Average Error: 3.7 → 2.3
Time: 8.9s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 1.39247876717738 \cdot 10^{167}:\\ \;\;\;\;\frac{\frac{\frac{\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 \cdot \frac{1}{{\beta}^{2}} + 1\right) - 1 \cdot \frac{1}{\beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]
\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}\;\beta \le 1.39247876717738 \cdot 10^{167}:\\
\;\;\;\;\frac{\frac{\frac{\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\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}\\

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

\end{array}
double f(double alpha, double beta) {
        double r116336 = alpha;
        double r116337 = beta;
        double r116338 = r116336 + r116337;
        double r116339 = r116337 * r116336;
        double r116340 = r116338 + r116339;
        double r116341 = 1.0;
        double r116342 = r116340 + r116341;
        double r116343 = 2.0;
        double r116344 = r116343 * r116341;
        double r116345 = r116338 + r116344;
        double r116346 = r116342 / r116345;
        double r116347 = r116346 / r116345;
        double r116348 = r116345 + r116341;
        double r116349 = r116347 / r116348;
        return r116349;
}
↓
double f(double alpha, double beta) {
        double r116350 = beta;
        double r116351 = 1.3924787671773796e+167;
        bool r116352 = r116350 <= r116351;
        double r116353 = alpha;
        double r116354 = r116350 * r116353;
        double r116355 = r116353 + r116350;
        double r116356 = r116354 + r116355;
        double r116357 = 1.0;
        double r116358 = r116356 + r116357;
        double r116359 = 2.0;
        double r116360 = r116359 * r116357;
        double r116361 = r116355 + r116360;
        double r116362 = r116358 / r116361;
        double r116363 = r116362 / r116361;
        double r116364 = r116361 + r116357;
        double r116365 = r116363 / r116364;
        double r116366 = 1.0;
        double r116367 = 2.0;
        double r116368 = pow(r116350, r116367);
        double r116369 = r116366 / r116368;
        double r116370 = r116359 * r116369;
        double r116371 = r116370 + r116366;
        double r116372 = r116366 / r116350;
        double r116373 = r116357 * r116372;
        double r116374 = r116371 - r116373;
        double r116375 = r116374 / r116361;
        double r116376 = r116375 / r116364;
        double r116377 = r116352 ? r116365 : r116376;
        return r116377;
}

Error

Bits error versus alpha

Bits error versus beta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if beta < 1.3924787671773796e+167

    1. Initial program 1.4

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

    3. Applied +-commutative1.4

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

    if 1.3924787671773796e+167 < beta

    1. Initial program 16.9

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

    3. Applied +-commutative16.9

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

    4. Taylor expanded around inf 7.6

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

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.39247876717738 \cdot 10^{167}:\\ \;\;\;\;\frac{\frac{\frac{\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 \cdot \frac{1}{{\beta}^{2}} + 1\right) - 1 \cdot \frac{1}{\beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(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)))