Average Error: 3.8 → 2.8
Time: 47.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 7.118628879859003 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt[3]{1.0 + \left(\mathsf{fma}\left(\beta, \alpha, \alpha\right) + \beta\right)}}{2 + \left(\beta + \alpha\right)}}{\sqrt{2 + \left(\beta + \alpha\right)}}}{\sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}} \cdot \frac{\frac{\sqrt[3]{1.0 + \left(\mathsf{fma}\left(\beta, \alpha, \alpha\right) + \beta\right)} \cdot \sqrt[3]{1.0 + \left(\mathsf{fma}\left(\beta, \alpha, \alpha\right) + \beta\right)}}{\sqrt{2 + \left(\beta + \alpha\right)}}}{\sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0} \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(2.0, \left(\frac{1}{\beta \cdot \beta}\right), 1\right) - 1.0 \cdot \frac{1}{\beta}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\\ \end{array}\]
\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 7.118628879859003 \cdot 10^{+119}:\\
\;\;\;\;\frac{\frac{\frac{\sqrt[3]{1.0 + \left(\mathsf{fma}\left(\beta, \alpha, \alpha\right) + \beta\right)}}{2 + \left(\beta + \alpha\right)}}{\sqrt{2 + \left(\beta + \alpha\right)}}}{\sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}} \cdot \frac{\frac{\sqrt[3]{1.0 + \left(\mathsf{fma}\left(\beta, \alpha, \alpha\right) + \beta\right)} \cdot \sqrt[3]{1.0 + \left(\mathsf{fma}\left(\beta, \alpha, \alpha\right) + \beta\right)}}{\sqrt{2 + \left(\beta + \alpha\right)}}}{\sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0} \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(2.0, \left(\frac{1}{\beta \cdot \beta}\right), 1\right) - 1.0 \cdot \frac{1}{\beta}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\\

\end{array}
double f(double alpha, double beta) {
        double r3947356 = alpha;
        double r3947357 = beta;
        double r3947358 = r3947356 + r3947357;
        double r3947359 = r3947357 * r3947356;
        double r3947360 = r3947358 + r3947359;
        double r3947361 = 1.0;
        double r3947362 = r3947360 + r3947361;
        double r3947363 = 2.0;
        double r3947364 = 1.0;
        double r3947365 = r3947363 * r3947364;
        double r3947366 = r3947358 + r3947365;
        double r3947367 = r3947362 / r3947366;
        double r3947368 = r3947367 / r3947366;
        double r3947369 = r3947366 + r3947361;
        double r3947370 = r3947368 / r3947369;
        return r3947370;
}
↓
double f(double alpha, double beta) {
        double r3947371 = beta;
        double r3947372 = 7.118628879859003e+119;
        bool r3947373 = r3947371 <= r3947372;
        double r3947374 = 1.0;
        double r3947375 = alpha;
        double r3947376 = fma(r3947371, r3947375, r3947375);
        double r3947377 = r3947376 + r3947371;
        double r3947378 = r3947374 + r3947377;
        double r3947379 = cbrt(r3947378);
        double r3947380 = 2.0;
        double r3947381 = r3947371 + r3947375;
        double r3947382 = r3947380 + r3947381;
        double r3947383 = r3947379 / r3947382;
        double r3947384 = sqrt(r3947382);
        double r3947385 = r3947383 / r3947384;
        double r3947386 = r3947382 + r3947374;
        double r3947387 = cbrt(r3947386);
        double r3947388 = r3947385 / r3947387;
        double r3947389 = r3947379 * r3947379;
        double r3947390 = r3947389 / r3947384;
        double r3947391 = r3947387 * r3947387;
        double r3947392 = r3947390 / r3947391;
        double r3947393 = r3947388 * r3947392;
        double r3947394 = 2.0;
        double r3947395 = 1.0;
        double r3947396 = r3947371 * r3947371;
        double r3947397 = r3947395 / r3947396;
        double r3947398 = fma(r3947394, r3947397, r3947395);
        double r3947399 = r3947395 / r3947371;
        double r3947400 = r3947374 * r3947399;
        double r3947401 = r3947398 - r3947400;
        double r3947402 = r3947401 / r3947382;
        double r3947403 = r3947402 / r3947386;
        double r3947404 = r3947373 ? r3947393 : r3947403;
        return r3947404;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if beta < 7.118628879859003e+119

    1. Initial program 0.8

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

    2. Simplified0.8

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

    4. Applied add-cube-cbrt1.1

      \[\leadsto \frac{\frac{\frac{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\color{blue}{\left(\sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0} \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\right) \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}}\]

    5. Applied add-sqr-sqrt1.6

      \[\leadsto \frac{\frac{\frac{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{\color{blue}{\sqrt{2 + \left(\beta + \alpha\right)} \cdot \sqrt{2 + \left(\beta + \alpha\right)}}}}{\left(\sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0} \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\right) \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}\]

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

      \[\leadsto \frac{\frac{\frac{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}{\color{blue}{1 \cdot \left(2 + \left(\beta + \alpha\right)\right)}}}{\sqrt{2 + \left(\beta + \alpha\right)} \cdot \sqrt{2 + \left(\beta + \alpha\right)}}}{\left(\sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0} \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\right) \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}\]

    7. Applied add-cube-cbrt1.5

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\sqrt[3]{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)} \cdot \sqrt[3]{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}\right) \cdot \sqrt[3]{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}}}{1 \cdot \left(2 + \left(\beta + \alpha\right)\right)}}{\sqrt{2 + \left(\beta + \alpha\right)} \cdot \sqrt{2 + \left(\beta + \alpha\right)}}}{\left(\sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0} \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\right) \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}\]

    8. Applied times-frac1.5

      \[\leadsto \frac{\frac{\color{blue}{\frac{\sqrt[3]{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)} \cdot \sqrt[3]{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}}{1} \cdot \frac{\sqrt[3]{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}}{2 + \left(\beta + \alpha\right)}}}{\sqrt{2 + \left(\beta + \alpha\right)} \cdot \sqrt{2 + \left(\beta + \alpha\right)}}}{\left(\sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0} \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\right) \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}\]

    9. Applied times-frac1.5

      \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt[3]{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)} \cdot \sqrt[3]{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}}{1}}{\sqrt{2 + \left(\beta + \alpha\right)}} \cdot \frac{\frac{\sqrt[3]{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}}{2 + \left(\beta + \alpha\right)}}{\sqrt{2 + \left(\beta + \alpha\right)}}}}{\left(\sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0} \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\right) \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}\]

    10. Applied times-frac1.0

      \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt[3]{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)} \cdot \sqrt[3]{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}}{1}}{\sqrt{2 + \left(\beta + \alpha\right)}}}{\sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0} \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}} \cdot \frac{\frac{\frac{\sqrt[3]{1.0 + \left(\beta + \mathsf{fma}\left(\beta, \alpha, \alpha\right)\right)}}{2 + \left(\beta + \alpha\right)}}{\sqrt{2 + \left(\beta + \alpha\right)}}}{\sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}}\]

    if 7.118628879859003e+119 < beta

    1. Initial program 15.0

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

    2. Simplified15.0

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

    3. Taylor expanded around -inf 9.6

      \[\leadsto \frac{\frac{\color{blue}{\left(2.0 \cdot \frac{1}{{\beta}^{2}} + 1\right) - 1.0 \cdot \frac{1}{\beta}}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\]

    4. Simplified9.6

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(2.0, \left(\frac{1}{\beta \cdot \beta}\right), 1\right) - 1.0 \cdot \frac{1}{\beta}}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 7.118628879859003 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt[3]{1.0 + \left(\mathsf{fma}\left(\beta, \alpha, \alpha\right) + \beta\right)}}{2 + \left(\beta + \alpha\right)}}{\sqrt{2 + \left(\beta + \alpha\right)}}}{\sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}} \cdot \frac{\frac{\sqrt[3]{1.0 + \left(\mathsf{fma}\left(\beta, \alpha, \alpha\right) + \beta\right)} \cdot \sqrt[3]{1.0 + \left(\mathsf{fma}\left(\beta, \alpha, \alpha\right) + \beta\right)}}{\sqrt{2 + \left(\beta + \alpha\right)}}}{\sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0} \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(2.0, \left(\frac{1}{\beta \cdot \beta}\right), 1\right) - 1.0 \cdot \frac{1}{\beta}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 +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)))