Average Error: 3.7 → 2.4
Time: 7.8s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 6.1364080644475965 \cdot 10^{175}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\beta + \left(\alpha + \mathsf{fma}\left(1, 2, 1\right)\right)\right) \cdot \frac{1}{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}\right) \cdot \mathsf{fma}\left(1, 2, \alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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 6.1364080644475965 \cdot 10^{175}:\\
\;\;\;\;\frac{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\beta + \left(\alpha + \mathsf{fma}\left(1, 2, 1\right)\right)\right) \cdot \frac{1}{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}\right) \cdot \mathsf{fma}\left(1, 2, \alpha + \beta\right)}\\

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

\end{array}
double f(double alpha, double beta) {
        double r92431 = alpha;
        double r92432 = beta;
        double r92433 = r92431 + r92432;
        double r92434 = r92432 * r92431;
        double r92435 = r92433 + r92434;
        double r92436 = 1.0;
        double r92437 = r92435 + r92436;
        double r92438 = 2.0;
        double r92439 = r92438 * r92436;
        double r92440 = r92433 + r92439;
        double r92441 = r92437 / r92440;
        double r92442 = r92441 / r92440;
        double r92443 = r92440 + r92436;
        double r92444 = r92442 / r92443;
        return r92444;
}
↓
double f(double alpha, double beta) {
        double r92445 = alpha;
        double r92446 = 6.1364080644475965e+175;
        bool r92447 = r92445 <= r92446;
        double r92448 = beta;
        double r92449 = r92445 + r92448;
        double r92450 = r92448 * r92445;
        double r92451 = r92449 + r92450;
        double r92452 = 1.0;
        double r92453 = r92451 + r92452;
        double r92454 = sqrt(r92453);
        double r92455 = 2.0;
        double r92456 = r92455 * r92452;
        double r92457 = r92449 + r92456;
        double r92458 = sqrt(r92457);
        double r92459 = r92454 / r92458;
        double r92460 = r92459 / r92458;
        double r92461 = fma(r92452, r92455, r92452);
        double r92462 = r92445 + r92461;
        double r92463 = r92448 + r92462;
        double r92464 = 1.0;
        double r92465 = r92464 / r92454;
        double r92466 = r92463 * r92465;
        double r92467 = fma(r92452, r92455, r92449);
        double r92468 = r92466 * r92467;
        double r92469 = r92460 / r92468;
        double r92470 = 0.0;
        double r92471 = r92447 ? r92469 : r92470;
        return r92471;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 6.1364080644475965e+175

    1. Initial program 1.5

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

    3. Applied add-sqr-sqrt2.1

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

    4. Applied add-sqr-sqrt2.5

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

    5. Applied add-sqr-sqrt2.4

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

    6. Applied times-frac2.4

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

    7. Applied times-frac2.1

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

    8. Applied associate-/l*2.2

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

    9. Simplified1.6

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

    11. Applied div-inv1.6

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

    if 6.1364080644475965e+175 < alpha

    1. Initial program 16.8

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

    2. Taylor expanded around inf 7.3

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

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 6.1364080644475965 \cdot 10^{175}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\beta + \left(\alpha + \mathsf{fma}\left(1, 2, 1\right)\right)\right) \cdot \frac{1}{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}\right) \cdot \mathsf{fma}\left(1, 2, \alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +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)))