Average Error: 16.3 → 5.9
Time: 21.9s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 17302899711342.2890625:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\frac{1}{2 + \left(\beta + \alpha\right)} \cdot \alpha - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 17302899711342.2890625:\\
\;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\frac{1}{2 + \left(\beta + \alpha\right)} \cdot \alpha - 1\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 r4758498 = beta;
        double r4758499 = alpha;
        double r4758500 = r4758498 - r4758499;
        double r4758501 = r4758499 + r4758498;
        double r4758502 = 2.0;
        double r4758503 = r4758501 + r4758502;
        double r4758504 = r4758500 / r4758503;
        double r4758505 = 1.0;
        double r4758506 = r4758504 + r4758505;
        double r4758507 = r4758506 / r4758502;
        return r4758507;
}
↓
double f(double alpha, double beta) {
        double r4758508 = alpha;
        double r4758509 = 17302899711342.29;
        bool r4758510 = r4758508 <= r4758509;
        double r4758511 = beta;
        double r4758512 = 2.0;
        double r4758513 = r4758511 + r4758508;
        double r4758514 = r4758512 + r4758513;
        double r4758515 = r4758511 / r4758514;
        double r4758516 = 1.0;
        double r4758517 = r4758516 / r4758514;
        double r4758518 = r4758517 * r4758508;
        double r4758519 = 1.0;
        double r4758520 = r4758518 - r4758519;
        double r4758521 = r4758515 - r4758520;
        double r4758522 = r4758521 / r4758512;
        double r4758523 = 4.0;
        double r4758524 = r4758508 * r4758508;
        double r4758525 = r4758523 / r4758524;
        double r4758526 = r4758512 / r4758508;
        double r4758527 = r4758525 - r4758526;
        double r4758528 = 8.0;
        double r4758529 = r4758508 * r4758524;
        double r4758530 = r4758528 / r4758529;
        double r4758531 = r4758527 - r4758530;
        double r4758532 = r4758515 - r4758531;
        double r4758533 = r4758532 / r4758512;
        double r4758534 = r4758510 ? r4758522 : r4758533;
        return r4758534;
}

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 < 17302899711342.29

    1. Initial program 0.3

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

    3. Applied div-sub0.3

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

    4. Applied associate-+l-0.3

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

    6. Applied div-inv0.3

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

    if 17302899711342.29 < alpha

    1. Initial program 50.7

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

    3. Applied div-sub50.7

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

    4. Applied associate-+l-49.1

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

    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

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

  4. Final simplification5.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 17302899711342.2890625:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\frac{1}{2 + \left(\beta + \alpha\right)} \cdot \alpha - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(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))