Octave 3.8, jcobi/1

Average Error: 16.2 → 8.9

Time: 1.2m

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 7863989155318432839393331734927900672:\\ \;\;\;\;\frac{{e}^{\left(\log \left(\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)\right)\right)}}{{e}^{\left(\log 2\right)}}\\ \mathbf{elif}\;\alpha \le 1.526450071771057118621392850450502817566 \cdot 10^{113}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{\frac{8}{\alpha}}{\alpha \cdot \alpha} + \frac{2}{\alpha}\right)\right)}{2}\\ \mathbf{elif}\;\alpha \le 5.536806714946031575190662455815918850526 \cdot 10^{166}:\\ \;\;\;\;{e}^{\left(\sqrt[3]{\log \left(\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2}\right)} \cdot \left(\sqrt[3]{\log \left(\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2}\right)} \cdot \sqrt[3]{\log \left(\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2}\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{\frac{8}{\alpha}}{\alpha \cdot \alpha} + \frac{2}{\alpha}\right)\right)}{2}\\ \end{array}\]

C

double f(double alpha, double beta) {
        double r3925635 = beta;
        double r3925636 = alpha;
        double r3925637 = r3925635 - r3925636;
        double r3925638 = r3925636 + r3925635;
        double r3925639 = 2.0;
        double r3925640 = r3925638 + r3925639;
        double r3925641 = r3925637 / r3925640;
        double r3925642 = 1.0;
        double r3925643 = r3925641 + r3925642;
        double r3925644 = r3925643 / r3925639;
        return r3925644;
}

↓

double f(double alpha, double beta) {
        double r3925645 = alpha;
        double r3925646 = 7.863989155318433e+36;
        bool r3925647 = r3925645 <= r3925646;
        double r3925648 = exp(1.0);
        double r3925649 = beta;
        double r3925650 = r3925649 + r3925645;
        double r3925651 = 2.0;
        double r3925652 = r3925650 + r3925651;
        double r3925653 = r3925649 / r3925652;
        double r3925654 = r3925645 / r3925652;
        double r3925655 = 1.0;
        double r3925656 = r3925654 - r3925655;
        double r3925657 = r3925653 - r3925656;
        double r3925658 = log(r3925657);
        double r3925659 = pow(r3925648, r3925658);
        double r3925660 = log(r3925651);
        double r3925661 = pow(r3925648, r3925660);
        double r3925662 = r3925659 / r3925661;
        double r3925663 = 1.5264500717710571e+113;
        bool r3925664 = r3925645 <= r3925663;
        double r3925665 = 4.0;
        double r3925666 = r3925645 * r3925645;
        double r3925667 = r3925665 / r3925666;
        double r3925668 = 8.0;
        double r3925669 = r3925668 / r3925645;
        double r3925670 = r3925669 / r3925666;
        double r3925671 = r3925651 / r3925645;
        double r3925672 = r3925670 + r3925671;
        double r3925673 = r3925667 - r3925672;
        double r3925674 = r3925653 - r3925673;
        double r3925675 = r3925674 / r3925651;
        double r3925676 = 5.5368067149460316e+166;
        bool r3925677 = r3925645 <= r3925676;
        double r3925678 = r3925657 / r3925651;
        double r3925679 = log(r3925678);
        double r3925680 = cbrt(r3925679);
        double r3925681 = r3925680 * r3925680;
        double r3925682 = r3925680 * r3925681;
        double r3925683 = pow(r3925648, r3925682);
        double r3925684 = r3925677 ? r3925683 : r3925675;
        double r3925685 = r3925664 ? r3925675 : r3925684;
        double r3925686 = r3925647 ? r3925662 : r3925685;
        return r3925686;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 3 regimes

2. if alpha < 7.863989155318433e+36

   1. Initial program 1.6

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

   3. Applied div-sub 1.6

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

   4. Applied associate-+l- 1.6

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

   6. Applied add-exp-log 1.6

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

   7. Applied add-exp-log 1.6

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

   8. Applied div-exp 1.7

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

   9. Simplified 1.6

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

   11. Applied pow1 1.6

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

   12. Applied log-pow 1.6

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

   13. Applied exp-prod 1.7

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

   14. Simplified 1.7

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

   16. Applied log-div 1.7

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

   17. Applied pow-sub 2.3

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

   if 7.863989155318433e+36 < alpha < 1.5264500717710571e+113 or 5.5368067149460316e+166 < alpha

   1. Initial program 51.0

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

   3. Applied div-sub 51.0

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

   4. Applied associate-+l- 49.3

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

   5. Taylor expanded around inf 18.7

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

   6. Simplified 18.7

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

   if 1.5264500717710571e+113 < alpha < 5.5368067149460316e+166

   1. Initial program 49.8

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

   3. Applied div-sub 49.8

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

   4. Applied associate-+l- 48.6

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

   6. Applied add-exp-log 48.6

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

   7. Applied add-exp-log 49.2

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

   8. Applied div-exp 49.2

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

   9. Simplified 49.2

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

   11. Applied pow1 49.2

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

   12. Applied log-pow 49.2

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

   13. Applied exp-prod 49.2

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

   14. Simplified 49.2

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

   16. Applied add-cube-cbrt 49.2

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

4. Final simplification 8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 7863989155318432839393331734927900672:\\ \;\;\;\;\frac{{e}^{\left(\log \left(\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)\right)\right)}}{{e}^{\left(\log 2\right)}}\\ \mathbf{elif}\;\alpha \le 1.526450071771057118621392850450502817566 \cdot 10^{113}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{\frac{8}{\alpha}}{\alpha \cdot \alpha} + \frac{2}{\alpha}\right)\right)}{2}\\ \mathbf{elif}\;\alpha \le 5.536806714946031575190662455815918850526 \cdot 10^{166}:\\ \;\;\;\;{e}^{\left(\sqrt[3]{\log \left(\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2}\right)} \cdot \left(\sqrt[3]{\log \left(\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2}\right)} \cdot \sqrt[3]{\log \left(\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2}\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{\frac{8}{\alpha}}{\alpha \cdot \alpha} + \frac{2}{\alpha}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))