Average Error: 23.9 → 7.7
Time: 28.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{2 \cdot i + \left(\beta + \alpha\right)}}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} \le -0.9999999999999963:\\ \;\;\;\;\frac{\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} + \left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)} \cdot \left(\beta + \alpha\right)}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} + 1.0}{2.0}\\ \end{array}\]
\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}\;\frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{2 \cdot i + \left(\beta + \alpha\right)}}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} \le -0.9999999999999963:\\
\;\;\;\;\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}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)} \cdot \left(\beta + \alpha\right)}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} + 1.0}{2.0}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r5252590 = alpha;
        double r5252591 = beta;
        double r5252592 = r5252590 + r5252591;
        double r5252593 = r5252591 - r5252590;
        double r5252594 = r5252592 * r5252593;
        double r5252595 = 2.0;
        double r5252596 = i;
        double r5252597 = r5252595 * r5252596;
        double r5252598 = r5252592 + r5252597;
        double r5252599 = r5252594 / r5252598;
        double r5252600 = 2.0;
        double r5252601 = r5252598 + r5252600;
        double r5252602 = r5252599 / r5252601;
        double r5252603 = 1.0;
        double r5252604 = r5252602 + r5252603;
        double r5252605 = r5252604 / r5252600;
        return r5252605;
}
↓
double f(double alpha, double beta, double i) {
        double r5252606 = beta;
        double r5252607 = alpha;
        double r5252608 = r5252606 + r5252607;
        double r5252609 = r5252606 - r5252607;
        double r5252610 = r5252608 * r5252609;
        double r5252611 = 2.0;
        double r5252612 = i;
        double r5252613 = r5252611 * r5252612;
        double r5252614 = r5252613 + r5252608;
        double r5252615 = r5252610 / r5252614;
        double r5252616 = 2.0;
        double r5252617 = r5252616 + r5252614;
        double r5252618 = r5252615 / r5252617;
        double r5252619 = -0.9999999999999963;
        bool r5252620 = r5252618 <= r5252619;
        double r5252621 = 8.0;
        double r5252622 = r5252607 * r5252607;
        double r5252623 = r5252607 * r5252622;
        double r5252624 = r5252621 / r5252623;
        double r5252625 = r5252616 / r5252607;
        double r5252626 = 4.0;
        double r5252627 = r5252626 / r5252622;
        double r5252628 = r5252625 - r5252627;
        double r5252629 = r5252624 + r5252628;
        double r5252630 = r5252629 / r5252616;
        double r5252631 = r5252609 / r5252614;
        double r5252632 = r5252631 * r5252608;
        double r5252633 = r5252632 / r5252617;
        double r5252634 = 1.0;
        double r5252635 = r5252633 + r5252634;
        double r5252636 = r5252635 / r5252616;
        double r5252637 = r5252620 ? r5252630 : r5252636;
        return r5252637;
}

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 beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) < -0.9999999999999963

    1. Initial program 62.7

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

    2. Taylor expanded around inf 32.7

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

    3. Simplified32.7

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

    if -0.9999999999999963 < (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))

    1. Initial program 12.5

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

    3. Applied *-un-lft-identity12.5

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

    4. Applied *-un-lft-identity12.5

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

    5. Applied times-frac0.3

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

    6. Applied times-frac0.3

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

    7. Simplified0.3

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

    9. Applied associate-*r/0.3

      \[\leadsto \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification7.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{2 \cdot i + \left(\beta + \alpha\right)}}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} \le -0.9999999999999963:\\ \;\;\;\;\frac{\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} + \left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)} \cdot \left(\beta + \alpha\right)}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} + 1.0}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(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))