Average Error: 3.5 → 1.2
Time: 1.5m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.5246833345897584 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right)}{\left(\alpha + 2\right) + \beta} \cdot \frac{1}{\left(\alpha + 2\right) + \beta}}{1.0 + \left(\left(\alpha + 2\right) + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\frac{\beta}{\alpha} + \left(\frac{\alpha}{\beta} + 2\right)\right)}\\ \end{array}\]
\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 2.5246833345897584 \cdot 10^{+155}:\\
\;\;\;\;\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right)}{\left(\alpha + 2\right) + \beta} \cdot \frac{1}{\left(\alpha + 2\right) + \beta}}{1.0 + \left(\left(\alpha + 2\right) + \beta\right)}\\

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

\end{array}
double f(double alpha, double beta) {
        double r3784336 = alpha;
        double r3784337 = beta;
        double r3784338 = r3784336 + r3784337;
        double r3784339 = r3784337 * r3784336;
        double r3784340 = r3784338 + r3784339;
        double r3784341 = 1.0;
        double r3784342 = r3784340 + r3784341;
        double r3784343 = 2.0;
        double r3784344 = 1.0;
        double r3784345 = r3784343 * r3784344;
        double r3784346 = r3784338 + r3784345;
        double r3784347 = r3784342 / r3784346;
        double r3784348 = r3784347 / r3784346;
        double r3784349 = r3784346 + r3784341;
        double r3784350 = r3784348 / r3784349;
        return r3784350;
}
↓
double f(double alpha, double beta) {
        double r3784351 = alpha;
        double r3784352 = 2.5246833345897584e+155;
        bool r3784353 = r3784351 <= r3784352;
        double r3784354 = 1.0;
        double r3784355 = beta;
        double r3784356 = r3784355 * r3784351;
        double r3784357 = r3784351 + r3784355;
        double r3784358 = r3784356 + r3784357;
        double r3784359 = r3784354 + r3784358;
        double r3784360 = 2.0;
        double r3784361 = r3784351 + r3784360;
        double r3784362 = r3784361 + r3784355;
        double r3784363 = r3784359 / r3784362;
        double r3784364 = 1.0;
        double r3784365 = r3784364 / r3784362;
        double r3784366 = r3784363 * r3784365;
        double r3784367 = r3784354 + r3784362;
        double r3784368 = r3784366 / r3784367;
        double r3784369 = r3784357 + r3784360;
        double r3784370 = r3784355 / r3784351;
        double r3784371 = r3784351 / r3784355;
        double r3784372 = r3784371 + r3784360;
        double r3784373 = r3784370 + r3784372;
        double r3784374 = r3784369 * r3784373;
        double r3784375 = r3784364 / r3784374;
        double r3784376 = r3784353 ? r3784368 : r3784375;
        return r3784376;
}

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 alpha < 2.5246833345897584e+155

    1. Initial program 1.2

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

    2. Simplified1.2

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

    4. Applied div-inv1.2

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

    if 2.5246833345897584e+155 < alpha

    1. Initial program 15.9

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

    2. Simplified15.9

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

    4. Applied div-inv15.9

      \[\leadsto \frac{\color{blue}{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{\left(\alpha + 2\right) + \beta} \cdot \frac{1}{\left(\alpha + 2\right) + \beta}}}{\left(\left(\alpha + 2\right) + \beta\right) + 1.0}\]
    5. Using strategy rm

    6. Applied clear-num16.6

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

    7. Simplified16.6

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

    8. Taylor expanded around inf 1.2

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.5246833345897584 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right)}{\left(\alpha + 2\right) + \beta} \cdot \frac{1}{\left(\alpha + 2\right) + \beta}}{1.0 + \left(\left(\alpha + 2\right) + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\frac{\beta}{\alpha} + \left(\frac{\alpha}{\beta} + 2\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(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)))