Average Error: 3.6 → 2.8
Time: 1.9m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 1.3353325288374214 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 4}}{\frac{1.0 + \left(\left(\alpha + \beta\right) + 2\right)}{\left(\alpha + \beta\right) - 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(0.5 + \beta \cdot 0.25\right) + \alpha \cdot 0.25}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 4}}{\frac{1}{\frac{\left(\alpha + \beta\right) - 2}{1.0 + \left(\left(\alpha + \beta\right) + 2\right)}}}\\ \end{array}\]
\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{1.0 + \left(\left(\alpha + \beta\right) + 2\right)}{\left(\alpha + \beta\right) - 2}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\left(0.5 + \beta \cdot 0.25\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 r7542158 = alpha;
        double r7542159 = beta;
        double r7542160 = r7542158 + r7542159;
        double r7542161 = r7542159 * r7542158;
        double r7542162 = r7542160 + r7542161;
        double r7542163 = 1.0;
        double r7542164 = r7542162 + r7542163;
        double r7542165 = 2.0;
        double r7542166 = 1.0;
        double r7542167 = r7542165 * r7542166;
        double r7542168 = r7542160 + r7542167;
        double r7542169 = r7542164 / r7542168;
        double r7542170 = r7542169 / r7542168;
        double r7542171 = r7542168 + r7542163;
        double r7542172 = r7542170 / r7542171;
        return r7542172;
}
↓
double f(double alpha, double beta) {
        double r7542173 = beta;
        double r7542174 = 1.3353325288374214e+154;
        bool r7542175 = r7542173 <= r7542174;
        double r7542176 = 1.0;
        double r7542177 = alpha;
        double r7542178 = r7542177 * r7542173;
        double r7542179 = r7542177 + r7542173;
        double r7542180 = r7542178 + r7542179;
        double r7542181 = r7542176 + r7542180;
        double r7542182 = 2.0;
        double r7542183 = r7542179 + r7542182;
        double r7542184 = r7542181 / r7542183;
        double r7542185 = r7542179 * r7542179;
        double r7542186 = 4.0;
        double r7542187 = r7542185 - r7542186;
        double r7542188 = r7542184 / r7542187;
        double r7542189 = r7542176 + r7542183;
        double r7542190 = r7542179 - r7542182;
        double r7542191 = r7542189 / r7542190;
        double r7542192 = r7542188 / r7542191;
        double r7542193 = 0.5;
        double r7542194 = 0.25;
        double r7542195 = r7542173 * r7542194;
        double r7542196 = r7542193 + r7542195;
        double r7542197 = r7542177 * r7542194;
        double r7542198 = r7542196 + r7542197;
        double r7542199 = r7542198 / r7542187;
        double r7542200 = 1.0;
        double r7542201 = r7542190 / r7542189;
        double r7542202 = r7542200 / r7542201;
        double r7542203 = r7542199 / r7542202;
        double r7542204 = r7542175 ? r7542192 : r7542203;
        return r7542204;
}

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

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

    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

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

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)))