Average Error: 16.5 → 7.1
Time: 29.7s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 9.40148130873499 \cdot 10^{+18}:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)\right)}}{2.0}\\ \mathbf{elif}\;\alpha \le 1.8715997178975197 \cdot 10^{+96}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2.0}\\ \mathbf{elif}\;\alpha \le 2.9789869640725355 \cdot 10^{+116}:\\ \;\;\;\;\frac{\frac{1}{\frac{2.0 + \left(\beta + \alpha\right)}{\beta}} - \log \left(\frac{e^{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}}}{e^{1.0}}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 9.40148130873499 \cdot 10^{+18}:\\
\;\;\;\;\frac{e^{\log \left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)\right)}}{2.0}\\

\mathbf{elif}\;\alpha \le 1.8715997178975197 \cdot 10^{+96}:\\
\;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2.0}\\

\mathbf{elif}\;\alpha \le 2.9789869640725355 \cdot 10^{+116}:\\
\;\;\;\;\frac{\frac{1}{\frac{2.0 + \left(\beta + \alpha\right)}{\beta}} - \log \left(\frac{e^{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}}}{e^{1.0}}\right)}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2.0}\\

\end{array}
double f(double alpha, double beta) {
        double r1965467 = beta;
        double r1965468 = alpha;
        double r1965469 = r1965467 - r1965468;
        double r1965470 = r1965468 + r1965467;
        double r1965471 = 2.0;
        double r1965472 = r1965470 + r1965471;
        double r1965473 = r1965469 / r1965472;
        double r1965474 = 1.0;
        double r1965475 = r1965473 + r1965474;
        double r1965476 = r1965475 / r1965471;
        return r1965476;
}
↓
double f(double alpha, double beta) {
        double r1965477 = alpha;
        double r1965478 = 9.40148130873499e+18;
        bool r1965479 = r1965477 <= r1965478;
        double r1965480 = beta;
        double r1965481 = 2.0;
        double r1965482 = r1965480 + r1965477;
        double r1965483 = r1965481 + r1965482;
        double r1965484 = r1965480 / r1965483;
        double r1965485 = r1965477 / r1965483;
        double r1965486 = 1.0;
        double r1965487 = r1965485 - r1965486;
        double r1965488 = r1965484 - r1965487;
        double r1965489 = log(r1965488);
        double r1965490 = exp(r1965489);
        double r1965491 = r1965490 / r1965481;
        double r1965492 = 1.8715997178975197e+96;
        bool r1965493 = r1965477 <= r1965492;
        double r1965494 = 4.0;
        double r1965495 = r1965477 * r1965477;
        double r1965496 = r1965494 / r1965495;
        double r1965497 = r1965481 / r1965477;
        double r1965498 = r1965496 - r1965497;
        double r1965499 = 8.0;
        double r1965500 = r1965477 * r1965495;
        double r1965501 = r1965499 / r1965500;
        double r1965502 = r1965498 - r1965501;
        double r1965503 = r1965484 - r1965502;
        double r1965504 = r1965503 / r1965481;
        double r1965505 = 2.9789869640725355e+116;
        bool r1965506 = r1965477 <= r1965505;
        double r1965507 = 1.0;
        double r1965508 = r1965483 / r1965480;
        double r1965509 = r1965507 / r1965508;
        double r1965510 = exp(r1965485);
        double r1965511 = exp(r1965486);
        double r1965512 = r1965510 / r1965511;
        double r1965513 = log(r1965512);
        double r1965514 = r1965509 - r1965513;
        double r1965515 = r1965514 / r1965481;
        double r1965516 = r1965506 ? r1965515 : r1965504;
        double r1965517 = r1965493 ? r1965504 : r1965516;
        double r1965518 = r1965479 ? r1965491 : r1965517;
        return r1965518;
}

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 3 regimes

  2. if alpha < 9.40148130873499e+18

    1. Initial program 0.6

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

    3. Applied div-sub0.6

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

    4. Applied associate-+l-0.6

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

    6. Applied add-exp-log0.6

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

    if 9.40148130873499e+18 < alpha < 1.8715997178975197e+96 or 2.9789869640725355e+116 < alpha

    1. Initial program 50.4

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

    3. Applied div-sub50.4

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

    4. Applied associate-+l-48.7

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

    5. Taylor expanded around inf 18.9

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

    6. Simplified18.9

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

    if 1.8715997178975197e+96 < alpha < 2.9789869640725355e+116

    1. Initial program 46.6

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

    3. Applied div-sub46.6

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

    4. Applied associate-+l-45.2

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

    6. Applied *-un-lft-identity45.2

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \beta}}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]

    7. Applied associate-/l*45.2

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\left(\alpha + \beta\right) + 2.0}{\beta}}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]
    8. Using strategy rm

    9. Applied add-log-exp45.2

      \[\leadsto \frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2.0}{\beta}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\log \left(e^{1.0}\right)}\right)}{2.0}\]

    10. Applied add-log-exp45.2

      \[\leadsto \frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2.0}{\beta}} - \left(\color{blue}{\log \left(e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}}\right)} - \log \left(e^{1.0}\right)\right)}{2.0}\]

    11. Applied diff-log45.2

      \[\leadsto \frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2.0}{\beta}} - \color{blue}{\log \left(\frac{e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}}}{e^{1.0}}\right)}}{2.0}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification7.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 9.40148130873499 \cdot 10^{+18}:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)\right)}}{2.0}\\ \mathbf{elif}\;\alpha \le 1.8715997178975197 \cdot 10^{+96}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2.0}\\ \mathbf{elif}\;\alpha \le 2.9789869640725355 \cdot 10^{+116}:\\ \;\;\;\;\frac{\frac{1}{\frac{2.0 + \left(\beta + \alpha\right)}{\beta}} - \log \left(\frac{e^{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}}}{e^{1.0}}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 +o rules:numerics
(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))