Average Error: 23.8 → 11.3
Time: 35.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 9.65511625266840693009196306264539464884 \cdot 10^{212}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{\sqrt{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}{\sqrt{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} \cdot \left(\beta + \alpha\right) + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4}{\alpha \cdot \alpha}\right) + \frac{2}{\alpha}}{2}\\ \end{array}\]
\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 9.65511625266840693009196306264539464884 \cdot 10^{212}:\\
\;\;\;\;\frac{\frac{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{\sqrt{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}{\sqrt{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} \cdot \left(\beta + \alpha\right) + 1}{2}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r5003448 = alpha;
        double r5003449 = beta;
        double r5003450 = r5003448 + r5003449;
        double r5003451 = r5003449 - r5003448;
        double r5003452 = r5003450 * r5003451;
        double r5003453 = 2.0;
        double r5003454 = i;
        double r5003455 = r5003453 * r5003454;
        double r5003456 = r5003450 + r5003455;
        double r5003457 = r5003452 / r5003456;
        double r5003458 = r5003456 + r5003453;
        double r5003459 = r5003457 / r5003458;
        double r5003460 = 1.0;
        double r5003461 = r5003459 + r5003460;
        double r5003462 = r5003461 / r5003453;
        return r5003462;
}
↓
double f(double alpha, double beta, double i) {
        double r5003463 = alpha;
        double r5003464 = 9.655116252668407e+212;
        bool r5003465 = r5003463 <= r5003464;
        double r5003466 = beta;
        double r5003467 = r5003466 - r5003463;
        double r5003468 = i;
        double r5003469 = 2.0;
        double r5003470 = r5003468 * r5003469;
        double r5003471 = r5003466 + r5003463;
        double r5003472 = r5003470 + r5003471;
        double r5003473 = r5003467 / r5003472;
        double r5003474 = r5003469 + r5003472;
        double r5003475 = sqrt(r5003474);
        double r5003476 = r5003473 / r5003475;
        double r5003477 = r5003476 / r5003475;
        double r5003478 = r5003477 * r5003471;
        double r5003479 = 1.0;
        double r5003480 = r5003478 + r5003479;
        double r5003481 = r5003480 / r5003469;
        double r5003482 = 8.0;
        double r5003483 = r5003463 * r5003463;
        double r5003484 = r5003463 * r5003483;
        double r5003485 = r5003482 / r5003484;
        double r5003486 = 4.0;
        double r5003487 = r5003486 / r5003483;
        double r5003488 = r5003485 - r5003487;
        double r5003489 = r5003469 / r5003463;
        double r5003490 = r5003488 + r5003489;
        double r5003491 = r5003490 / r5003469;
        double r5003492 = r5003465 ? r5003481 : r5003491;
        return r5003492;
}

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 < 9.655116252668407e+212

    1. Initial program 19.4

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

    3. Applied *-un-lft-identity19.4

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

    4. Applied *-un-lft-identity19.4

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

    5. Applied times-frac7.9

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

    6. Applied times-frac7.9

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

    7. Simplified7.9

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

    9. Applied add-sqr-sqrt7.9

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

    10. Applied associate-/r*7.9

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

    if 9.655116252668407e+212 < 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 41.9

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

    3. Simplified41.9

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

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 9.65511625266840693009196306264539464884 \cdot 10^{212}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{\sqrt{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}{\sqrt{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} \cdot \left(\beta + \alpha\right) + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4}{\alpha \cdot \alpha}\right) + \frac{2}{\alpha}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(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))