Average Error: 3.5 → 2.2
Time: 30.5s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 5.248579992166405 \cdot 10^{+165}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}\right)\right)}{3.0 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - \frac{1.0}{\alpha}\right) + \frac{2.0}{\alpha \cdot \alpha}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \end{array}\]
\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 5.248579992166405 \cdot 10^{+165}:\\
\;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}\right)\right)}{3.0 + \left(\beta + \alpha\right)}\\

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

\end{array}
double f(double alpha, double beta) {
        double r3517407 = alpha;
        double r3517408 = beta;
        double r3517409 = r3517407 + r3517408;
        double r3517410 = r3517408 * r3517407;
        double r3517411 = r3517409 + r3517410;
        double r3517412 = 1.0;
        double r3517413 = r3517411 + r3517412;
        double r3517414 = 2.0;
        double r3517415 = 1.0;
        double r3517416 = r3517414 * r3517415;
        double r3517417 = r3517409 + r3517416;
        double r3517418 = r3517413 / r3517417;
        double r3517419 = r3517418 / r3517417;
        double r3517420 = r3517417 + r3517412;
        double r3517421 = r3517419 / r3517420;
        return r3517421;
}
↓
double f(double alpha, double beta) {
        double r3517422 = alpha;
        double r3517423 = 5.248579992166405e+165;
        bool r3517424 = r3517422 <= r3517423;
        double r3517425 = 1.0;
        double r3517426 = beta;
        double r3517427 = r3517426 * r3517422;
        double r3517428 = r3517426 + r3517422;
        double r3517429 = r3517427 + r3517428;
        double r3517430 = r3517425 + r3517429;
        double r3517431 = 2.0;
        double r3517432 = r3517428 + r3517431;
        double r3517433 = r3517430 / r3517432;
        double r3517434 = r3517433 / r3517432;
        double r3517435 = log1p(r3517434);
        double r3517436 = expm1(r3517435);
        double r3517437 = 3.0;
        double r3517438 = r3517437 + r3517428;
        double r3517439 = r3517436 / r3517438;
        double r3517440 = 1.0;
        double r3517441 = r3517425 / r3517422;
        double r3517442 = r3517440 - r3517441;
        double r3517443 = 2.0;
        double r3517444 = r3517422 * r3517422;
        double r3517445 = r3517443 / r3517444;
        double r3517446 = r3517442 + r3517445;
        double r3517447 = r3517446 / r3517432;
        double r3517448 = r3517425 + r3517432;
        double r3517449 = r3517447 / r3517448;
        double r3517450 = r3517424 ? r3517439 : r3517449;
        return r3517450;
}

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 < 5.248579992166405e+165

    1. Initial program 1.3

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

    2. Taylor expanded around 0 1.3

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\alpha + \left(\beta + 3.0\right)}}\]

    3. Simplified1.3

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\alpha + \beta\right) + 3.0}}\]
    4. Using strategy rm

    5. Applied expm1-log1p-u1.3

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)\right)}}{\left(\alpha + \beta\right) + 3.0}\]

    if 5.248579992166405e+165 < alpha

    1. Initial program 15.7

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

    2. Taylor expanded around inf 7.4

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

    3. Simplified7.4

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

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 5.248579992166405 \cdot 10^{+165}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}\right)\right)}{3.0 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - \frac{1.0}{\alpha}\right) + \frac{2.0}{\alpha \cdot \alpha}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +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)))