Average Error: 52.3 → 35.1
Time: 43.9s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 6.630518678732338 \cdot 10^{+208}:\\ \;\;\;\;\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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}\;\alpha \le 6.630518678732338 \cdot 10^{+208}:\\
\;\;\;\;\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot i} \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)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r5336465 = i;
        double r5336466 = alpha;
        double r5336467 = beta;
        double r5336468 = r5336466 + r5336467;
        double r5336469 = r5336468 + r5336465;
        double r5336470 = r5336465 * r5336469;
        double r5336471 = r5336467 * r5336466;
        double r5336472 = r5336471 + r5336470;
        double r5336473 = r5336470 * r5336472;
        double r5336474 = 2.0;
        double r5336475 = r5336474 * r5336465;
        double r5336476 = r5336468 + r5336475;
        double r5336477 = r5336476 * r5336476;
        double r5336478 = r5336473 / r5336477;
        double r5336479 = 1.0;
        double r5336480 = r5336477 - r5336479;
        double r5336481 = r5336478 / r5336480;
        return r5336481;
}
↓
double f(double alpha, double beta, double i) {
        double r5336482 = alpha;
        double r5336483 = 6.630518678732338e+208;
        bool r5336484 = r5336482 <= r5336483;
        double r5336485 = i;
        double r5336486 = beta;
        double r5336487 = r5336482 + r5336486;
        double r5336488 = r5336485 + r5336487;
        double r5336489 = r5336485 * r5336488;
        double r5336490 = r5336486 * r5336482;
        double r5336491 = r5336489 + r5336490;
        double r5336492 = 2.0;
        double r5336493 = r5336492 * r5336485;
        double r5336494 = r5336487 + r5336493;
        double r5336495 = r5336491 / r5336494;
        double r5336496 = r5336489 / r5336494;
        double r5336497 = 1.0;
        double r5336498 = sqrt(r5336497);
        double r5336499 = r5336498 + r5336494;
        double r5336500 = r5336496 / r5336499;
        double r5336501 = r5336495 * r5336500;
        double r5336502 = r5336494 - r5336498;
        double r5336503 = r5336501 / r5336502;
        double r5336504 = 0.0;
        double r5336505 = r5336484 ? r5336503 : r5336504;
        return r5336505;
}

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 < 6.630518678732338e+208

    1. Initial program 51.0

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

    3. Applied add-sqr-sqrt51.0

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

    4. Applied difference-of-squares51.0

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

    5. Applied times-frac36.2

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

    6. Applied times-frac33.9

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

    8. Applied associate-*r/33.9

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

    if 6.630518678732338e+208 < alpha

    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. Using strategy rm

    3. Applied add-sqr-sqrt62.6

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

    4. Applied difference-of-squares62.6

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

    5. Applied times-frac56.4

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

    6. Applied times-frac55.0

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

    7. Taylor expanded around -inf 44.5

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

  4. Final simplification35.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 6.630518678732338 \cdot 10^{+208}:\\ \;\;\;\;\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 
(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)))