Octave 3.8, jcobi/1

Average Error: 16.4 → 6.4

Time: 24.2s

Precision: 64

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

Math

\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 1485119.434541507:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{\left(\beta + \alpha\right) + 2.0} \cdot \left(\frac{\beta}{\left(\beta + \alpha\right) + 2.0} \cdot \frac{\beta}{\left(\beta + \alpha\right) + 2.0}\right) - \left(\left(\frac{\alpha}{\left(\beta + \alpha\right) + 2.0} - 1.0\right) \cdot \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2.0} - 1.0\right)\right) \cdot \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2.0} - 1.0\right)\right) - \log \left(\mathsf{fma}\left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2.0}\right), \left(\frac{\beta}{\left(\beta + \alpha\right) + 2.0}\right), \left(\left(\frac{\alpha}{\left(\beta + \alpha\right) + 2.0} - 1.0\right) \cdot \left(\frac{\beta}{\left(\beta + \alpha\right) + 2.0} + \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2.0} - 1.0\right)\right)\right)\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\beta + \alpha\right) + 2.0}} \cdot \frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\beta + \alpha\right) + 2.0} \cdot \sqrt[3]{\left(\beta + \alpha\right) + 2.0}} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]

C

double f(double alpha, double beta) {
        double r3238706 = beta;
        double r3238707 = alpha;
        double r3238708 = r3238706 - r3238707;
        double r3238709 = r3238707 + r3238706;
        double r3238710 = 2.0;
        double r3238711 = r3238709 + r3238710;
        double r3238712 = r3238708 / r3238711;
        double r3238713 = 1.0;
        double r3238714 = r3238712 + r3238713;
        double r3238715 = r3238714 / r3238710;
        return r3238715;
}

↓

double f(double alpha, double beta) {
        double r3238716 = alpha;
        double r3238717 = 1485119.434541507;
        bool r3238718 = r3238716 <= r3238717;
        double r3238719 = beta;
        double r3238720 = r3238719 + r3238716;
        double r3238721 = 2.0;
        double r3238722 = r3238720 + r3238721;
        double r3238723 = r3238719 / r3238722;
        double r3238724 = r3238723 * r3238723;
        double r3238725 = r3238723 * r3238724;
        double r3238726 = r3238716 / r3238722;
        double r3238727 = 1.0;
        double r3238728 = r3238726 - r3238727;
        double r3238729 = r3238728 * r3238728;
        double r3238730 = r3238729 * r3238728;
        double r3238731 = r3238725 - r3238730;
        double r3238732 = log(r3238731);
        double r3238733 = r3238723 + r3238728;
        double r3238734 = r3238728 * r3238733;
        double r3238735 = fma(r3238723, r3238723, r3238734);
        double r3238736 = log(r3238735);
        double r3238737 = r3238732 - r3238736;
        double r3238738 = exp(r3238737);
        double r3238739 = r3238738 / r3238721;
        double r3238740 = cbrt(r3238719);
        double r3238741 = cbrt(r3238722);
        double r3238742 = r3238740 / r3238741;
        double r3238743 = r3238740 * r3238740;
        double r3238744 = r3238741 * r3238741;
        double r3238745 = r3238743 / r3238744;
        double r3238746 = r3238742 * r3238745;
        double r3238747 = 4.0;
        double r3238748 = r3238716 * r3238716;
        double r3238749 = r3238747 / r3238748;
        double r3238750 = r3238721 / r3238716;
        double r3238751 = r3238749 - r3238750;
        double r3238752 = 8.0;
        double r3238753 = r3238752 / r3238716;
        double r3238754 = r3238753 / r3238748;
        double r3238755 = r3238751 - r3238754;
        double r3238756 = r3238746 - r3238755;
        double r3238757 = r3238756 / r3238721;
        double r3238758 = r3238718 ? r3238739 : r3238757;
        return r3238758;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if alpha < 1485119.434541507

   1. Initial program 0.1

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
   2. Using strategy rm

   3. Applied div-sub 0.1

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)} + 1.0}{2.0}\]

   4. Applied associate-+l- 0.1

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]
   5. Using strategy rm

   6. Applied add-exp-log 0.1

      \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right)}}}{2.0}\]
   7. Using strategy rm

   8. Applied flip3-- 0.1

      \[\leadsto \frac{e^{\log \color{blue}{\left(\frac{{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}^{3}}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0} + \left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right) + \frac{\beta}{\left(\alpha + \beta\right) + 2.0} \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right)}\right)}}}{2.0}\]

   9. Applied log-div 0.1

      \[\leadsto \frac{e^{\color{blue}{\log \left({\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}^{3}\right) - \log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0} + \left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right) + \frac{\beta}{\left(\alpha + \beta\right) + 2.0} \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right)\right)}}}{2.0}\]

   10. Simplified 0.1

       \[\leadsto \frac{e^{\color{blue}{\log \left(\left(\frac{\beta}{2.0 + \left(\alpha + \beta\right)} \cdot \frac{\beta}{2.0 + \left(\alpha + \beta\right)}\right) \cdot \frac{\beta}{2.0 + \left(\alpha + \beta\right)} - \left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)} - 1.0\right) \cdot \left(\left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)} - 1.0\right) \cdot \left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)} - 1.0\right)\right)\right)} - \log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0} + \left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right) + \frac{\beta}{\left(\alpha + \beta\right) + 2.0} \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right)\right)}}{2.0}\]

   11. Simplified 0.1

       \[\leadsto \frac{e^{\log \left(\left(\frac{\beta}{2.0 + \left(\alpha + \beta\right)} \cdot \frac{\beta}{2.0 + \left(\alpha + \beta\right)}\right) \cdot \frac{\beta}{2.0 + \left(\alpha + \beta\right)} - \left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)} - 1.0\right) \cdot \left(\left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)} - 1.0\right) \cdot \left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)} - 1.0\right)\right)\right) - \color{blue}{\log \left(\mathsf{fma}\left(\left(\frac{\beta}{2.0 + \left(\alpha + \beta\right)}\right), \left(\frac{\beta}{2.0 + \left(\alpha + \beta\right)}\right), \left(\left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)} - 1.0\right) \cdot \left(\frac{\beta}{2.0 + \left(\alpha + \beta\right)} + \left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)} - 1.0\right)\right)\right)\right)\right)}}}{2.0}\]

   if 1485119.434541507 < alpha

   1. Initial program 49.0

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
   2. Using strategy rm

   3. Applied div-sub 49.0

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)} + 1.0}{2.0}\]

   4. Applied associate-+l- 47.4

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 47.5

      \[\leadsto \frac{\frac{\beta}{\color{blue}{\left(\sqrt[3]{\left(\alpha + \beta\right) + 2.0} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2.0}\right) \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2.0}}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]

   7. Applied add-cube-cbrt 47.4

      \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}\right) \cdot \sqrt[3]{\beta}}}{\left(\sqrt[3]{\left(\alpha + \beta\right) + 2.0} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2.0}\right) \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2.0}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]

   8. Applied times-frac 47.4

      \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2.0} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2.0}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2.0}}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]

   9. Taylor expanded around -inf 18.9

      \[\leadsto \frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2.0} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2.0}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2.0}} - \color{blue}{\left(4.0 \cdot \frac{1}{{\alpha}^{2}} - \left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2.0}\]

   10. Simplified 18.9

       \[\leadsto \frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2.0} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2.0}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2.0}} - \color{blue}{\left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right)}}{2.0}\]
3. Recombined 2 regimes into one program.

4. Final simplification 6.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1485119.434541507:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{\left(\beta + \alpha\right) + 2.0} \cdot \left(\frac{\beta}{\left(\beta + \alpha\right) + 2.0} \cdot \frac{\beta}{\left(\beta + \alpha\right) + 2.0}\right) - \left(\left(\frac{\alpha}{\left(\beta + \alpha\right) + 2.0} - 1.0\right) \cdot \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2.0} - 1.0\right)\right) \cdot \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2.0} - 1.0\right)\right) - \log \left(\mathsf{fma}\left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2.0}\right), \left(\frac{\beta}{\left(\beta + \alpha\right) + 2.0}\right), \left(\left(\frac{\alpha}{\left(\beta + \alpha\right) + 2.0} - 1.0\right) \cdot \left(\frac{\beta}{\left(\beta + \alpha\right) + 2.0} + \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2.0} - 1.0\right)\right)\right)\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\beta + \alpha\right) + 2.0}} \cdot \frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\beta + \alpha\right) + 2.0} \cdot \sqrt[3]{\left(\beta + \alpha\right) + 2.0}} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))