Average Error: 3.6 → 2.4
Time: 2.4m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 4.0898973557336655 \cdot 10^{+168}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right) + 1.0}{\left(\alpha + \beta\right) + 2}}{\left(2 + 1.0\right) + \left(\alpha + \beta\right)} \cdot \frac{1}{\left(\alpha + \beta\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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}\;\alpha \le 4.0898973557336655 \cdot 10^{+168}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right) + 1.0}{\left(\alpha + \beta\right) + 2}}{\left(2 + 1.0\right) + \left(\alpha + \beta\right)} \cdot \frac{1}{\left(\alpha + \beta\right) + 2}\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}
double f(double alpha, double beta) {
        double r6215343 = alpha;
        double r6215344 = beta;
        double r6215345 = r6215343 + r6215344;
        double r6215346 = r6215344 * r6215343;
        double r6215347 = r6215345 + r6215346;
        double r6215348 = 1.0;
        double r6215349 = r6215347 + r6215348;
        double r6215350 = 2.0;
        double r6215351 = 1.0;
        double r6215352 = r6215350 * r6215351;
        double r6215353 = r6215345 + r6215352;
        double r6215354 = r6215349 / r6215353;
        double r6215355 = r6215354 / r6215353;
        double r6215356 = r6215353 + r6215348;
        double r6215357 = r6215355 / r6215356;
        return r6215357;
}
↓
double f(double alpha, double beta) {
        double r6215358 = alpha;
        double r6215359 = 4.0898973557336655e+168;
        bool r6215360 = r6215358 <= r6215359;
        double r6215361 = beta;
        double r6215362 = r6215358 + r6215361;
        double r6215363 = fma(r6215358, r6215361, r6215362);
        double r6215364 = 1.0;
        double r6215365 = r6215363 + r6215364;
        double r6215366 = 2.0;
        double r6215367 = r6215362 + r6215366;
        double r6215368 = r6215365 / r6215367;
        double r6215369 = r6215366 + r6215364;
        double r6215370 = r6215369 + r6215362;
        double r6215371 = r6215368 / r6215370;
        double r6215372 = 1.0;
        double r6215373 = r6215372 / r6215367;
        double r6215374 = r6215371 * r6215373;
        double r6215375 = 0.0;
        double r6215376 = r6215360 ? r6215374 : r6215375;
        return r6215376;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 4.0898973557336655e+168

    1. Initial program 1.4

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

    2. Simplified1.4

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

    4. Applied *-un-lft-identity1.4

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

    5. Applied *-un-lft-identity1.4

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

    6. Applied *-un-lft-identity1.4

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

    7. Applied distribute-lft-out1.4

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

    8. Applied distribute-lft-out1.4

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

    9. Applied add-sqr-sqrt2.0

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

    10. Applied add-sqr-sqrt2.4

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

    11. Applied *-un-lft-identity2.4

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

    12. Applied *-un-lft-identity2.4

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

    13. Applied distribute-lft-out2.4

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

    14. Applied times-frac2.4

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

    15. Applied times-frac2.1

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

    16. Applied times-frac2.1

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

    17. Simplified1.6

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

    18. Simplified1.5

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

    if 4.0898973557336655e+168 < alpha

    1. Initial program 16.1

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

    2. Simplified16.1

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

    3. Taylor expanded around inf 7.3

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 4.0898973557336655 \cdot 10^{+168}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right) + 1.0}{\left(\alpha + \beta\right) + 2}}{\left(2 + 1.0\right) + \left(\alpha + \beta\right)} \cdot \frac{1}{\left(\alpha + \beta\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 +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)))