Average Error: 16.4 → 3.6
Time: 14.9s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.999999999999966:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{\frac{8.0}{\alpha \cdot \alpha}}{\alpha}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\log \left(e^{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}\right)} \cdot \left(\sqrt[3]{\log \left(e^{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}\right)} \cdot \sqrt[3]{\log \left(e^{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}\right)}\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.999999999999966:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{\frac{8.0}{\alpha \cdot \alpha}}{\alpha}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{\log \left(e^{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}\right)} \cdot \left(\sqrt[3]{\log \left(e^{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}\right)} \cdot \sqrt[3]{\log \left(e^{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}\right)}\right)}{2.0}\\

\end{array}
double f(double alpha, double beta) {
        double r2549185 = beta;
        double r2549186 = alpha;
        double r2549187 = r2549185 - r2549186;
        double r2549188 = r2549186 + r2549185;
        double r2549189 = 2.0;
        double r2549190 = r2549188 + r2549189;
        double r2549191 = r2549187 / r2549190;
        double r2549192 = 1.0;
        double r2549193 = r2549191 + r2549192;
        double r2549194 = r2549193 / r2549189;
        return r2549194;
}
↓
double f(double alpha, double beta) {
        double r2549195 = beta;
        double r2549196 = alpha;
        double r2549197 = r2549195 - r2549196;
        double r2549198 = r2549196 + r2549195;
        double r2549199 = 2.0;
        double r2549200 = r2549198 + r2549199;
        double r2549201 = r2549197 / r2549200;
        double r2549202 = -0.999999999999966;
        bool r2549203 = r2549201 <= r2549202;
        double r2549204 = r2549195 / r2549200;
        double r2549205 = 4.0;
        double r2549206 = r2549196 * r2549196;
        double r2549207 = r2549205 / r2549206;
        double r2549208 = 8.0;
        double r2549209 = r2549208 / r2549206;
        double r2549210 = r2549209 / r2549196;
        double r2549211 = r2549207 - r2549210;
        double r2549212 = r2549199 / r2549196;
        double r2549213 = r2549211 - r2549212;
        double r2549214 = r2549204 - r2549213;
        double r2549215 = r2549214 / r2549199;
        double r2549216 = 1.0;
        double r2549217 = r2549216 + r2549201;
        double r2549218 = exp(r2549217);
        double r2549219 = log(r2549218);
        double r2549220 = cbrt(r2549219);
        double r2549221 = r2549220 * r2549220;
        double r2549222 = r2549220 * r2549221;
        double r2549223 = r2549222 / r2549199;
        double r2549224 = r2549203 ? r2549215 : r2549223;
        return r2549224;
}

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 (/ (- beta alpha) (+ (+ alpha beta) 2.0)) < -0.999999999999966

    1. Initial program 60.5

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

    3. Applied div-sub60.5

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

    4. Applied associate-+l-58.6

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

    5. Taylor expanded around -inf 10.6

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

    6. Simplified10.6

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

    if -0.999999999999966 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

    1. Initial program 0.4

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

    3. Applied add-log-exp0.4

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

    4. Applied add-log-exp0.4

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

    5. Applied sum-log0.5

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

    6. Simplified0.4

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

    8. Applied add-cube-cbrt1.1

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\log \left(e^{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}\right)} \cdot \sqrt[3]{\log \left(e^{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}\right)}}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.999999999999966:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{\frac{8.0}{\alpha \cdot \alpha}}{\alpha}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\log \left(e^{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}\right)} \cdot \left(\sqrt[3]{\log \left(e^{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}\right)} \cdot \sqrt[3]{\log \left(e^{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}\right)}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 
(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))