Average Error: 24.1 → 12.1
Time: 37.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1154993415963887.25 \lor \neg \left(\alpha \le 4.279679450822490201695791715974354991479 \cdot 10^{61} \lor \neg \left(\alpha \le 2.988624513793906253585647747365693470749 \cdot 10^{144}\right)\right):\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \end{array}\]
\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 1154993415963887.25 \lor \neg \left(\alpha \le 4.279679450822490201695791715974354991479 \cdot 10^{61} \lor \neg \left(\alpha \le 2.988624513793906253585647747365693470749 \cdot 10^{144}\right)\right):\\
\;\;\;\;\frac{\sqrt[3]{{\left(\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1\right)}^{3}}}{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 r95407 = alpha;
        double r95408 = beta;
        double r95409 = r95407 + r95408;
        double r95410 = r95408 - r95407;
        double r95411 = r95409 * r95410;
        double r95412 = 2.0;
        double r95413 = i;
        double r95414 = r95412 * r95413;
        double r95415 = r95409 + r95414;
        double r95416 = r95411 / r95415;
        double r95417 = r95415 + r95412;
        double r95418 = r95416 / r95417;
        double r95419 = 1.0;
        double r95420 = r95418 + r95419;
        double r95421 = r95420 / r95412;
        return r95421;
}
↓
double f(double alpha, double beta, double i) {
        double r95422 = alpha;
        double r95423 = 1154993415963887.2;
        bool r95424 = r95422 <= r95423;
        double r95425 = 4.27967945082249e+61;
        bool r95426 = r95422 <= r95425;
        double r95427 = 2.9886245137939063e+144;
        bool r95428 = r95422 <= r95427;
        double r95429 = !r95428;
        bool r95430 = r95426 || r95429;
        double r95431 = !r95430;
        bool r95432 = r95424 || r95431;
        double r95433 = beta;
        double r95434 = r95422 + r95433;
        double r95435 = 2.0;
        double r95436 = i;
        double r95437 = r95435 * r95436;
        double r95438 = r95434 + r95437;
        double r95439 = r95438 + r95435;
        double r95440 = sqrt(r95439);
        double r95441 = r95434 / r95440;
        double r95442 = r95433 - r95422;
        double r95443 = r95442 / r95438;
        double r95444 = r95443 / r95440;
        double r95445 = r95441 * r95444;
        double r95446 = 1.0;
        double r95447 = r95445 + r95446;
        double r95448 = 3.0;
        double r95449 = pow(r95447, r95448);
        double r95450 = cbrt(r95449);
        double r95451 = r95450 / r95435;
        double r95452 = 1.0;
        double r95453 = r95452 / r95422;
        double r95454 = r95435 * r95453;
        double r95455 = 8.0;
        double r95456 = pow(r95422, r95448);
        double r95457 = r95452 / r95456;
        double r95458 = r95455 * r95457;
        double r95459 = r95454 + r95458;
        double r95460 = 4.0;
        double r95461 = 2.0;
        double r95462 = pow(r95422, r95461);
        double r95463 = r95452 / r95462;
        double r95464 = r95460 * r95463;
        double r95465 = r95459 - r95464;
        double r95466 = r95465 / r95435;
        double r95467 = r95432 ? r95451 : r95466;
        return r95467;
}

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 < 1154993415963887.2 or 4.27967945082249e+61 < alpha < 2.9886245137939063e+144

    1. Initial program 14.5

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

    3. Applied add-sqr-sqrt14.6

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

    4. Applied *-un-lft-identity14.6

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

    5. Applied times-frac3.5

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

    6. Applied times-frac3.5

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

    7. Simplified3.5

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

    9. Applied add-cbrt-cube3.5

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1\right) \cdot \left(\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1\right)\right) \cdot \left(\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1\right)}}}{2}\]

    10. Simplified3.5

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

    if 1154993415963887.2 < alpha < 4.27967945082249e+61 or 2.9886245137939063e+144 < alpha

    1. Initial program 56.7

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

    2. Taylor expanded around inf 41.5

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

  4. Final simplification12.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1154993415963887.25 \lor \neg \left(\alpha \le 4.279679450822490201695791715974354991479 \cdot 10^{61} \lor \neg \left(\alpha \le 2.988624513793906253585647747365693470749 \cdot 10^{144}\right)\right):\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(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))