8. Applied add-cube-cbrt 0.2

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

   9. Applied times-frac 0.2

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

   10. Applied prod-diff 0.2

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

   11. Simplified 0.1

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

   12. Simplified 0.1

       \[\leadsto \frac{\left({\left(\frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2}}\right)}^{3} - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)\right) + \color{blue}{0}}{2}\]
   13. Using strategy rm

   14. Applied pow1/3 1.8

       \[\leadsto \frac{\left({\left(\frac{\sqrt[3]{\beta}}{\color{blue}{{\left(\left(\alpha + \beta\right) + 2\right)}^{\frac{1}{3}}}}\right)}^{3} - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)\right) + 0}{2}\]
   15. Using strategy rm

   16. Applied cube-div 1.8

       \[\leadsto \frac{\left(\color{blue}{\frac{{\left(\sqrt[3]{\beta}\right)}^{3}}{{\left({\left(\left(\alpha + \beta\right) + 2\right)}^{\frac{1}{3}}\right)}^{3}}} - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)\right) + 0}{2}\]

   17. Simplified 1.8

       \[\leadsto \frac{\left(\frac{\color{blue}{\beta}}{{\left({\left(\left(\alpha + \beta\right) + 2\right)}^{\frac{1}{3}}\right)}^{3}} - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)\right) + 0}{2}\]

   18. Simplified 0.1

       \[\leadsto \frac{\left(\frac{\beta}{\color{blue}{\left(2 + \alpha\right) + \beta}} - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)\right) + 0}{2}\]

   if 307451316.1096792 < alpha

   1. Initial program 49.9

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

   3. Applied div-sub 49.9

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

   4. Applied associate-+l- 48.3

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

   6. Applied add-cube-cbrt 48.4

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

   7. Applied add-cube-cbrt 48.5

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

   8. Applied add-cube-cbrt 48.4

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

   9. Applied times-frac 48.4

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

   10. Applied prod-diff 48.4

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

   11. Simplified 48.4

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

   12. Simplified 48.4

       \[\leadsto \frac{\left({\left(\frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2}}\right)}^{3} - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)\right) + \color{blue}{0}}{2}\]
   13. Using strategy rm

   14. Applied pow1/3 49.3

       \[\leadsto \frac{\left({\left(\frac{\sqrt[3]{\beta}}{\color{blue}{{\left(\left(\alpha + \beta\right) + 2\right)}^{\frac{1}{3}}}}\right)}^{3} - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)\right) + 0}{2}\]

   15. Taylor expanded around inf 17.8

       \[\leadsto \frac{\left({\left(\frac{\sqrt[3]{\beta}}{{\left(\left(\alpha + \beta\right) + 2\right)}^{\frac{1}{3}}}\right)}^{3} - \color{blue}{\left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}\right) + 0}{2}\]

   16. Simplified 17.8

       \[\leadsto \frac{\left({\left(\frac{\sqrt[3]{\beta}}{{\left(\left(\alpha + \beta\right) + 2\right)}^{\frac{1}{3}}}\right)}^{3} - \color{blue}{\left(\left(\frac{4}{{\alpha}^{2}} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}\right) + 0}{2}\]
3. Recombined 2 regimes into one program.

4. Final simplification 5.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 307451316.10967922210693359375:\\ \;\;\;\;\frac{\frac{\beta}{\left(2 + \alpha\right) + \beta} - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\beta}}{{\left(\left(\alpha + \beta\right) + 2\right)}^{\frac{1}{3}}}\right)}^{3} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +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))