Average Error: 24.0 → 11.3
Time: 13.6s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.336842127561784635835223312819898508058 \cdot 10^{115}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - \frac{\frac{\alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right) + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \end{array}\]
\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 1.336842127561784635835223312819898508058 \cdot 10^{115}:\\
\;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - \frac{\frac{\alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right) + 1}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r93314 = alpha;
        double r93315 = beta;
        double r93316 = r93314 + r93315;
        double r93317 = r93315 - r93314;
        double r93318 = r93316 * r93317;
        double r93319 = 2.0;
        double r93320 = i;
        double r93321 = r93319 * r93320;
        double r93322 = r93316 + r93321;
        double r93323 = r93318 / r93322;
        double r93324 = r93322 + r93319;
        double r93325 = r93323 / r93324;
        double r93326 = 1.0;
        double r93327 = r93325 + r93326;
        double r93328 = r93327 / r93319;
        return r93328;
}
↓
double f(double alpha, double beta, double i) {
        double r93329 = alpha;
        double r93330 = 1.3368421275617846e+115;
        bool r93331 = r93329 <= r93330;
        double r93332 = beta;
        double r93333 = r93329 + r93332;
        double r93334 = 2.0;
        double r93335 = i;
        double r93336 = r93334 * r93335;
        double r93337 = r93333 + r93336;
        double r93338 = r93332 / r93337;
        double r93339 = r93337 + r93334;
        double r93340 = r93338 / r93339;
        double r93341 = r93329 / r93337;
        double r93342 = r93341 / r93339;
        double r93343 = r93340 - r93342;
        double r93344 = r93333 * r93343;
        double r93345 = 1.0;
        double r93346 = r93344 + r93345;
        double r93347 = r93346 / r93334;
        double r93348 = 1.0;
        double r93349 = r93348 / r93329;
        double r93350 = r93334 * r93349;
        double r93351 = 8.0;
        double r93352 = 3.0;
        double r93353 = pow(r93329, r93352);
        double r93354 = r93348 / r93353;
        double r93355 = r93351 * r93354;
        double r93356 = r93350 + r93355;
        double r93357 = 4.0;
        double r93358 = 2.0;
        double r93359 = pow(r93329, r93358);
        double r93360 = r93348 / r93359;
        double r93361 = r93357 * r93360;
        double r93362 = r93356 - r93361;
        double r93363 = r93362 / r93334;
        double r93364 = r93331 ? r93347 : r93363;
        return r93364;
}

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 < 1.3368421275617846e+115

    1. Initial program 14.2

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

    3. Applied *-un-lft-identity14.2

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

    4. Applied *-un-lft-identity14.2

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

    5. Applied times-frac3.7

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

    6. Applied times-frac3.7

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

    7. Simplified3.7

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

    9. Applied div-sub3.7

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

    10. Applied div-sub3.7

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

    if 1.3368421275617846e+115 < alpha

    1. Initial program 60.3

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

    2. Taylor expanded around inf 39.4

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.336842127561784635835223312819898508058 \cdot 10^{115}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - \frac{\frac{\alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right) + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(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))