Average Error: 16.7 → 6.1
Time: 24.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 457427713.59383404:\\ \;\;\;\;\frac{e^{\log \left(\frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \beta - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2.0} - 1.0\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\beta + \alpha\right) + 2.0} \cdot \sqrt[3]{\left(\beta + \alpha\right) + 2.0}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\beta + \alpha\right) + 2.0}} - \left(\frac{\frac{1}{\alpha}}{\alpha} \cdot \left(4.0 - \frac{8.0}{\alpha}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 457427713.59383404:\\
\;\;\;\;\frac{e^{\log \left(\frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \beta - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2.0} - 1.0\right)\right)}}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\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}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\beta + \alpha\right) + 2.0}} - \left(\frac{\frac{1}{\alpha}}{\alpha} \cdot \left(4.0 - \frac{8.0}{\alpha}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\

\end{array}
double f(double alpha, double beta) {
        double r3068138 = beta;
        double r3068139 = alpha;
        double r3068140 = r3068138 - r3068139;
        double r3068141 = r3068139 + r3068138;
        double r3068142 = 2.0;
        double r3068143 = r3068141 + r3068142;
        double r3068144 = r3068140 / r3068143;
        double r3068145 = 1.0;
        double r3068146 = r3068144 + r3068145;
        double r3068147 = r3068146 / r3068142;
        return r3068147;
}
↓
double f(double alpha, double beta) {
        double r3068148 = alpha;
        double r3068149 = 457427713.59383404;
        bool r3068150 = r3068148 <= r3068149;
        double r3068151 = 1.0;
        double r3068152 = beta;
        double r3068153 = r3068152 + r3068148;
        double r3068154 = 2.0;
        double r3068155 = r3068153 + r3068154;
        double r3068156 = r3068151 / r3068155;
        double r3068157 = r3068156 * r3068152;
        double r3068158 = r3068148 / r3068155;
        double r3068159 = 1.0;
        double r3068160 = r3068158 - r3068159;
        double r3068161 = r3068157 - r3068160;
        double r3068162 = log(r3068161);
        double r3068163 = exp(r3068162);
        double r3068164 = r3068163 / r3068154;
        double r3068165 = cbrt(r3068152);
        double r3068166 = r3068165 * r3068165;
        double r3068167 = cbrt(r3068155);
        double r3068168 = r3068167 * r3068167;
        double r3068169 = r3068166 / r3068168;
        double r3068170 = r3068165 / r3068167;
        double r3068171 = r3068169 * r3068170;
        double r3068172 = r3068151 / r3068148;
        double r3068173 = r3068172 / r3068148;
        double r3068174 = 4.0;
        double r3068175 = 8.0;
        double r3068176 = r3068175 / r3068148;
        double r3068177 = r3068174 - r3068176;
        double r3068178 = r3068173 * r3068177;
        double r3068179 = r3068154 / r3068148;
        double r3068180 = r3068178 - r3068179;
        double r3068181 = r3068171 - r3068180;
        double r3068182 = r3068181 / r3068154;
        double r3068183 = r3068150 ? r3068164 : r3068182;
        return r3068183;
}

Error

Bits error versus alpha

Bits error versus beta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if alpha < 457427713.59383404

    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-sub0.1

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

    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-log0.1

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

    8. Applied div-inv0.1

      \[\leadsto \frac{e^{\log \left(\color{blue}{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right)}}{2.0}\]

    if 457427713.59383404 < alpha

    1. Initial program 49.9

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

    3. Applied div-sub49.9

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

    4. Applied associate-+l-48.3

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

    6. Applied add-cube-cbrt48.4

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

    7. Applied add-cube-cbrt48.3

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

    8. Applied times-frac48.3

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

    9. Taylor expanded around -inf 18.2

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

    10. Simplified18.2

      \[\leadsto \frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2.0} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2.0}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2.0}} - \color{blue}{\left(\frac{\frac{1}{\alpha}}{\alpha} \cdot \left(4.0 - \frac{8.0}{\alpha}\right) - \frac{2.0}{\alpha}\right)}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 457427713.59383404:\\ \;\;\;\;\frac{e^{\log \left(\frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \beta - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2.0} - 1.0\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\beta + \alpha\right) + 2.0} \cdot \sqrt[3]{\left(\beta + \alpha\right) + 2.0}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\beta + \alpha\right) + 2.0}} - \left(\frac{\frac{1}{\alpha}}{\alpha} \cdot \left(4.0 - \frac{8.0}{\alpha}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 
(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))