Octave 3.8, jcobi/1

Average Error: 16.1 → 6.2

Time: 6.9s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 44214.9985027534785:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} + \left(1 \cdot 1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right) - \left(\left(\alpha + \beta\right) + 2\right) \cdot \left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {1}^{3}\right)}{\left(1 \cdot \left(1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) + \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 2 \cdot 2}\right) \cdot \left(\left(\alpha + \beta\right) - 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\frac{4}{\alpha}}{\alpha} - \left(\frac{2}{\alpha} - \frac{-8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

C

double f(double alpha, double beta) {
        double r179677 = beta;
        double r179678 = alpha;
        double r179679 = r179677 - r179678;
        double r179680 = r179678 + r179677;
        double r179681 = 2.0;
        double r179682 = r179680 + r179681;
        double r179683 = r179679 / r179682;
        double r179684 = 1.0;
        double r179685 = r179683 + r179684;
        double r179686 = r179685 / r179681;
        return r179686;
}

↓

double f(double alpha, double beta) {
        double r179687 = alpha;
        double r179688 = 44214.99850275348;
        bool r179689 = r179687 <= r179688;
        double r179690 = beta;
        double r179691 = r179687 + r179690;
        double r179692 = 2.0;
        double r179693 = r179691 + r179692;
        double r179694 = r179687 / r179693;
        double r179695 = r179694 * r179694;
        double r179696 = 1.0;
        double r179697 = r179696 * r179696;
        double r179698 = r179694 * r179696;
        double r179699 = r179697 + r179698;
        double r179700 = r179695 + r179699;
        double r179701 = r179690 * r179700;
        double r179702 = 3.0;
        double r179703 = pow(r179694, r179702);
        double r179704 = pow(r179696, r179702);
        double r179705 = r179703 - r179704;
        double r179706 = r179693 * r179705;
        double r179707 = r179701 - r179706;
        double r179708 = r179696 + r179694;
        double r179709 = r179696 * r179708;
        double r179710 = r179691 * r179691;
        double r179711 = r179692 * r179692;
        double r179712 = r179710 - r179711;
        double r179713 = r179687 / r179712;
        double r179714 = r179694 * r179713;
        double r179715 = r179691 - r179692;
        double r179716 = r179714 * r179715;
        double r179717 = r179709 + r179716;
        double r179718 = r179717 * r179693;
        double r179719 = r179707 / r179718;
        double r179720 = r179719 / r179692;
        double r179721 = r179690 / r179693;
        double r179722 = 4.0;
        double r179723 = r179722 / r179687;
        double r179724 = r179723 / r179687;
        double r179725 = r179692 / r179687;
        double r179726 = 8.0;
        double r179727 = -r179726;
        double r179728 = pow(r179687, r179702);
        double r179729 = r179727 / r179728;
        double r179730 = r179725 - r179729;
        double r179731 = r179724 - r179730;
        double r179732 = r179721 - r179731;
        double r179733 = r179732 / r179692;
        double r179734 = r179689 ? r179720 : r179733;
        return r179734;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if alpha < 44214.99850275348

   1. Initial program 0.0

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

   3. Applied div-sub 0.0

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

   4. Applied associate-+l- 0.0

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

   6. Applied flip3-- 0.0

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

   7. Applied frac-sub 0.1

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

   8. Simplified 0.1

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

   10. Applied flip-+ 0.1

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

   11. Applied associate-/r/ 0.1

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

   12. Applied associate-*r* 0.1

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

   if 44214.99850275348 < alpha

   1. Initial program 48.8

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

   3. Applied div-sub 48.8

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

   4. Applied associate-+l- 47.3

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

   6. Applied add-log-exp 47.3

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

   7. Applied add-log-exp 47.4

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

   8. Applied diff-log 47.4

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

   9. Simplified 47.3

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

   10. Taylor expanded around inf 18.8

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

   11. Simplified 18.8

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

4. Final simplification 6.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 44214.9985027534785:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} + \left(1 \cdot 1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right) - \left(\left(\alpha + \beta\right) + 2\right) \cdot \left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {1}^{3}\right)}{\left(1 \cdot \left(1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) + \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 2 \cdot 2}\right) \cdot \left(\left(\alpha + \beta\right) - 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\frac{4}{\alpha}}{\alpha} - \left(\frac{2}{\alpha} - \frac{-8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))