Average Error: 24.3 → 12.3
Time: 10.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 7.81955504868517278 \cdot 10^{64}:\\ \;\;\;\;\frac{\frac{\frac{{\left(\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right)}^{1}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right) + \frac{2}{\alpha}}{2}\\ \end{array}\]
\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 7.81955504868517278 \cdot 10^{64}:\\
\;\;\;\;\frac{\frac{\frac{{\left(\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right)}^{1}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right) + \frac{2}{\alpha}}{2}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r134375 = alpha;
        double r134376 = beta;
        double r134377 = r134375 + r134376;
        double r134378 = r134376 - r134375;
        double r134379 = r134377 * r134378;
        double r134380 = 2.0;
        double r134381 = i;
        double r134382 = r134380 * r134381;
        double r134383 = r134377 + r134382;
        double r134384 = r134379 / r134383;
        double r134385 = r134383 + r134380;
        double r134386 = r134384 / r134385;
        double r134387 = 1.0;
        double r134388 = r134386 + r134387;
        double r134389 = r134388 / r134380;
        return r134389;
}
↓
double f(double alpha, double beta, double i) {
        double r134390 = alpha;
        double r134391 = 7.819555048685173e+64;
        bool r134392 = r134390 <= r134391;
        double r134393 = beta;
        double r134394 = r134390 + r134393;
        double r134395 = r134393 - r134390;
        double r134396 = 2.0;
        double r134397 = i;
        double r134398 = r134396 * r134397;
        double r134399 = r134394 + r134398;
        double r134400 = r134395 / r134399;
        double r134401 = r134394 * r134400;
        double r134402 = 1.0;
        double r134403 = pow(r134401, r134402);
        double r134404 = r134399 + r134396;
        double r134405 = sqrt(r134404);
        double r134406 = r134403 / r134405;
        double r134407 = r134406 / r134405;
        double r134408 = 1.0;
        double r134409 = r134407 + r134408;
        double r134410 = r134409 / r134396;
        double r134411 = 8.0;
        double r134412 = 3.0;
        double r134413 = pow(r134390, r134412);
        double r134414 = r134402 / r134413;
        double r134415 = r134411 * r134414;
        double r134416 = 4.0;
        double r134417 = 2.0;
        double r134418 = pow(r134390, r134417);
        double r134419 = r134402 / r134418;
        double r134420 = r134416 * r134419;
        double r134421 = r134415 - r134420;
        double r134422 = r134396 / r134390;
        double r134423 = r134421 + r134422;
        double r134424 = r134423 / r134396;
        double r134425 = r134392 ? r134410 : r134424;
        return r134425;
}

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 < 7.819555048685173e+64

    1. Initial program 12.8

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

    3. Applied *-un-lft-identity12.8

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

    4. Applied times-frac1.8

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

    5. Simplified1.8

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

    7. Applied add-sqr-sqrt1.8

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

    8. Applied associate-/r*1.8

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

    10. Applied pow11.8

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

    11. Applied pow11.8

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

    12. Applied pow-prod-down1.8

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

    if 7.819555048685173e+64 < alpha

    1. Initial program 55.7

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

    3. Applied *-un-lft-identity55.7

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

    4. Applied times-frac41.4

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

    5. Simplified41.4

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

    6. Taylor expanded around inf 40.8

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

    7. Simplified40.8

      \[\leadsto \frac{\color{blue}{\left(8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right) + \frac{2}{\alpha}}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification12.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 7.81955504868517278 \cdot 10^{64}:\\ \;\;\;\;\frac{\frac{\frac{{\left(\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right)}^{1}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right) + \frac{2}{\alpha}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2)) 1) 2))