Average Error: 52.8 → 35.8
Time: 34.5s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 1.605441052792236 \cdot 10^{+216}:\\ \;\;\;\;\frac{\frac{\frac{\alpha \cdot \beta + i \cdot \left(\left(i + \beta\right) + \alpha\right)}{2 \cdot i + \left(\beta + \alpha\right)} \cdot \frac{i \cdot \left(\left(i + \beta\right) + \alpha\right)}{2 \cdot i + \left(\beta + \alpha\right)}}{\left(2 \cdot i + \left(\beta + \alpha\right)\right) - \sqrt{1.0}}}{\sqrt{1.0} + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}
↓
\begin{array}{l}
\mathbf{if}\;\beta \le 1.605441052792236 \cdot 10^{+216}:\\
\;\;\;\;\frac{\frac{\frac{\alpha \cdot \beta + i \cdot \left(\left(i + \beta\right) + \alpha\right)}{2 \cdot i + \left(\beta + \alpha\right)} \cdot \frac{i \cdot \left(\left(i + \beta\right) + \alpha\right)}{2 \cdot i + \left(\beta + \alpha\right)}}{\left(2 \cdot i + \left(\beta + \alpha\right)\right) - \sqrt{1.0}}}{\sqrt{1.0} + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}
double f(double alpha, double beta, double i) {
        double r3780592 = i;
        double r3780593 = alpha;
        double r3780594 = beta;
        double r3780595 = r3780593 + r3780594;
        double r3780596 = r3780595 + r3780592;
        double r3780597 = r3780592 * r3780596;
        double r3780598 = r3780594 * r3780593;
        double r3780599 = r3780598 + r3780597;
        double r3780600 = r3780597 * r3780599;
        double r3780601 = 2.0;
        double r3780602 = r3780601 * r3780592;
        double r3780603 = r3780595 + r3780602;
        double r3780604 = r3780603 * r3780603;
        double r3780605 = r3780600 / r3780604;
        double r3780606 = 1.0;
        double r3780607 = r3780604 - r3780606;
        double r3780608 = r3780605 / r3780607;
        return r3780608;
}
↓
double f(double alpha, double beta, double i) {
        double r3780609 = beta;
        double r3780610 = 1.605441052792236e+216;
        bool r3780611 = r3780609 <= r3780610;
        double r3780612 = alpha;
        double r3780613 = r3780612 * r3780609;
        double r3780614 = i;
        double r3780615 = r3780614 + r3780609;
        double r3780616 = r3780615 + r3780612;
        double r3780617 = r3780614 * r3780616;
        double r3780618 = r3780613 + r3780617;
        double r3780619 = 2.0;
        double r3780620 = r3780619 * r3780614;
        double r3780621 = r3780609 + r3780612;
        double r3780622 = r3780620 + r3780621;
        double r3780623 = r3780618 / r3780622;
        double r3780624 = r3780617 / r3780622;
        double r3780625 = r3780623 * r3780624;
        double r3780626 = 1.0;
        double r3780627 = sqrt(r3780626);
        double r3780628 = r3780622 - r3780627;
        double r3780629 = r3780625 / r3780628;
        double r3780630 = r3780627 + r3780622;
        double r3780631 = r3780629 / r3780630;
        double r3780632 = 0.0;
        double r3780633 = r3780611 ? r3780631 : r3780632;
        return r3780633;
}

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 beta < 1.605441052792236e+216

    1. Initial program 51.8

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

    3. Applied add-sqr-sqrt51.8

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

    4. Applied difference-of-squares51.8

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

    5. Applied times-frac37.6

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

    6. Applied times-frac35.3

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

    8. Applied *-un-lft-identity35.3

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

    9. Applied add-sqr-sqrt35.3

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

    10. Applied times-frac35.3

      \[\leadsto \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\color{blue}{\frac{\sqrt{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}}{1} \cdot \frac{\sqrt{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}\]
    11. Using strategy rm

    12. Applied associate-*l/35.3

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

    13. Simplified35.3

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

    if 1.605441052792236e+216 < beta

    1. Initial program 62.7

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

    2. Taylor expanded around inf 41.0

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification35.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.605441052792236 \cdot 10^{+216}:\\ \;\;\;\;\frac{\frac{\frac{\alpha \cdot \beta + i \cdot \left(\left(i + \beta\right) + \alpha\right)}{2 \cdot i + \left(\beta + \alpha\right)} \cdot \frac{i \cdot \left(\left(i + \beta\right) + \alpha\right)}{2 \cdot i + \left(\beta + \alpha\right)}}{\left(2 \cdot i + \left(\beta + \alpha\right)\right) - \sqrt{1.0}}}{\sqrt{1.0} + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (> alpha -1) (> beta -1) (> i 1))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)))