Average Error: 16.3 → 6.1
Time: 5.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 800075579838946432:\\ \;\;\;\;\frac{\frac{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right) \cdot \left(\beta - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} + \frac{-8}{{\alpha}^{3}}\right) + \frac{-2}{\alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 800075579838946432:\\
\;\;\;\;\frac{\frac{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right) \cdot \left(\beta - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\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 r91333 = beta;
        double r91334 = alpha;
        double r91335 = r91333 - r91334;
        double r91336 = r91334 + r91333;
        double r91337 = 2.0;
        double r91338 = r91336 + r91337;
        double r91339 = r91335 / r91338;
        double r91340 = 1.0;
        double r91341 = r91339 + r91340;
        double r91342 = r91341 / r91337;
        return r91342;
}
↓
double f(double alpha, double beta) {
        double r91343 = alpha;
        double r91344 = 8.000755798389464e+17;
        bool r91345 = r91343 <= r91344;
        double r91346 = beta;
        double r91347 = r91343 + r91346;
        double r91348 = 2.0;
        double r91349 = r91347 + r91348;
        double r91350 = r91343 / r91349;
        double r91351 = 1.0;
        double r91352 = r91350 + r91351;
        double r91353 = r91350 - r91351;
        double r91354 = r91353 * r91349;
        double r91355 = r91346 - r91354;
        double r91356 = r91352 * r91355;
        double r91357 = r91349 * r91352;
        double r91358 = r91356 / r91357;
        double r91359 = r91358 / r91348;
        double r91360 = r91346 / r91349;
        double r91361 = 4.0;
        double r91362 = r91361 / r91343;
        double r91363 = r91362 / r91343;
        double r91364 = 8.0;
        double r91365 = -r91364;
        double r91366 = 3.0;
        double r91367 = pow(r91343, r91366);
        double r91368 = r91365 / r91367;
        double r91369 = r91363 + r91368;
        double r91370 = -r91348;
        double r91371 = r91370 / r91343;
        double r91372 = r91369 + r91371;
        double r91373 = r91360 - r91372;
        double r91374 = r91373 / r91348;
        double r91375 = r91345 ? r91359 : r91374;
        return r91375;
}

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 < 8.000755798389464e+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

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

    8. Applied flip--0.5

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

    9. Applied frac-sub0.6

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

    10. Applied log-div2.2

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

    11. Applied exp-diff2.2

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

    12. Simplified2.5

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

    13. Simplified0.6

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

    if 8.000755798389464e+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. Taylor expanded around inf 18.2

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

    6. Simplified18.2

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 800075579838946432:\\ \;\;\;\;\frac{\frac{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right) \cdot \left(\beta - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} + \frac{-8}{{\alpha}^{3}}\right) + \frac{-2}{\alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))