Average Error: 24.5 → 11.5
Time: 53.8s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 5.320698544242960562668364882622880621523 \cdot 10^{193}:\\ \;\;\;\;\frac{1 + \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \left(\alpha + \beta\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{8}{\left(\alpha \cdot \alpha\right) \cdot \alpha} + \frac{2}{\alpha}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \end{array}\]
\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 5.320698544242960562668364882622880621523 \cdot 10^{193}:\\
\;\;\;\;\frac{1 + \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \left(\alpha + \beta\right)\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\frac{8}{\left(\alpha \cdot \alpha\right) \cdot \alpha} + \frac{2}{\alpha}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r10665407 = alpha;
        double r10665408 = beta;
        double r10665409 = r10665407 + r10665408;
        double r10665410 = r10665408 - r10665407;
        double r10665411 = r10665409 * r10665410;
        double r10665412 = 2.0;
        double r10665413 = i;
        double r10665414 = r10665412 * r10665413;
        double r10665415 = r10665409 + r10665414;
        double r10665416 = r10665411 / r10665415;
        double r10665417 = r10665415 + r10665412;
        double r10665418 = r10665416 / r10665417;
        double r10665419 = 1.0;
        double r10665420 = r10665418 + r10665419;
        double r10665421 = r10665420 / r10665412;
        return r10665421;
}
↓
double f(double alpha, double beta, double i) {
        double r10665422 = alpha;
        double r10665423 = 5.3206985442429606e+193;
        bool r10665424 = r10665422 <= r10665423;
        double r10665425 = 1.0;
        double r10665426 = 1.0;
        double r10665427 = beta;
        double r10665428 = r10665422 + r10665427;
        double r10665429 = 2.0;
        double r10665430 = i;
        double r10665431 = r10665429 * r10665430;
        double r10665432 = r10665428 + r10665431;
        double r10665433 = r10665432 + r10665429;
        double r10665434 = r10665426 / r10665433;
        double r10665435 = r10665427 - r10665422;
        double r10665436 = r10665435 / r10665432;
        double r10665437 = r10665436 * r10665428;
        double r10665438 = r10665434 * r10665437;
        double r10665439 = r10665425 + r10665438;
        double r10665440 = r10665439 / r10665429;
        double r10665441 = 8.0;
        double r10665442 = r10665422 * r10665422;
        double r10665443 = r10665442 * r10665422;
        double r10665444 = r10665441 / r10665443;
        double r10665445 = r10665429 / r10665422;
        double r10665446 = r10665444 + r10665445;
        double r10665447 = 4.0;
        double r10665448 = r10665447 / r10665442;
        double r10665449 = r10665446 - r10665448;
        double r10665450 = r10665449 / r10665429;
        double r10665451 = r10665424 ? r10665440 : r10665450;
        return r10665451;
}

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 < 5.3206985442429606e+193

    1. Initial program 18.8

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

    3. Applied *-un-lft-identity18.8

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

    4. Applied times-frac7.5

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

    5. Simplified7.5

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

    7. Applied div-inv7.4

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

    if 5.3206985442429606e+193 < alpha

    1. Initial program 64.0

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

    2. Taylor expanded around inf 39.2

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}}{2}\]

    3. Simplified39.2

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

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 5.320698544242960562668364882622880621523 \cdot 10^{193}:\\ \;\;\;\;\frac{1 + \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \left(\alpha + \beta\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{8}{\left(\alpha \cdot \alpha\right) \cdot \alpha} + \frac{2}{\alpha}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))