Average Error: 53.0 → 36.5
Time: 42.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 5.313821326912652 \cdot 10^{+172}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right) + \alpha \cdot \beta}{\left(\beta + \alpha\right) + 2 \cdot i}}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) - \sqrt{1.0}}} \cdot \left(\sqrt[3]{\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right) + \alpha \cdot \beta}{\left(\beta + \alpha\right) + 2 \cdot i}}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) - \sqrt{1.0}}} \cdot \sqrt[3]{\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right) + \alpha \cdot \beta}{\left(\beta + \alpha\right) + 2 \cdot i}}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) - \sqrt{1.0}}}\right)\right) \cdot \frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{1.0} + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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 5.313821326912652 \cdot 10^{+172}:\\
\;\;\;\;\left(\sqrt[3]{\frac{\frac{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}}} \cdot \left(\sqrt[3]{\frac{\frac{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}}} \cdot \sqrt[3]{\frac{\frac{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}}}\right)\right) \cdot \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)}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r3632381 = i;
        double r3632382 = alpha;
        double r3632383 = beta;
        double r3632384 = r3632382 + r3632383;
        double r3632385 = r3632384 + r3632381;
        double r3632386 = r3632381 * r3632385;
        double r3632387 = r3632383 * r3632382;
        double r3632388 = r3632387 + r3632386;
        double r3632389 = r3632386 * r3632388;
        double r3632390 = 2.0;
        double r3632391 = r3632390 * r3632381;
        double r3632392 = r3632384 + r3632391;
        double r3632393 = r3632392 * r3632392;
        double r3632394 = r3632389 / r3632393;
        double r3632395 = 1.0;
        double r3632396 = r3632393 - r3632395;
        double r3632397 = r3632394 / r3632396;
        return r3632397;
}
↓
double f(double alpha, double beta, double i) {
        double r3632398 = beta;
        double r3632399 = 5.313821326912652e+172;
        bool r3632400 = r3632398 <= r3632399;
        double r3632401 = i;
        double r3632402 = alpha;
        double r3632403 = r3632398 + r3632402;
        double r3632404 = r3632401 + r3632403;
        double r3632405 = r3632401 * r3632404;
        double r3632406 = r3632402 * r3632398;
        double r3632407 = r3632405 + r3632406;
        double r3632408 = 2.0;
        double r3632409 = r3632408 * r3632401;
        double r3632410 = r3632403 + r3632409;
        double r3632411 = r3632407 / r3632410;
        double r3632412 = 1.0;
        double r3632413 = sqrt(r3632412);
        double r3632414 = r3632410 - r3632413;
        double r3632415 = r3632411 / r3632414;
        double r3632416 = cbrt(r3632415);
        double r3632417 = r3632416 * r3632416;
        double r3632418 = r3632416 * r3632417;
        double r3632419 = r3632405 / r3632410;
        double r3632420 = r3632413 + r3632410;
        double r3632421 = r3632419 / r3632420;
        double r3632422 = r3632418 * r3632421;
        double r3632423 = 0.0;
        double r3632424 = r3632400 ? r3632422 : r3632423;
        return r3632424;
}

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 < 5.313821326912652e+172

    1. Initial program 51.4

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

    3. Applied add-sqr-sqrt51.4

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

    4. Applied difference-of-squares51.4

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

    5. Applied times-frac36.6

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

    6. Applied times-frac34.8

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

    8. Applied add-cube-cbrt34.9

      \[\leadsto \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}} \cdot \sqrt[3]{\frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}}\right) \cdot \sqrt[3]{\frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}}\right)}\]

    if 5.313821326912652e+172 < beta

    1. Initial program 62.6

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

    2. Taylor expanded around -inf 45.6

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

  4. Final simplification36.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 5.313821326912652 \cdot 10^{+172}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right) + \alpha \cdot \beta}{\left(\beta + \alpha\right) + 2 \cdot i}}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) - \sqrt{1.0}}} \cdot \left(\sqrt[3]{\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right) + \alpha \cdot \beta}{\left(\beta + \alpha\right) + 2 \cdot i}}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) - \sqrt{1.0}}} \cdot \sqrt[3]{\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right) + \alpha \cdot \beta}{\left(\beta + \alpha\right) + 2 \cdot i}}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) - \sqrt{1.0}}}\right)\right) \cdot \frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{1.0} + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019137 
(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)))