Average Error: 24.4 → 11.3
Time: 34.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.58003310586896986495188632827235655524 \cdot 10^{209}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\left(\alpha + \beta\right) \cdot \left(\frac{1}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}\right) + 1\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{8}{{\alpha}^{3}} + \frac{2}{\alpha}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \end{array}\]
\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 1.58003310586896986495188632827235655524 \cdot 10^{209}:\\
\;\;\;\;\frac{\sqrt[3]{{\left(\left(\alpha + \beta\right) \cdot \left(\frac{1}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}\right) + 1\right)}^{3}}}{2}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r87276 = alpha;
        double r87277 = beta;
        double r87278 = r87276 + r87277;
        double r87279 = r87277 - r87276;
        double r87280 = r87278 * r87279;
        double r87281 = 2.0;
        double r87282 = i;
        double r87283 = r87281 * r87282;
        double r87284 = r87278 + r87283;
        double r87285 = r87280 / r87284;
        double r87286 = r87284 + r87281;
        double r87287 = r87285 / r87286;
        double r87288 = 1.0;
        double r87289 = r87287 + r87288;
        double r87290 = r87289 / r87281;
        return r87290;
}
↓
double f(double alpha, double beta, double i) {
        double r87291 = alpha;
        double r87292 = 1.5800331058689699e+209;
        bool r87293 = r87291 <= r87292;
        double r87294 = beta;
        double r87295 = r87291 + r87294;
        double r87296 = 1.0;
        double r87297 = 2.0;
        double r87298 = i;
        double r87299 = r87297 * r87298;
        double r87300 = r87295 + r87299;
        double r87301 = r87300 + r87297;
        double r87302 = sqrt(r87301);
        double r87303 = r87296 / r87302;
        double r87304 = r87294 - r87291;
        double r87305 = r87304 / r87300;
        double r87306 = r87305 / r87302;
        double r87307 = r87303 * r87306;
        double r87308 = r87295 * r87307;
        double r87309 = 1.0;
        double r87310 = r87308 + r87309;
        double r87311 = 3.0;
        double r87312 = pow(r87310, r87311);
        double r87313 = cbrt(r87312);
        double r87314 = r87313 / r87297;
        double r87315 = 8.0;
        double r87316 = pow(r87291, r87311);
        double r87317 = r87315 / r87316;
        double r87318 = r87297 / r87291;
        double r87319 = r87317 + r87318;
        double r87320 = 4.0;
        double r87321 = r87291 * r87291;
        double r87322 = r87320 / r87321;
        double r87323 = r87319 - r87322;
        double r87324 = r87323 / r87297;
        double r87325 = r87293 ? r87314 : r87324;
        return r87325;
}

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 < 1.5800331058689699e+209

    1. Initial program 19.7

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

    3. Applied *-un-lft-identity19.7

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

    4. Applied *-un-lft-identity19.7

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

    5. Applied times-frac7.5

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

    6. Applied times-frac7.5

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

    7. Simplified7.5

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

    9. Applied add-cbrt-cube7.5

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

    10. Simplified7.5

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

    12. Applied add-sqr-sqrt7.5

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

    13. Applied *-un-lft-identity7.5

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

    14. Applied *-un-lft-identity7.5

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

    15. Applied times-frac7.5

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

    16. Applied times-frac7.5

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

    17. Simplified7.5

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

    if 1.5800331058689699e+209 < 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 43.3

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

    3. Simplified43.3

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

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.58003310586896986495188632827235655524 \cdot 10^{209}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\left(\alpha + \beta\right) \cdot \left(\frac{1}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}\right) + 1\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{8}{{\alpha}^{3}} + \frac{2}{\alpha}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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))