Average Error: 23.7 → 12.4
Time: 30.5s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 4.246640781335346 \cdot 10^{+79}:\\ \;\;\;\;\frac{e^{\log \left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2.0}{\alpha} + \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right) - \frac{4.0}{\alpha \cdot \alpha}}{2.0}\\ \end{array}\]
\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 4.246640781335346 \cdot 10^{+79}:\\
\;\;\;\;\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{\left(\frac{2.0}{\alpha} + \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right) - \frac{4.0}{\alpha \cdot \alpha}}{2.0}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r2084129 = alpha;
        double r2084130 = beta;
        double r2084131 = r2084129 + r2084130;
        double r2084132 = r2084130 - r2084129;
        double r2084133 = r2084131 * r2084132;
        double r2084134 = 2.0;
        double r2084135 = i;
        double r2084136 = r2084134 * r2084135;
        double r2084137 = r2084131 + r2084136;
        double r2084138 = r2084133 / r2084137;
        double r2084139 = 2.0;
        double r2084140 = r2084137 + r2084139;
        double r2084141 = r2084138 / r2084140;
        double r2084142 = 1.0;
        double r2084143 = r2084141 + r2084142;
        double r2084144 = r2084143 / r2084139;
        return r2084144;
}
↓
double f(double alpha, double beta, double i) {
        double r2084145 = alpha;
        double r2084146 = 4.246640781335346e+79;
        bool r2084147 = r2084145 <= r2084146;
        double r2084148 = beta;
        double r2084149 = r2084148 + r2084145;
        double r2084150 = r2084148 - r2084145;
        double r2084151 = i;
        double r2084152 = 2.0;
        double r2084153 = r2084151 * r2084152;
        double r2084154 = r2084153 + r2084149;
        double r2084155 = r2084150 / r2084154;
        double r2084156 = 2.0;
        double r2084157 = r2084156 + r2084154;
        double r2084158 = r2084155 / r2084157;
        double r2084159 = r2084149 * r2084158;
        double r2084160 = 1.0;
        double r2084161 = r2084159 + r2084160;
        double r2084162 = log(r2084161);
        double r2084163 = exp(r2084162);
        double r2084164 = r2084163 / r2084156;
        double r2084165 = r2084156 / r2084145;
        double r2084166 = 8.0;
        double r2084167 = r2084145 * r2084145;
        double r2084168 = r2084167 * r2084145;
        double r2084169 = r2084166 / r2084168;
        double r2084170 = r2084165 + r2084169;
        double r2084171 = 4.0;
        double r2084172 = r2084171 / r2084167;
        double r2084173 = r2084170 - r2084172;
        double r2084174 = r2084173 / r2084156;
        double r2084175 = r2084147 ? r2084164 : r2084174;
        return r2084175;
}

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 < 4.246640781335346e+79

    1. Initial program 12.4

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

    3. Applied *-un-lft-identity12.4

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

    4. Applied *-un-lft-identity12.4

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

    5. Applied distribute-lft-out12.4

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

    6. Applied *-un-lft-identity12.4

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

    7. Applied times-frac2.3

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

    8. Applied times-frac2.3

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

    9. Simplified2.3

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

    11. Applied add-exp-log2.3

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

    if 4.246640781335346e+79 < alpha

    1. Initial program 55.9

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

    3. Applied *-un-lft-identity55.9

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

    4. Applied *-un-lft-identity55.9

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

    5. Applied distribute-lft-out55.9

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

    6. Applied *-un-lft-identity55.9

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

    7. Applied times-frac42.3

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

    8. Applied times-frac42.2

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

    9. Simplified42.2

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

    11. Applied add-exp-log42.2

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

    12. Taylor expanded around -inf 41.6

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

    13. Simplified41.6

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

  4. Final simplification12.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 4.246640781335346 \cdot 10^{+79}:\\ \;\;\;\;\frac{e^{\log \left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2.0}{\alpha} + \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right) - \frac{4.0}{\alpha \cdot \alpha}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(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))