Average Error: 3.5 → 0.9
Time: 1.7m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 5.874077370608373 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{1.0 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}}}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 2\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\left(\frac{1}{\beta} - \frac{1}{\beta \cdot \beta}\right) + \frac{1}{\alpha}}}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 2\right) + 1.0}\\ \end{array}\]
\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}
↓
\begin{array}{l}
\mathbf{if}\;\beta \le 5.874077370608373 \cdot 10^{+160}:\\
\;\;\;\;\frac{\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{1.0 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}}}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 2\right) + 1.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{1}{\left(\frac{1}{\beta} - \frac{1}{\beta \cdot \beta}\right) + \frac{1}{\alpha}}}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 2\right) + 1.0}\\

\end{array}
double f(double alpha, double beta) {
        double r5188339 = alpha;
        double r5188340 = beta;
        double r5188341 = r5188339 + r5188340;
        double r5188342 = r5188340 * r5188339;
        double r5188343 = r5188341 + r5188342;
        double r5188344 = 1.0;
        double r5188345 = r5188343 + r5188344;
        double r5188346 = 2.0;
        double r5188347 = 1.0;
        double r5188348 = r5188346 * r5188347;
        double r5188349 = r5188341 + r5188348;
        double r5188350 = r5188345 / r5188349;
        double r5188351 = r5188350 / r5188349;
        double r5188352 = r5188349 + r5188344;
        double r5188353 = r5188351 / r5188352;
        return r5188353;
}
↓
double f(double alpha, double beta) {
        double r5188354 = beta;
        double r5188355 = 5.874077370608373e+160;
        bool r5188356 = r5188354 <= r5188355;
        double r5188357 = 1.0;
        double r5188358 = alpha;
        double r5188359 = r5188358 + r5188354;
        double r5188360 = 2.0;
        double r5188361 = r5188359 + r5188360;
        double r5188362 = 1.0;
        double r5188363 = fma(r5188354, r5188358, r5188359);
        double r5188364 = r5188362 + r5188363;
        double r5188365 = r5188361 / r5188364;
        double r5188366 = r5188357 / r5188365;
        double r5188367 = r5188366 / r5188361;
        double r5188368 = r5188361 + r5188362;
        double r5188369 = r5188367 / r5188368;
        double r5188370 = r5188357 / r5188354;
        double r5188371 = r5188354 * r5188354;
        double r5188372 = r5188357 / r5188371;
        double r5188373 = r5188370 - r5188372;
        double r5188374 = r5188357 / r5188358;
        double r5188375 = r5188373 + r5188374;
        double r5188376 = r5188357 / r5188375;
        double r5188377 = r5188376 / r5188361;
        double r5188378 = r5188377 / r5188368;
        double r5188379 = r5188356 ? r5188369 : r5188378;
        return r5188379;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if beta < 5.874077370608373e+160

    1. Initial program 1.0

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    2. Simplified1.0

      \[\leadsto \color{blue}{\frac{\frac{\frac{1.0 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}\]
    3. Using strategy rm

    4. Applied clear-num1.1

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\frac{2 + \left(\beta + \alpha\right)}{1.0 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}}}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\]

    if 5.874077370608373e+160 < beta

    1. Initial program 16.3

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    2. Simplified16.3

      \[\leadsto \color{blue}{\frac{\frac{\frac{1.0 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}\]
    3. Using strategy rm

    4. Applied clear-num16.3

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\frac{2 + \left(\beta + \alpha\right)}{1.0 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}}}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\]

    5. Taylor expanded around inf 0.1

      \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\frac{1}{\alpha} + \frac{1}{\beta}\right) - \frac{1}{{\beta}^{2}}}}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\]

    6. Simplified0.1

      \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\frac{1}{\alpha} + \left(\frac{1}{\beta} - \frac{1}{\beta \cdot \beta}\right)}}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 5.874077370608373 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{1.0 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}}}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 2\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\left(\frac{1}{\beta} - \frac{1}{\beta \cdot \beta}\right) + \frac{1}{\alpha}}}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 2\right) + 1.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0)))