Average Error: 3.6 → 2.3
Time: 23.0s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
\[\begin{array}{l} \mathbf{if}\;\alpha \le 8.836463279649086 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - \frac{1.0}{\alpha}\right) + \frac{2.0}{\alpha \cdot \alpha}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \end{array}\]
\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 8.836463279649086 \cdot 10^{+162}:\\
\;\;\;\;\frac{\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\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 r2302962 = alpha;
        double r2302963 = beta;
        double r2302964 = r2302962 + r2302963;
        double r2302965 = r2302963 * r2302962;
        double r2302966 = r2302964 + r2302965;
        double r2302967 = 1.0;
        double r2302968 = r2302966 + r2302967;
        double r2302969 = 2.0;
        double r2302970 = 1.0;
        double r2302971 = r2302969 * r2302970;
        double r2302972 = r2302964 + r2302971;
        double r2302973 = r2302968 / r2302972;
        double r2302974 = r2302973 / r2302972;
        double r2302975 = r2302972 + r2302967;
        double r2302976 = r2302974 / r2302975;
        return r2302976;
}

double f(double alpha, double beta) {
        double r2302977 = alpha;
        double r2302978 = 8.836463279649086e+162;
        bool r2302979 = r2302977 <= r2302978;
        double r2302980 = 1.0;
        double r2302981 = beta;
        double r2302982 = r2302981 * r2302977;
        double r2302983 = r2302981 + r2302977;
        double r2302984 = r2302982 + r2302983;
        double r2302985 = r2302980 + r2302984;
        double r2302986 = 2.0;
        double r2302987 = r2302983 + r2302986;
        double r2302988 = r2302985 / r2302987;
        double r2302989 = r2302988 / r2302987;
        double r2302990 = r2302980 + r2302987;
        double r2302991 = r2302989 / r2302990;
        double r2302992 = 1.0;
        double r2302993 = r2302980 / r2302977;
        double r2302994 = r2302992 - r2302993;
        double r2302995 = 2.0;
        double r2302996 = r2302977 * r2302977;
        double r2302997 = r2302995 / r2302996;
        double r2302998 = r2302994 + r2302997;
        double r2302999 = r2302998 / r2302987;
        double r2303000 = r2302999 / r2302990;
        double r2303001 = r2302979 ? r2302991 : r2303000;
        return r2303001;
}

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 < 8.836463279649086e+162

    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. Using strategy rm
    3. Applied +-commutative1.3

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

    if 8.836463279649086e+162 < alpha

    1. Initial program 16.3

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
    2. Taylor expanded around inf 8.0

      \[\leadsto \frac{\frac{\color{blue}{\left(2.0 \cdot \frac{1}{{\alpha}^{2}} + 1\right) - 1.0 \cdot \frac{1}{\alpha}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
    3. Simplified8.0

      \[\leadsto \frac{\frac{\color{blue}{\frac{2.0}{\alpha \cdot \alpha} + \left(1 - \frac{1.0}{\alpha}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
  3. Recombined 2 regimes into one program.
  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 8.836463279649086 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - \frac{1.0}{\alpha}\right) + \frac{2.0}{\alpha \cdot \alpha}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 
(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)))