Average Error: 23.6 → 11.3
Time: 13.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.60680517543021152 \cdot 10^{177}:\\ \;\;\;\;\frac{e^{\log \left(\frac{\alpha + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}} + 1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{8}{{\alpha}^{3}} + \left(\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]
\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} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1.60680517543021152 \cdot 10^{177}:\\
\;\;\;\;\frac{e^{\log \left(\frac{\alpha + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}} + 1\right)}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{8}{{\alpha}^{3}} + \left(\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r149241 = alpha;
        double r149242 = beta;
        double r149243 = r149241 + r149242;
        double r149244 = r149242 - r149241;
        double r149245 = r149243 * r149244;
        double r149246 = 2.0;
        double r149247 = i;
        double r149248 = r149246 * r149247;
        double r149249 = r149243 + r149248;
        double r149250 = r149245 / r149249;
        double r149251 = r149249 + r149246;
        double r149252 = r149250 / r149251;
        double r149253 = 1.0;
        double r149254 = r149252 + r149253;
        double r149255 = r149254 / r149246;
        return r149255;
}
↓
double f(double alpha, double beta, double i) {
        double r149256 = alpha;
        double r149257 = 1.6068051754302115e+177;
        bool r149258 = r149256 <= r149257;
        double r149259 = beta;
        double r149260 = r149256 + r149259;
        double r149261 = 2.0;
        double r149262 = i;
        double r149263 = r149261 * r149262;
        double r149264 = r149260 + r149263;
        double r149265 = r149264 + r149261;
        double r149266 = r149259 - r149256;
        double r149267 = r149264 / r149266;
        double r149268 = r149265 * r149267;
        double r149269 = r149260 / r149268;
        double r149270 = 1.0;
        double r149271 = r149269 + r149270;
        double r149272 = log(r149271);
        double r149273 = exp(r149272);
        double r149274 = r149273 / r149261;
        double r149275 = 8.0;
        double r149276 = 3.0;
        double r149277 = pow(r149256, r149276);
        double r149278 = r149275 / r149277;
        double r149279 = r149261 / r149256;
        double r149280 = 4.0;
        double r149281 = r149256 * r149256;
        double r149282 = r149280 / r149281;
        double r149283 = r149279 - r149282;
        double r149284 = r149278 + r149283;
        double r149285 = r149284 / r149261;
        double r149286 = r149258 ? r149274 : r149285;
        return r149286;
}

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 < 1.6068051754302115e+177

    1. Initial program 17.0

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

    3. Applied associate-/l*6.3

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

    5. Applied associate-/l/6.3

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

    7. Applied add-exp-log6.3

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

    if 1.6068051754302115e+177 < alpha

    1. Initial program 64.0

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

    3. Applied associate-/l*49.2

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

    5. Applied associate-/l/49.2

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

    7. Applied add-exp-log49.2

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

    8. Taylor expanded around inf 42.2

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}}{2}\]

    9. Simplified42.2

      \[\leadsto \frac{\color{blue}{\frac{8}{{\alpha}^{3}} + \left(\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.60680517543021152 \cdot 10^{177}:\\ \;\;\;\;\frac{e^{\log \left(\frac{\alpha + \beta}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}} + 1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{8}{{\alpha}^{3}} + \left(\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2)) 1) 2))