Average Error: 16.7 → 6.3
Time: 17.7s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 427088978407551744:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\left(\frac{12}{\alpha \cdot \alpha} - \frac{4}{\alpha}\right) - \frac{32}{{\alpha}^{3}}}{\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1}}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 427088978407551744:\\
\;\;\;\;\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} - \frac{\left(\frac{12}{\alpha \cdot \alpha} - \frac{4}{\alpha}\right) - \frac{32}{{\alpha}^{3}}}{\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1}}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r118419 = beta;
        double r118420 = alpha;
        double r118421 = r118419 - r118420;
        double r118422 = r118420 + r118419;
        double r118423 = 2.0;
        double r118424 = r118422 + r118423;
        double r118425 = r118421 / r118424;
        double r118426 = 1.0;
        double r118427 = r118425 + r118426;
        double r118428 = r118427 / r118423;
        return r118428;
}
↓
double f(double alpha, double beta) {
        double r118429 = alpha;
        double r118430 = 4.2708897840755174e+17;
        bool r118431 = r118429 <= r118430;
        double r118432 = beta;
        double r118433 = r118429 + r118432;
        double r118434 = 2.0;
        double r118435 = r118433 + r118434;
        double r118436 = r118432 / r118435;
        double r118437 = r118429 / r118435;
        double r118438 = 1.0;
        double r118439 = r118437 - r118438;
        double r118440 = r118436 - r118439;
        double r118441 = log(r118440);
        double r118442 = exp(r118441);
        double r118443 = r118442 / r118434;
        double r118444 = 12.0;
        double r118445 = r118429 * r118429;
        double r118446 = r118444 / r118445;
        double r118447 = 4.0;
        double r118448 = r118447 / r118429;
        double r118449 = r118446 - r118448;
        double r118450 = 32.0;
        double r118451 = 3.0;
        double r118452 = pow(r118429, r118451);
        double r118453 = r118450 / r118452;
        double r118454 = r118449 - r118453;
        double r118455 = r118437 + r118438;
        double r118456 = r118454 / r118455;
        double r118457 = r118436 - r118456;
        double r118458 = r118457 / r118434;
        double r118459 = r118431 ? r118443 : r118458;
        return r118459;
}

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 < 4.2708897840755174e+17

    1. Initial program 0.5

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

    3. Applied div-sub0.5

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

    4. Applied associate-+l-0.5

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

    6. Applied add-exp-log0.5

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

    if 4.2708897840755174e+17 < alpha

    1. Initial program 50.5

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

    3. Applied div-sub50.5

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

    4. Applied associate-+l-48.9

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

    6. Applied flip--48.9

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

    7. Taylor expanded around inf 18.3

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\color{blue}{12 \cdot \frac{1}{{\alpha}^{2}} - \left(4 \cdot \frac{1}{\alpha} + 32 \cdot \frac{1}{{\alpha}^{3}}\right)}}{\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1}}{2}\]

    8. Simplified18.3

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\color{blue}{\left(\frac{12}{\alpha \cdot \alpha} - \frac{4}{\alpha}\right) - \frac{32}{{\alpha}^{3}}}}{\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 427088978407551744:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\left(\frac{12}{\alpha \cdot \alpha} - \frac{4}{\alpha}\right) - \frac{32}{{\alpha}^{3}}}{\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))