Average Error: 16.3 → 6.2
Time: 25.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 3145284642582392593186816:\\ \;\;\;\;\frac{e^{\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta}{2 + \left(\beta + \alpha\right)}\right)\right) - \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 3145284642582392593186816:\\
\;\;\;\;\frac{e^{\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta}{2 + \left(\beta + \alpha\right)}\right)\right) - \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)\right)}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r3513522 = beta;
        double r3513523 = alpha;
        double r3513524 = r3513522 - r3513523;
        double r3513525 = r3513523 + r3513522;
        double r3513526 = 2.0;
        double r3513527 = r3513525 + r3513526;
        double r3513528 = r3513524 / r3513527;
        double r3513529 = 1.0;
        double r3513530 = r3513528 + r3513529;
        double r3513531 = r3513530 / r3513526;
        return r3513531;
}
↓
double f(double alpha, double beta) {
        double r3513532 = alpha;
        double r3513533 = 3.1452846425823926e+24;
        bool r3513534 = r3513532 <= r3513533;
        double r3513535 = beta;
        double r3513536 = 2.0;
        double r3513537 = r3513535 + r3513532;
        double r3513538 = r3513536 + r3513537;
        double r3513539 = r3513535 / r3513538;
        double r3513540 = log1p(r3513539);
        double r3513541 = expm1(r3513540);
        double r3513542 = r3513532 / r3513538;
        double r3513543 = 1.0;
        double r3513544 = r3513542 - r3513543;
        double r3513545 = r3513541 - r3513544;
        double r3513546 = log(r3513545);
        double r3513547 = exp(r3513546);
        double r3513548 = r3513547 / r3513536;
        double r3513549 = 4.0;
        double r3513550 = r3513532 * r3513532;
        double r3513551 = r3513549 / r3513550;
        double r3513552 = r3513536 / r3513532;
        double r3513553 = r3513551 - r3513552;
        double r3513554 = 8.0;
        double r3513555 = r3513532 * r3513550;
        double r3513556 = r3513554 / r3513555;
        double r3513557 = r3513553 - r3513556;
        double r3513558 = r3513539 - r3513557;
        double r3513559 = r3513558 / r3513536;
        double r3513560 = r3513534 ? r3513548 : r3513559;
        return r3513560;
}

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 < 3.1452846425823926e+24

    1. Initial program 1.0

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

    3. Applied div-sub1.0

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

    4. Applied associate-+l-1.0

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

    6. Applied add-exp-log1.0

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

    8. Applied expm1-log1p-u1.0

      \[\leadsto \frac{e^{\log \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta}{\left(\alpha + \beta\right) + 2}\right)\right)} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}}{2}\]

    if 3.1452846425823926e+24 < alpha

    1. Initial program 51.1

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

    3. Applied div-sub51.1

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

    4. Applied associate-+l-49.5

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

    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

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

  4. Final simplification6.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3145284642582392593186816:\\ \;\;\;\;\frac{e^{\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta}{2 + \left(\beta + \alpha\right)}\right)\right) - \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))