Average Error: 23.4 → 11.3
Time: 38.0s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 3.7195490550080187 \cdot 10^{+118}:\\ \;\;\;\;\frac{1.0 + \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \left(\frac{1}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \left(\beta + \alpha\right)\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} + \left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]
\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 3.7195490550080187 \cdot 10^{+118}:\\
\;\;\;\;\frac{1.0 + \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \left(\frac{1}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \left(\beta + \alpha\right)\right)}{2.0}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r4225530 = alpha;
        double r4225531 = beta;
        double r4225532 = r4225530 + r4225531;
        double r4225533 = r4225531 - r4225530;
        double r4225534 = r4225532 * r4225533;
        double r4225535 = 2.0;
        double r4225536 = i;
        double r4225537 = r4225535 * r4225536;
        double r4225538 = r4225532 + r4225537;
        double r4225539 = r4225534 / r4225538;
        double r4225540 = 2.0;
        double r4225541 = r4225538 + r4225540;
        double r4225542 = r4225539 / r4225541;
        double r4225543 = 1.0;
        double r4225544 = r4225542 + r4225543;
        double r4225545 = r4225544 / r4225540;
        return r4225545;
}
↓
double f(double alpha, double beta, double i) {
        double r4225546 = alpha;
        double r4225547 = 3.7195490550080187e+118;
        bool r4225548 = r4225546 <= r4225547;
        double r4225549 = 1.0;
        double r4225550 = beta;
        double r4225551 = r4225550 - r4225546;
        double r4225552 = r4225550 + r4225546;
        double r4225553 = 2.0;
        double r4225554 = i;
        double r4225555 = r4225553 * r4225554;
        double r4225556 = r4225552 + r4225555;
        double r4225557 = r4225551 / r4225556;
        double r4225558 = 2.0;
        double r4225559 = r4225558 + r4225556;
        double r4225560 = sqrt(r4225559);
        double r4225561 = r4225557 / r4225560;
        double r4225562 = 1.0;
        double r4225563 = r4225562 / r4225560;
        double r4225564 = r4225563 * r4225552;
        double r4225565 = r4225561 * r4225564;
        double r4225566 = r4225549 + r4225565;
        double r4225567 = r4225566 / r4225558;
        double r4225568 = 8.0;
        double r4225569 = r4225546 * r4225546;
        double r4225570 = r4225546 * r4225569;
        double r4225571 = r4225568 / r4225570;
        double r4225572 = r4225558 / r4225546;
        double r4225573 = 4.0;
        double r4225574 = r4225573 / r4225569;
        double r4225575 = r4225572 - r4225574;
        double r4225576 = r4225571 + r4225575;
        double r4225577 = r4225576 / r4225558;
        double r4225578 = r4225548 ? r4225567 : r4225577;
        return r4225578;
}

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 < 3.7195490550080187e+118

    1. Initial program 14.4

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

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

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

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

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

    5. Applied times-frac3.9

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

    6. Applied times-frac3.9

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

    7. Simplified3.9

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

    9. Applied add-sqr-sqrt3.9

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

    10. Applied *-un-lft-identity3.9

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

    11. Applied times-frac3.9

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

    12. Applied associate-*r*3.9

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

    if 3.7195490550080187e+118 < alpha

    1. Initial program 59.4

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

    2. Taylor expanded around inf 40.9

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

    3. Simplified40.9

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

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.7195490550080187 \cdot 10^{+118}:\\ \;\;\;\;\frac{1.0 + \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \left(\frac{1}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \left(\beta + \alpha\right)\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} + \left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019137 
(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))