Average Error: 16.2 → 5.9
Time: 5.0s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 16045536794042094:\\ \;\;\;\;\frac{e^{\log \left(\frac{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right) \cdot \left(\beta - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} + \frac{-8}{{\alpha}^{3}}\right) + \frac{-2}{\alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 16045536794042094:\\
\;\;\;\;\frac{e^{\log \left(\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)}\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 r157287 = beta;
        double r157288 = alpha;
        double r157289 = r157287 - r157288;
        double r157290 = r157288 + r157287;
        double r157291 = 2.0;
        double r157292 = r157290 + r157291;
        double r157293 = r157289 / r157292;
        double r157294 = 1.0;
        double r157295 = r157293 + r157294;
        double r157296 = r157295 / r157291;
        return r157296;
}
↓
double f(double alpha, double beta) {
        double r157297 = alpha;
        double r157298 = 16045536794042094.0;
        bool r157299 = r157297 <= r157298;
        double r157300 = beta;
        double r157301 = r157297 + r157300;
        double r157302 = 2.0;
        double r157303 = r157301 + r157302;
        double r157304 = r157297 / r157303;
        double r157305 = 1.0;
        double r157306 = r157304 + r157305;
        double r157307 = r157304 - r157305;
        double r157308 = r157307 * r157303;
        double r157309 = r157300 - r157308;
        double r157310 = r157306 * r157309;
        double r157311 = r157303 * r157306;
        double r157312 = r157310 / r157311;
        double r157313 = log(r157312);
        double r157314 = exp(r157313);
        double r157315 = r157314 / r157302;
        double r157316 = r157300 / r157303;
        double r157317 = 4.0;
        double r157318 = r157317 / r157297;
        double r157319 = r157318 / r157297;
        double r157320 = 8.0;
        double r157321 = -r157320;
        double r157322 = 3.0;
        double r157323 = pow(r157297, r157322);
        double r157324 = r157321 / r157323;
        double r157325 = r157319 + r157324;
        double r157326 = -r157302;
        double r157327 = r157326 / r157297;
        double r157328 = r157325 + r157327;
        double r157329 = r157316 - r157328;
        double r157330 = r157329 / r157302;
        double r157331 = r157299 ? r157315 : r157330;
        return r157331;
}

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 < 16045536794042094.0

    1. Initial program 0.4

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

    3. Applied div-sub0.4

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

    4. Applied associate-+l-0.4

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

    6. Applied add-exp-log0.4

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

    8. Applied flip--0.4

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

    9. Applied frac-sub0.5

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

    10. Simplified0.4

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

    if 16045536794042094.0 < alpha

    1. Initial program 51.0

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

    3. Applied div-sub51.0

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

    4. Applied associate-+l-49.4

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

    5. Taylor expanded around inf 18.0

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

    6. Simplified18.0

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

  4. Final simplification5.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 16045536794042094:\\ \;\;\;\;\frac{e^{\log \left(\frac{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right) \cdot \left(\beta - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} + \frac{-8}{{\alpha}^{3}}\right) + \frac{-2}{\alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))