Average Error: 23.7 → 11.4
Time: 32.8s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 3.234277159331003 \cdot 10^{+135}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(1.0 + \frac{\beta + \alpha}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right) \cdot \left(1.0 + \frac{\beta + \alpha}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right)\right) \cdot \frac{{1.0}^{3} + {\left(\frac{\beta + \alpha}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right)}^{3}}{\left(1.0 \cdot 1.0 - \left(\frac{\beta + \alpha}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right) \cdot 1.0\right) + \left(\frac{\beta + \alpha}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right) \cdot \left(\frac{\beta + \alpha}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right)}}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \left(\frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]
\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) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 3.234277159331003 \cdot 10^{+135}:\\
\;\;\;\;\frac{\sqrt[3]{\left(\left(1.0 + \frac{\beta + \alpha}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right) \cdot \left(1.0 + \frac{\beta + \alpha}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right)\right) \cdot \frac{{1.0}^{3} + {\left(\frac{\beta + \alpha}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right)}^{3}}{\left(1.0 \cdot 1.0 - \left(\frac{\beta + \alpha}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right) \cdot 1.0\right) + \left(\frac{\beta + \alpha}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right) \cdot \left(\frac{\beta + \alpha}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right)}}}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{2.0}{\alpha} + \left(\frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r4193454 = alpha;
        double r4193455 = beta;
        double r4193456 = r4193454 + r4193455;
        double r4193457 = r4193455 - r4193454;
        double r4193458 = r4193456 * r4193457;
        double r4193459 = 2.0;
        double r4193460 = i;
        double r4193461 = r4193459 * r4193460;
        double r4193462 = r4193456 + r4193461;
        double r4193463 = r4193458 / r4193462;
        double r4193464 = 2.0;
        double r4193465 = r4193462 + r4193464;
        double r4193466 = r4193463 / r4193465;
        double r4193467 = 1.0;
        double r4193468 = r4193466 + r4193467;
        double r4193469 = r4193468 / r4193464;
        return r4193469;
}
↓
double f(double alpha, double beta, double i) {
        double r4193470 = alpha;
        double r4193471 = 3.234277159331003e+135;
        bool r4193472 = r4193470 <= r4193471;
        double r4193473 = 1.0;
        double r4193474 = beta;
        double r4193475 = r4193474 + r4193470;
        double r4193476 = 2.0;
        double r4193477 = 2.0;
        double r4193478 = i;
        double r4193479 = r4193477 * r4193478;
        double r4193480 = r4193475 + r4193479;
        double r4193481 = r4193476 + r4193480;
        double r4193482 = sqrt(r4193481);
        double r4193483 = r4193475 / r4193482;
        double r4193484 = r4193474 - r4193470;
        double r4193485 = r4193484 / r4193480;
        double r4193486 = r4193485 / r4193482;
        double r4193487 = r4193483 * r4193486;
        double r4193488 = r4193473 + r4193487;
        double r4193489 = r4193488 * r4193488;
        double r4193490 = 3.0;
        double r4193491 = pow(r4193473, r4193490);
        double r4193492 = pow(r4193487, r4193490);
        double r4193493 = r4193491 + r4193492;
        double r4193494 = r4193473 * r4193473;
        double r4193495 = r4193487 * r4193473;
        double r4193496 = r4193494 - r4193495;
        double r4193497 = r4193487 * r4193487;
        double r4193498 = r4193496 + r4193497;
        double r4193499 = r4193493 / r4193498;
        double r4193500 = r4193489 * r4193499;
        double r4193501 = cbrt(r4193500);
        double r4193502 = r4193501 / r4193476;
        double r4193503 = r4193476 / r4193470;
        double r4193504 = 8.0;
        double r4193505 = r4193504 / r4193470;
        double r4193506 = r4193470 * r4193470;
        double r4193507 = r4193505 / r4193506;
        double r4193508 = 4.0;
        double r4193509 = r4193508 / r4193506;
        double r4193510 = r4193507 - r4193509;
        double r4193511 = r4193503 + r4193510;
        double r4193512 = r4193511 / r4193476;
        double r4193513 = r4193472 ? r4193502 : r4193512;
        return r4193513;
}

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 < 3.234277159331003e+135

    1. Initial program 15.1

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt15.0

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} + 1.0}{2.0}\]

    4. Applied *-un-lft-identity15.0

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

    5. Applied times-frac4.6

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

    6. Applied times-frac4.6

      \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{1}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} + 1.0}{2.0}\]
    7. Using strategy rm

    8. Applied add-cbrt-cube4.6

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\frac{\frac{\alpha + \beta}{1}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0\right) \cdot \left(\frac{\frac{\alpha + \beta}{1}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0\right)\right) \cdot \left(\frac{\frac{\alpha + \beta}{1}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0\right)}}}{2.0}\]
    9. Using strategy rm

    10. Applied flip3-+4.6

      \[\leadsto \frac{\sqrt[3]{\left(\left(\frac{\frac{\alpha + \beta}{1}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0\right) \cdot \left(\frac{\frac{\alpha + \beta}{1}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0\right)\right) \cdot \color{blue}{\frac{{\left(\frac{\frac{\alpha + \beta}{1}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right)}^{3} + {1.0}^{3}}{\left(\frac{\frac{\alpha + \beta}{1}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right) \cdot \left(\frac{\frac{\alpha + \beta}{1}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right) + \left(1.0 \cdot 1.0 - \left(\frac{\frac{\alpha + \beta}{1}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right) \cdot 1.0\right)}}}}{2.0}\]

    if 3.234277159331003e+135 < alpha

    1. Initial program 60.9

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied add-log-exp60.9

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + \color{blue}{\log \left(e^{1.0}\right)}}{2.0}\]

    4. Applied add-log-exp60.9

      \[\leadsto \frac{\color{blue}{\log \left(e^{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right)} + \log \left(e^{1.0}\right)}{2.0}\]

    5. Applied sum-log60.9

      \[\leadsto \frac{\color{blue}{\log \left(e^{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot e^{1.0}\right)}}{2.0}\]

    6. Simplified45.7

      \[\leadsto \frac{\log \color{blue}{\left(e^{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{\left(2 \cdot i + 2.0\right) + \left(\alpha + \beta\right)} + 1.0}\right)}}{2.0}\]

    7. Taylor expanded around inf 40.7

      \[\leadsto \frac{\color{blue}{\left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right) - 4.0 \cdot \frac{1}{{\alpha}^{2}}}}{2.0}\]

    8. Simplified40.7

      \[\leadsto \frac{\color{blue}{\left(\frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{2.0}{\alpha}}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.234277159331003 \cdot 10^{+135}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(1.0 + \frac{\beta + \alpha}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right) \cdot \left(1.0 + \frac{\beta + \alpha}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right)\right) \cdot \frac{{1.0}^{3} + {\left(\frac{\beta + \alpha}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right)}^{3}}{\left(1.0 \cdot 1.0 - \left(\frac{\beta + \alpha}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right) \cdot 1.0\right) + \left(\frac{\beta + \alpha}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right) \cdot \left(\frac{\beta + \alpha}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right)}}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \left(\frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))