Average Error: 23.7 → 11.7
Time: 30.5s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.0312336573895782 \cdot 10^{+150}:\\ \;\;\;\;\frac{e^{\log \left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \left(\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]
\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1.0312336573895782 \cdot 10^{+150}:\\
\;\;\;\;\frac{e^{\log \left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right)}}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{2.0}{\alpha} + \left(\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r2354164 = alpha;
        double r2354165 = beta;
        double r2354166 = r2354164 + r2354165;
        double r2354167 = r2354165 - r2354164;
        double r2354168 = r2354166 * r2354167;
        double r2354169 = 2.0;
        double r2354170 = i;
        double r2354171 = r2354169 * r2354170;
        double r2354172 = r2354166 + r2354171;
        double r2354173 = r2354168 / r2354172;
        double r2354174 = 2.0;
        double r2354175 = r2354172 + r2354174;
        double r2354176 = r2354173 / r2354175;
        double r2354177 = 1.0;
        double r2354178 = r2354176 + r2354177;
        double r2354179 = r2354178 / r2354174;
        return r2354179;
}
↓
double f(double alpha, double beta, double i) {
        double r2354180 = alpha;
        double r2354181 = 1.0312336573895782e+150;
        bool r2354182 = r2354180 <= r2354181;
        double r2354183 = beta;
        double r2354184 = r2354183 + r2354180;
        double r2354185 = r2354183 - r2354180;
        double r2354186 = i;
        double r2354187 = 2.0;
        double r2354188 = r2354186 * r2354187;
        double r2354189 = r2354188 + r2354184;
        double r2354190 = r2354185 / r2354189;
        double r2354191 = 2.0;
        double r2354192 = r2354191 + r2354189;
        double r2354193 = r2354190 / r2354192;
        double r2354194 = r2354184 * r2354193;
        double r2354195 = 1.0;
        double r2354196 = r2354194 + r2354195;
        double r2354197 = log(r2354196);
        double r2354198 = exp(r2354197);
        double r2354199 = r2354198 / r2354191;
        double r2354200 = r2354191 / r2354180;
        double r2354201 = 8.0;
        double r2354202 = r2354180 * r2354180;
        double r2354203 = r2354180 * r2354202;
        double r2354204 = r2354201 / r2354203;
        double r2354205 = 4.0;
        double r2354206 = r2354205 / r2354202;
        double r2354207 = r2354204 - r2354206;
        double r2354208 = r2354200 + r2354207;
        double r2354209 = r2354208 / r2354191;
        double r2354210 = r2354182 ? r2354199 : r2354209;
        return r2354210;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if alpha < 1.0312336573895782e+150

    1. Initial program 15.6

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity15.6

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

    4. Applied *-un-lft-identity15.6

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

    5. Applied distribute-lft-out15.6

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

    6. Applied *-un-lft-identity15.6

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

    7. Applied times-frac5.6

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

    8. Applied times-frac5.5

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

    9. Simplified5.5

      \[\leadsto \frac{\color{blue}{\left(\alpha + \beta\right)} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    10. Using strategy rm

    11. Applied add-exp-log5.5

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

    if 1.0312336573895782e+150 < alpha

    1. Initial program 62.9

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity62.9

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

    4. Applied *-un-lft-identity62.9

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

    5. Applied distribute-lft-out62.9

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

    6. Applied *-un-lft-identity62.9

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

    7. Applied times-frac47.9

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

    8. Applied times-frac47.9

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

    9. Simplified47.9

      \[\leadsto \frac{\color{blue}{\left(\alpha + \beta\right)} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    10. Using strategy rm

    11. Applied add-exp-log47.9

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

    12. Taylor expanded around -inf 41.3

      \[\leadsto \frac{\color{blue}{\left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right) - 4.0 \cdot \frac{1}{{\alpha}^{2}}}}{2.0}\]

    13. Simplified41.3

      \[\leadsto \frac{\color{blue}{\frac{2.0}{\alpha} + \left(\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4.0}{\alpha \cdot \alpha}\right)}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.0312336573895782 \cdot 10^{+150}:\\ \;\;\;\;\frac{e^{\log \left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \left(\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))