Average Error: 52.8 → 37.1
Time: 1.1m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.0321689507084963 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{\sqrt{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\sqrt{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}} \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(i \cdot i\right) \cdot \frac{1}{4}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}\\ \end{array}\]
\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}\;\alpha \le 1.0321689507084963 \cdot 10^{+99}:\\
\;\;\;\;\frac{\frac{\sqrt{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\sqrt{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}} \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\left(i \cdot i\right) \cdot \frac{1}{4}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r4784696 = i;
        double r4784697 = alpha;
        double r4784698 = beta;
        double r4784699 = r4784697 + r4784698;
        double r4784700 = r4784699 + r4784696;
        double r4784701 = r4784696 * r4784700;
        double r4784702 = r4784698 * r4784697;
        double r4784703 = r4784702 + r4784701;
        double r4784704 = r4784701 * r4784703;
        double r4784705 = 2.0;
        double r4784706 = r4784705 * r4784696;
        double r4784707 = r4784699 + r4784706;
        double r4784708 = r4784707 * r4784707;
        double r4784709 = r4784704 / r4784708;
        double r4784710 = 1.0;
        double r4784711 = r4784708 - r4784710;
        double r4784712 = r4784709 / r4784711;
        return r4784712;
}
↓
double f(double alpha, double beta, double i) {
        double r4784713 = alpha;
        double r4784714 = 1.0321689507084963e+99;
        bool r4784715 = r4784713 <= r4784714;
        double r4784716 = i;
        double r4784717 = beta;
        double r4784718 = r4784713 + r4784717;
        double r4784719 = r4784716 + r4784718;
        double r4784720 = r4784716 * r4784719;
        double r4784721 = r4784717 * r4784713;
        double r4784722 = r4784720 + r4784721;
        double r4784723 = sqrt(r4784722);
        double r4784724 = 2.0;
        double r4784725 = r4784724 * r4784716;
        double r4784726 = r4784718 + r4784725;
        double r4784727 = r4784726 / r4784723;
        double r4784728 = r4784723 / r4784727;
        double r4784729 = 1.0;
        double r4784730 = sqrt(r4784729);
        double r4784731 = r4784726 - r4784730;
        double r4784732 = r4784728 / r4784731;
        double r4784733 = r4784720 / r4784726;
        double r4784734 = r4784730 + r4784726;
        double r4784735 = r4784733 / r4784734;
        double r4784736 = r4784732 * r4784735;
        double r4784737 = r4784716 * r4784716;
        double r4784738 = 0.25;
        double r4784739 = r4784737 * r4784738;
        double r4784740 = r4784739 / r4784734;
        double r4784741 = r4784740 / r4784731;
        double r4784742 = r4784715 ? r4784736 : r4784741;
        return r4784742;
}

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 < 1.0321689507084963e+99

    1. Initial program 50.1

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

    3. Applied add-sqr-sqrt50.1

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

    4. Applied difference-of-squares50.1

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

    5. Applied times-frac35.1

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

    6. Applied times-frac33.7

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

    8. Applied add-sqr-sqrt33.7

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

    9. Applied associate-/l*33.7

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

    if 1.0321689507084963e+99 < alpha

    1. Initial program 61.5

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

    2. Taylor expanded around inf 49.5

      \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot {i}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]

    3. Simplified49.5

      \[\leadsto \frac{\color{blue}{\left(i \cdot i\right) \cdot \frac{1}{4}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt49.5

      \[\leadsto \frac{\left(i \cdot i\right) \cdot \frac{1}{4}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \color{blue}{\sqrt{1.0} \cdot \sqrt{1.0}}}\]

    6. Applied difference-of-squares49.5

      \[\leadsto \frac{\left(i \cdot i\right) \cdot \frac{1}{4}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}\right)}}\]

    7. Applied associate-/r*48.4

      \[\leadsto \color{blue}{\frac{\frac{\left(i \cdot i\right) \cdot \frac{1}{4}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification37.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.0321689507084963 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{\sqrt{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\sqrt{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}} \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(i \cdot i\right) \cdot \frac{1}{4}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 
(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)))