Average Error: 15.7 → 5.9
Time: 5.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1290203807327944437661696:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} + \frac{-8}{{\alpha}^{3}}\right) + \frac{-2}{\alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1290203807327944437661696:\\
\;\;\;\;\frac{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} + \frac{-8}{{\alpha}^{3}}\right) + \frac{-2}{\alpha}\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r107374 = beta;
        double r107375 = alpha;
        double r107376 = r107374 - r107375;
        double r107377 = r107375 + r107374;
        double r107378 = 2.0;
        double r107379 = r107377 + r107378;
        double r107380 = r107376 / r107379;
        double r107381 = 1.0;
        double r107382 = r107380 + r107381;
        double r107383 = r107382 / r107378;
        return r107383;
}
↓
double f(double alpha, double beta) {
        double r107384 = alpha;
        double r107385 = 1.2902038073279444e+24;
        bool r107386 = r107384 <= r107385;
        double r107387 = beta;
        double r107388 = r107384 + r107387;
        double r107389 = 2.0;
        double r107390 = r107388 + r107389;
        double r107391 = r107387 / r107390;
        double r107392 = r107384 / r107390;
        double r107393 = 1.0;
        double r107394 = r107392 - r107393;
        double r107395 = r107391 - r107394;
        double r107396 = log(r107395);
        double r107397 = exp(r107396);
        double r107398 = r107397 / r107389;
        double r107399 = 4.0;
        double r107400 = r107399 / r107384;
        double r107401 = r107400 / r107384;
        double r107402 = 8.0;
        double r107403 = -r107402;
        double r107404 = 3.0;
        double r107405 = pow(r107384, r107404);
        double r107406 = r107403 / r107405;
        double r107407 = r107401 + r107406;
        double r107408 = -r107389;
        double r107409 = r107408 / r107384;
        double r107410 = r107407 + r107409;
        double r107411 = r107391 - r107410;
        double r107412 = r107411 / r107389;
        double r107413 = r107386 ? r107398 : r107412;
        return r107413;
}

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 < 1.2902038073279444e+24

    1. Initial program 0.7

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

    3. Applied div-sub0.7

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

    4. Applied associate-+l-0.7

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

    6. Applied add-exp-log0.7

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

    if 1.2902038073279444e+24 < alpha

    1. Initial program 50.1

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

    3. Applied div-sub50.1

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

    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. Taylor expanded around inf 17.8

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

    6. Simplified17.8

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

  4. Final simplification5.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1290203807327944437661696:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} + \frac{-8}{{\alpha}^{3}}\right) + \frac{-2}{\alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))