Average Error: 23.1 → 11.2
Time: 34.5s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.0677683986829125 \cdot 10^{+213}:\\ \;\;\;\;\frac{1.0 + \left(\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}\right) \cdot \frac{1}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}}{2.0}\\ \end{array}\]
\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 1.0677683986829125 \cdot 10^{+213}:\\
\;\;\;\;\frac{1.0 + \left(\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}\right) \cdot \frac{1}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}{2.0}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r4726238 = alpha;
        double r4726239 = beta;
        double r4726240 = r4726238 + r4726239;
        double r4726241 = r4726239 - r4726238;
        double r4726242 = r4726240 * r4726241;
        double r4726243 = 2.0;
        double r4726244 = i;
        double r4726245 = r4726243 * r4726244;
        double r4726246 = r4726240 + r4726245;
        double r4726247 = r4726242 / r4726246;
        double r4726248 = 2.0;
        double r4726249 = r4726246 + r4726248;
        double r4726250 = r4726247 / r4726249;
        double r4726251 = 1.0;
        double r4726252 = r4726250 + r4726251;
        double r4726253 = r4726252 / r4726248;
        return r4726253;
}
↓
double f(double alpha, double beta, double i) {
        double r4726254 = alpha;
        double r4726255 = 1.0677683986829125e+213;
        bool r4726256 = r4726254 <= r4726255;
        double r4726257 = 1.0;
        double r4726258 = beta;
        double r4726259 = r4726258 + r4726254;
        double r4726260 = r4726258 - r4726254;
        double r4726261 = i;
        double r4726262 = 2.0;
        double r4726263 = r4726261 * r4726262;
        double r4726264 = r4726263 + r4726259;
        double r4726265 = r4726260 / r4726264;
        double r4726266 = r4726259 * r4726265;
        double r4726267 = 1.0;
        double r4726268 = 2.0;
        double r4726269 = r4726268 + r4726264;
        double r4726270 = r4726267 / r4726269;
        double r4726271 = r4726266 * r4726270;
        double r4726272 = r4726257 + r4726271;
        double r4726273 = r4726272 / r4726268;
        double r4726274 = r4726268 / r4726254;
        double r4726275 = 4.0;
        double r4726276 = r4726254 * r4726254;
        double r4726277 = r4726275 / r4726276;
        double r4726278 = r4726274 - r4726277;
        double r4726279 = 8.0;
        double r4726280 = r4726276 * r4726254;
        double r4726281 = r4726279 / r4726280;
        double r4726282 = r4726278 + r4726281;
        double r4726283 = r4726282 / r4726268;
        double r4726284 = r4726256 ? r4726273 : r4726283;
        return r4726284;
}

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 < 1.0677683986829125e+213

    1. Initial program 18.5

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

    3. Applied *-un-lft-identity18.5

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

    4. Applied times-frac7.5

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

    5. Simplified7.5

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

    7. Applied div-inv7.5

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

    if 1.0677683986829125e+213 < alpha

    1. Initial program 63.2

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

    2. Taylor expanded around inf 43.1

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

    3. Simplified43.1

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

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.0677683986829125 \cdot 10^{+213}:\\ \;\;\;\;\frac{1.0 + \left(\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}\right) \cdot \frac{1}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(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))