Average Error: 3.6 → 2.8
Time: 2.0m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 1.3353325288374214 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 4}}{\frac{1}{\frac{\left(\alpha + \beta\right) - 2}{1.0 + \left(\left(\alpha + \beta\right) + 2\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(0.25 \cdot \beta + 0.5\right) + \alpha \cdot 0.25}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 4}}{\frac{1}{\frac{\left(\alpha + \beta\right) - 2}{1.0 + \left(\left(\alpha + \beta\right) + 2\right)}}}\\ \end{array}\]
\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}\;\beta \le 1.3353325288374214 \cdot 10^{+154}:\\
\;\;\;\;\frac{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 4}}{\frac{1}{\frac{\left(\alpha + \beta\right) - 2}{1.0 + \left(\left(\alpha + \beta\right) + 2\right)}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\left(0.25 \cdot \beta + 0.5\right) + \alpha \cdot 0.25}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 4}}{\frac{1}{\frac{\left(\alpha + \beta\right) - 2}{1.0 + \left(\left(\alpha + \beta\right) + 2\right)}}}\\

\end{array}
double f(double alpha, double beta) {
        double r7826753 = alpha;
        double r7826754 = beta;
        double r7826755 = r7826753 + r7826754;
        double r7826756 = r7826754 * r7826753;
        double r7826757 = r7826755 + r7826756;
        double r7826758 = 1.0;
        double r7826759 = r7826757 + r7826758;
        double r7826760 = 2.0;
        double r7826761 = 1.0;
        double r7826762 = r7826760 * r7826761;
        double r7826763 = r7826755 + r7826762;
        double r7826764 = r7826759 / r7826763;
        double r7826765 = r7826764 / r7826763;
        double r7826766 = r7826763 + r7826758;
        double r7826767 = r7826765 / r7826766;
        return r7826767;
}
↓
double f(double alpha, double beta) {
        double r7826768 = beta;
        double r7826769 = 1.3353325288374214e+154;
        bool r7826770 = r7826768 <= r7826769;
        double r7826771 = 1.0;
        double r7826772 = alpha;
        double r7826773 = r7826772 * r7826768;
        double r7826774 = r7826772 + r7826768;
        double r7826775 = r7826773 + r7826774;
        double r7826776 = r7826771 + r7826775;
        double r7826777 = 2.0;
        double r7826778 = r7826774 + r7826777;
        double r7826779 = r7826776 / r7826778;
        double r7826780 = r7826774 * r7826774;
        double r7826781 = 4.0;
        double r7826782 = r7826780 - r7826781;
        double r7826783 = r7826779 / r7826782;
        double r7826784 = 1.0;
        double r7826785 = r7826774 - r7826777;
        double r7826786 = r7826771 + r7826778;
        double r7826787 = r7826785 / r7826786;
        double r7826788 = r7826784 / r7826787;
        double r7826789 = r7826783 / r7826788;
        double r7826790 = 0.25;
        double r7826791 = r7826790 * r7826768;
        double r7826792 = 0.5;
        double r7826793 = r7826791 + r7826792;
        double r7826794 = r7826772 * r7826790;
        double r7826795 = r7826793 + r7826794;
        double r7826796 = r7826795 / r7826782;
        double r7826797 = r7826796 / r7826788;
        double r7826798 = r7826770 ? r7826789 : r7826797;
        return r7826798;
}

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 < 1.3353325288374214e+154

    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. Using strategy rm

    3. Applied flip-+1.6

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

    4. Applied associate-/r/1.6

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

    5. Applied associate-/l*1.6

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

    7. Applied clear-num1.6

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

    if 1.3353325288374214e+154 < beta

    1. Initial program 16.1

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

    3. Applied flip-+18.7

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

    4. Applied associate-/r/18.7

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

    5. Applied associate-/l*18.7

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

    7. Applied clear-num18.7

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

    8. Taylor expanded around 0 8.9

      \[\leadsto \frac{\frac{\color{blue}{0.25 \cdot \alpha + \left(0.25 \cdot \beta + 0.5\right)}}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - \left(2 \cdot 1\right) \cdot \left(2 \cdot 1\right)}}{\frac{1}{\frac{\left(\alpha + \beta\right) - 2 \cdot 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.3353325288374214 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 4}}{\frac{1}{\frac{\left(\alpha + \beta\right) - 2}{1.0 + \left(\left(\alpha + \beta\right) + 2\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(0.25 \cdot \beta + 0.5\right) + \alpha \cdot 0.25}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 4}}{\frac{1}{\frac{\left(\alpha + \beta\right) - 2}{1.0 + \left(\left(\alpha + \beta\right) + 2\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(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)))