Average Error: 52.8 → 38.4
Time: 53.0s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 5.675791616953813 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{1.0} + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)} \cdot \frac{\sqrt{i \cdot \left(i + \left(\beta + \alpha\right)\right) + \alpha \cdot \beta} \cdot \frac{\sqrt{i \cdot \left(i + \left(\beta + \alpha\right)\right) + \alpha \cdot \beta}}{\left(\beta + \alpha\right) + 2 \cdot i}}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) - \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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 5.675791616953813 \cdot 10^{+89}:\\
\;\;\;\;\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{1.0} + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)} \cdot \frac{\sqrt{i \cdot \left(i + \left(\beta + \alpha\right)\right) + \alpha \cdot \beta} \cdot \frac{\sqrt{i \cdot \left(i + \left(\beta + \alpha\right)\right) + \alpha \cdot \beta}}{\left(\beta + \alpha\right) + 2 \cdot i}}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) - \sqrt{1.0}}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r4609386 = i;
        double r4609387 = alpha;
        double r4609388 = beta;
        double r4609389 = r4609387 + r4609388;
        double r4609390 = r4609389 + r4609386;
        double r4609391 = r4609386 * r4609390;
        double r4609392 = r4609388 * r4609387;
        double r4609393 = r4609392 + r4609391;
        double r4609394 = r4609391 * r4609393;
        double r4609395 = 2.0;
        double r4609396 = r4609395 * r4609386;
        double r4609397 = r4609389 + r4609396;
        double r4609398 = r4609397 * r4609397;
        double r4609399 = r4609394 / r4609398;
        double r4609400 = 1.0;
        double r4609401 = r4609398 - r4609400;
        double r4609402 = r4609399 / r4609401;
        return r4609402;
}
↓
double f(double alpha, double beta, double i) {
        double r4609403 = beta;
        double r4609404 = 5.675791616953813e+89;
        bool r4609405 = r4609403 <= r4609404;
        double r4609406 = i;
        double r4609407 = alpha;
        double r4609408 = r4609403 + r4609407;
        double r4609409 = r4609406 + r4609408;
        double r4609410 = r4609406 * r4609409;
        double r4609411 = 2.0;
        double r4609412 = r4609411 * r4609406;
        double r4609413 = r4609408 + r4609412;
        double r4609414 = r4609410 / r4609413;
        double r4609415 = 1.0;
        double r4609416 = sqrt(r4609415);
        double r4609417 = r4609416 + r4609413;
        double r4609418 = r4609414 / r4609417;
        double r4609419 = r4609407 * r4609403;
        double r4609420 = r4609410 + r4609419;
        double r4609421 = sqrt(r4609420);
        double r4609422 = r4609421 / r4609413;
        double r4609423 = r4609421 * r4609422;
        double r4609424 = r4609413 - r4609416;
        double r4609425 = r4609423 / r4609424;
        double r4609426 = r4609418 * r4609425;
        double r4609427 = 0.0;
        double r4609428 = r4609405 ? r4609426 : r4609427;
        return r4609428;
}

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 < 5.675791616953813e+89

    1. Initial program 50.3

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

    3. Applied add-sqr-sqrt50.3

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

    4. Applied difference-of-squares50.3

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

    5. Applied times-frac35.6

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

    6. Applied times-frac34.3

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

    8. Applied *-un-lft-identity34.3

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

    9. Applied add-sqr-sqrt34.4

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

    10. Applied times-frac34.3

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

    if 5.675791616953813e+89 < beta

    1. Initial program 61.2

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

    2. Taylor expanded around -inf 51.9

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

  4. Final simplification38.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 5.675791616953813 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{1.0} + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)} \cdot \frac{\sqrt{i \cdot \left(i + \left(\beta + \alpha\right)\right) + \alpha \cdot \beta} \cdot \frac{\sqrt{i \cdot \left(i + \left(\beta + \alpha\right)\right) + \alpha \cdot \beta}}{\left(\beta + \alpha\right) + 2 \cdot i}}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) - \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(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)))