Average Error: 16.1 → 6.0
Time: 22.0s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 417035.8774050206993706524372100830078125:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\left(\alpha + \beta\right) + 2}, 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 417035.8774050206993706524372100830078125:\\
\;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\left(\alpha + \beta\right) + 2}, 1\right)}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r7830525 = beta;
        double r7830526 = alpha;
        double r7830527 = r7830525 - r7830526;
        double r7830528 = r7830526 + r7830525;
        double r7830529 = 2.0;
        double r7830530 = r7830528 + r7830529;
        double r7830531 = r7830527 / r7830530;
        double r7830532 = 1.0;
        double r7830533 = r7830531 + r7830532;
        double r7830534 = r7830533 / r7830529;
        return r7830534;
}
↓
double f(double alpha, double beta) {
        double r7830535 = alpha;
        double r7830536 = 417035.8774050207;
        bool r7830537 = r7830535 <= r7830536;
        double r7830538 = beta;
        double r7830539 = r7830538 - r7830535;
        double r7830540 = 1.0;
        double r7830541 = r7830535 + r7830538;
        double r7830542 = 2.0;
        double r7830543 = r7830541 + r7830542;
        double r7830544 = r7830540 / r7830543;
        double r7830545 = 1.0;
        double r7830546 = fma(r7830539, r7830544, r7830545);
        double r7830547 = r7830546 / r7830542;
        double r7830548 = r7830538 / r7830543;
        double r7830549 = 4.0;
        double r7830550 = r7830535 * r7830535;
        double r7830551 = r7830549 / r7830550;
        double r7830552 = r7830542 / r7830535;
        double r7830553 = r7830551 - r7830552;
        double r7830554 = 8.0;
        double r7830555 = r7830550 * r7830535;
        double r7830556 = r7830554 / r7830555;
        double r7830557 = r7830553 - r7830556;
        double r7830558 = r7830548 - r7830557;
        double r7830559 = r7830558 / r7830542;
        double r7830560 = r7830537 ? r7830547 : r7830559;
        return r7830560;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 417035.8774050207

    1. Initial program 0.0

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

    3. Applied div-inv0.0

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

    4. Applied fma-def0.0

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

    if 417035.8774050207 < alpha

    1. Initial program 48.7

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

    3. Applied div-sub48.7

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

    4. Applied associate-+l-47.3

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

    5. Taylor expanded around inf 18.1

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

    6. Simplified18.1

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

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 417035.8774050206993706524372100830078125:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\left(\alpha + \beta\right) + 2}, 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +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))