Average Error: 16.5 → 7.1
Time: 28.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 9.40148130873499 \cdot 10^{+18}:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)\right)}}{2.0}\\ \mathbf{elif}\;\alpha \le 1.8715997178975197 \cdot 10^{+96}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{\frac{4.0}{\alpha}}{\alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right)}{2.0}\\ \mathbf{elif}\;\alpha \le 2.9789869640725355 \cdot 10^{+116}:\\ \;\;\;\;\frac{\frac{1}{\frac{2.0 + \left(\beta + \alpha\right)}{\beta}} - \log \left(\frac{e^{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}}}{e^{1.0}}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{\frac{4.0}{\alpha}}{\alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right)}{2.0}\\ \end{array}\]
\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{\frac{4.0}{\alpha}}{\alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\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{\frac{4.0}{\alpha}}{\alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right)}{2.0}\\

\end{array}
double f(double alpha, double beta) {
        double r3817510 = beta;
        double r3817511 = alpha;
        double r3817512 = r3817510 - r3817511;
        double r3817513 = r3817511 + r3817510;
        double r3817514 = 2.0;
        double r3817515 = r3817513 + r3817514;
        double r3817516 = r3817512 / r3817515;
        double r3817517 = 1.0;
        double r3817518 = r3817516 + r3817517;
        double r3817519 = r3817518 / r3817514;
        return r3817519;
}
↓
double f(double alpha, double beta) {
        double r3817520 = alpha;
        double r3817521 = 9.40148130873499e+18;
        bool r3817522 = r3817520 <= r3817521;
        double r3817523 = beta;
        double r3817524 = 2.0;
        double r3817525 = r3817523 + r3817520;
        double r3817526 = r3817524 + r3817525;
        double r3817527 = r3817523 / r3817526;
        double r3817528 = r3817520 / r3817526;
        double r3817529 = 1.0;
        double r3817530 = r3817528 - r3817529;
        double r3817531 = r3817527 - r3817530;
        double r3817532 = log(r3817531);
        double r3817533 = exp(r3817532);
        double r3817534 = r3817533 / r3817524;
        double r3817535 = 1.8715997178975197e+96;
        bool r3817536 = r3817520 <= r3817535;
        double r3817537 = 4.0;
        double r3817538 = r3817537 / r3817520;
        double r3817539 = r3817538 / r3817520;
        double r3817540 = r3817524 / r3817520;
        double r3817541 = r3817539 - r3817540;
        double r3817542 = 8.0;
        double r3817543 = r3817520 * r3817520;
        double r3817544 = r3817543 * r3817520;
        double r3817545 = r3817542 / r3817544;
        double r3817546 = r3817541 - r3817545;
        double r3817547 = r3817527 - r3817546;
        double r3817548 = r3817547 / r3817524;
        double r3817549 = 2.9789869640725355e+116;
        bool r3817550 = r3817520 <= r3817549;
        double r3817551 = 1.0;
        double r3817552 = r3817526 / r3817523;
        double r3817553 = r3817551 / r3817552;
        double r3817554 = exp(r3817528);
        double r3817555 = exp(r3817529);
        double r3817556 = r3817554 / r3817555;
        double r3817557 = log(r3817556);
        double r3817558 = r3817553 - r3817557;
        double r3817559 = r3817558 / r3817524;
        double r3817560 = r3817550 ? r3817559 : r3817548;
        double r3817561 = r3817536 ? r3817548 : r3817560;
        double r3817562 = r3817522 ? r3817534 : r3817561;
        return r3817562;
}

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

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

    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

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

Reproduce

herbie shell --seed 2019139 
(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))