Average Error: 3.8 → 1.3
Time: 37.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 6.628222884241031799960737125913462886879 \cdot 10^{153}:\\ \;\;\;\;\frac{\frac{\frac{1}{e^{\log \left(\frac{2 \cdot 1 + \left(\beta + \alpha\right)}{1 + \left(\left(\beta + \alpha \cdot \beta\right) + \alpha\right)}\right)}}}{2 \cdot 1 + \left(\beta + \alpha\right)}}{1 + \left(2 \cdot 1 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\left(\frac{1}{\beta} + \frac{1}{\alpha}\right) - \frac{1}{\alpha \cdot \alpha}}}{2 \cdot 1 + \left(\beta + \alpha\right)}}{1 + \left(2 \cdot 1 + \left(\beta + \alpha\right)\right)}\\ \end{array}\]
\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 6.628222884241031799960737125913462886879 \cdot 10^{153}:\\
\;\;\;\;\frac{\frac{\frac{1}{e^{\log \left(\frac{2 \cdot 1 + \left(\beta + \alpha\right)}{1 + \left(\left(\beta + \alpha \cdot \beta\right) + \alpha\right)}\right)}}}{2 \cdot 1 + \left(\beta + \alpha\right)}}{1 + \left(2 \cdot 1 + \left(\beta + \alpha\right)\right)}\\

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

\end{array}
double f(double alpha, double beta) {
        double r194410 = alpha;
        double r194411 = beta;
        double r194412 = r194410 + r194411;
        double r194413 = r194411 * r194410;
        double r194414 = r194412 + r194413;
        double r194415 = 1.0;
        double r194416 = r194414 + r194415;
        double r194417 = 2.0;
        double r194418 = r194417 * r194415;
        double r194419 = r194412 + r194418;
        double r194420 = r194416 / r194419;
        double r194421 = r194420 / r194419;
        double r194422 = r194419 + r194415;
        double r194423 = r194421 / r194422;
        return r194423;
}
↓
double f(double alpha, double beta) {
        double r194424 = beta;
        double r194425 = 6.628222884241032e+153;
        bool r194426 = r194424 <= r194425;
        double r194427 = 1.0;
        double r194428 = 2.0;
        double r194429 = 1.0;
        double r194430 = r194428 * r194429;
        double r194431 = alpha;
        double r194432 = r194424 + r194431;
        double r194433 = r194430 + r194432;
        double r194434 = r194431 * r194424;
        double r194435 = r194424 + r194434;
        double r194436 = r194435 + r194431;
        double r194437 = r194429 + r194436;
        double r194438 = r194433 / r194437;
        double r194439 = log(r194438);
        double r194440 = exp(r194439);
        double r194441 = r194427 / r194440;
        double r194442 = r194441 / r194433;
        double r194443 = r194429 + r194433;
        double r194444 = r194442 / r194443;
        double r194445 = r194427 / r194424;
        double r194446 = r194427 / r194431;
        double r194447 = r194445 + r194446;
        double r194448 = r194431 * r194431;
        double r194449 = r194427 / r194448;
        double r194450 = r194447 - r194449;
        double r194451 = r194427 / r194450;
        double r194452 = r194451 / r194433;
        double r194453 = r194452 / r194443;
        double r194454 = r194426 ? r194444 : r194453;
        return r194454;
}

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 < 6.628222884241032e+153

    1. Initial program 1.3

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

    3. Applied clear-num1.3

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

    4. Simplified1.3

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

    6. Applied add-exp-log2.6

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

    7. Applied add-exp-log1.6

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

    8. Applied div-exp1.6

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

    9. Simplified1.4

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

    if 6.628222884241032e+153 < beta

    1. Initial program 15.8

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

    3. Applied clear-num15.8

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

    4. Simplified15.8

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

    5. Taylor expanded around inf 0.8

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

    6. Simplified0.8

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 6.628222884241031799960737125913462886879 \cdot 10^{153}:\\ \;\;\;\;\frac{\frac{\frac{1}{e^{\log \left(\frac{2 \cdot 1 + \left(\beta + \alpha\right)}{1 + \left(\left(\beta + \alpha \cdot \beta\right) + \alpha\right)}\right)}}}{2 \cdot 1 + \left(\beta + \alpha\right)}}{1 + \left(2 \cdot 1 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\left(\frac{1}{\beta} + \frac{1}{\alpha}\right) - \frac{1}{\alpha \cdot \alpha}}}{2 \cdot 1 + \left(\beta + \alpha\right)}}{1 + \left(2 \cdot 1 + \left(\beta + \alpha\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))