Average Error: 3.8 → 1.4
Time: 13.5s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.234411931849943825494305932270323500516 \cdot 10^{159}:\\ \;\;\;\;\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{\frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)\right) \cdot \mathsf{fma}\left(1, 2, \alpha + \beta\right)}\\ \end{array}\]
\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}\;\alpha \le 2.234411931849943825494305932270323500516 \cdot 10^{159}:\\
\;\;\;\;\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{\frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}\\

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

\end{array}
double f(double alpha, double beta) {
        double r93330 = alpha;
        double r93331 = beta;
        double r93332 = r93330 + r93331;
        double r93333 = r93331 * r93330;
        double r93334 = r93332 + r93333;
        double r93335 = 1.0;
        double r93336 = r93334 + r93335;
        double r93337 = 2.0;
        double r93338 = r93337 * r93335;
        double r93339 = r93332 + r93338;
        double r93340 = r93336 / r93339;
        double r93341 = r93340 / r93339;
        double r93342 = r93339 + r93335;
        double r93343 = r93341 / r93342;
        return r93343;
}
↓
double f(double alpha, double beta) {
        double r93344 = alpha;
        double r93345 = 2.234411931849944e+159;
        bool r93346 = r93344 <= r93345;
        double r93347 = beta;
        double r93348 = r93344 + r93347;
        double r93349 = r93347 * r93344;
        double r93350 = r93348 + r93349;
        double r93351 = 1.0;
        double r93352 = r93350 + r93351;
        double r93353 = 2.0;
        double r93354 = r93353 * r93351;
        double r93355 = r93348 + r93354;
        double r93356 = r93352 / r93355;
        double r93357 = 1.0;
        double r93358 = r93355 + r93351;
        double r93359 = r93357 / r93358;
        double r93360 = fma(r93351, r93353, r93348);
        double r93361 = r93359 / r93360;
        double r93362 = r93356 * r93361;
        double r93363 = 2.0;
        double r93364 = r93347 / r93344;
        double r93365 = r93344 / r93347;
        double r93366 = r93364 + r93365;
        double r93367 = r93363 + r93366;
        double r93368 = r93367 * r93360;
        double r93369 = r93357 / r93368;
        double r93370 = r93346 ? r93362 : r93369;
        return r93370;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 2.234411931849944e+159

    1. Initial program 1.3

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

    3. Applied add-sqr-sqrt1.9

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

    4. Applied div-inv1.9

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

    5. Applied times-frac2.2

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

    6. Simplified2.2

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

    8. Applied div-inv2.2

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

    9. Applied associate-*l*2.5

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

    10. Simplified1.6

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

    if 2.234411931849944e+159 < alpha

    1. Initial program 16.0

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

    3. Applied *-un-lft-identity16.0

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

    4. Applied *-un-lft-identity16.0

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

    5. Applied times-frac16.0

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

    6. Applied associate-/l*16.2

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

    7. Simplified16.2

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

    8. Taylor expanded around inf 0.4

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.234411931849943825494305932270323500516 \cdot 10^{159}:\\ \;\;\;\;\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{\frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)\right) \cdot \mathsf{fma}\left(1, 2, \alpha + \beta\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +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)))