Average Error: 3.7 → 2.3
Time: 14.8s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 8.4856429570103299 \cdot 10^{177}:\\ \;\;\;\;\frac{\frac{\frac{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, 1, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, 1, 1\right)}\\ \end{array}\]
\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}
↓
\begin{array}{l}
\mathbf{if}\;\beta \le 8.4856429570103299 \cdot 10^{177}:\\
\;\;\;\;\frac{\frac{\frac{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, 1, 1\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{0}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, 1, 1\right)}\\

\end{array}
double f(double alpha, double beta) {
        double r119402 = alpha;
        double r119403 = beta;
        double r119404 = r119402 + r119403;
        double r119405 = r119403 * r119402;
        double r119406 = r119404 + r119405;
        double r119407 = 1.0;
        double r119408 = r119406 + r119407;
        double r119409 = 2.0;
        double r119410 = r119409 * r119407;
        double r119411 = r119404 + r119410;
        double r119412 = r119408 / r119411;
        double r119413 = r119412 / r119411;
        double r119414 = r119411 + r119407;
        double r119415 = r119413 / r119414;
        return r119415;
}
↓
double f(double alpha, double beta) {
        double r119416 = beta;
        double r119417 = 8.48564295701033e+177;
        bool r119418 = r119416 <= r119417;
        double r119419 = 1.0;
        double r119420 = alpha;
        double r119421 = r119420 * r119416;
        double r119422 = r119420 + r119416;
        double r119423 = r119421 + r119422;
        double r119424 = r119419 + r119423;
        double r119425 = 2.0;
        double r119426 = fma(r119419, r119425, r119422);
        double r119427 = r119424 / r119426;
        double r119428 = r119427 / r119426;
        double r119429 = fma(r119425, r119419, r119419);
        double r119430 = r119422 + r119429;
        double r119431 = r119428 / r119430;
        double r119432 = 0.0;
        double r119433 = r119432 / r119430;
        double r119434 = r119418 ? r119431 : r119433;
        return r119434;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if beta < 8.48564295701033e+177

    1. Initial program 1.6

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

    2. Simplified1.6

      \[\leadsto \color{blue}{\frac{\frac{\frac{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, 1, 1\right)}}\]
    3. Using strategy rm

    4. Applied fma-udef1.6

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

    if 8.48564295701033e+177 < beta

    1. Initial program 16.3

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

    2. Simplified16.3

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

    3. Taylor expanded around inf 6.4

      \[\leadsto \frac{\color{blue}{0}}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, 1, 1\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 8.4856429570103299 \cdot 10^{177}:\\ \;\;\;\;\frac{\frac{\frac{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, 1, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, 1, 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1)))