Average Error: 24.2 → 11.5
Time: 8.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.4714840476837656 \cdot 10^{148}:\\ \;\;\;\;\frac{\left(\left(\alpha + \beta\right) \cdot \frac{1}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}\right) \cdot \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \end{array}\]
\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 1.4714840476837656 \cdot 10^{148}:\\
\;\;\;\;\frac{\left(\left(\alpha + \beta\right) \cdot \frac{1}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}\right) \cdot \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 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 r104579 = alpha;
        double r104580 = beta;
        double r104581 = r104579 + r104580;
        double r104582 = r104580 - r104579;
        double r104583 = r104581 * r104582;
        double r104584 = 2.0;
        double r104585 = i;
        double r104586 = r104584 * r104585;
        double r104587 = r104581 + r104586;
        double r104588 = r104583 / r104587;
        double r104589 = r104587 + r104584;
        double r104590 = r104588 / r104589;
        double r104591 = 1.0;
        double r104592 = r104590 + r104591;
        double r104593 = r104592 / r104584;
        return r104593;
}
↓
double f(double alpha, double beta, double i) {
        double r104594 = alpha;
        double r104595 = 1.4714840476837656e+148;
        bool r104596 = r104594 <= r104595;
        double r104597 = beta;
        double r104598 = r104594 + r104597;
        double r104599 = 1.0;
        double r104600 = 2.0;
        double r104601 = i;
        double r104602 = r104600 * r104601;
        double r104603 = r104598 + r104602;
        double r104604 = r104597 - r104594;
        double r104605 = r104603 / r104604;
        double r104606 = r104599 / r104605;
        double r104607 = r104598 * r104606;
        double r104608 = r104603 + r104600;
        double r104609 = r104599 / r104608;
        double r104610 = r104607 * r104609;
        double r104611 = 1.0;
        double r104612 = r104610 + r104611;
        double r104613 = r104612 / r104600;
        double r104614 = r104599 / r104594;
        double r104615 = r104600 * r104614;
        double r104616 = 8.0;
        double r104617 = 3.0;
        double r104618 = pow(r104594, r104617);
        double r104619 = r104599 / r104618;
        double r104620 = r104616 * r104619;
        double r104621 = r104615 + r104620;
        double r104622 = 4.0;
        double r104623 = 2.0;
        double r104624 = pow(r104594, r104623);
        double r104625 = r104599 / r104624;
        double r104626 = r104622 * r104625;
        double r104627 = r104621 - r104626;
        double r104628 = r104627 / r104600;
        double r104629 = r104596 ? r104613 : r104628;
        return r104629;
}

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 < 1.4714840476837656e+148

    1. Initial program 15.8

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

    3. Applied associate-/l*4.9

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

    5. Applied div-inv4.9

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

    7. Applied div-inv4.9

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

    if 1.4714840476837656e+148 < alpha

    1. Initial program 63.4

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

    2. Taylor expanded around inf 42.7

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

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.4714840476837656 \cdot 10^{148}:\\ \;\;\;\;\frac{\left(\left(\alpha + \beta\right) \cdot \frac{1}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}\right) \cdot \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(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))