Average Error: 24.0 → 11.2
Time: 16.7s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.0395626917288666 \cdot 10^{182}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\mathsf{fma}\left(\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}, \frac{\frac{\sqrt[3]{\beta - \alpha}}{\frac{\left|\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right|}{\sqrt[3]{\beta - \alpha}}} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}, 1\right)\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \end{array}\]
\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 1.0395626917288666 \cdot 10^{182}:\\
\;\;\;\;\frac{\sqrt[3]{{\left(\mathsf{fma}\left(\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}, \frac{\frac{\sqrt[3]{\beta - \alpha}}{\frac{\left|\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right|}{\sqrt[3]{\beta - \alpha}}} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}, 1\right)\right)}^{3}}}{2}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r156242 = alpha;
        double r156243 = beta;
        double r156244 = r156242 + r156243;
        double r156245 = r156243 - r156242;
        double r156246 = r156244 * r156245;
        double r156247 = 2.0;
        double r156248 = i;
        double r156249 = r156247 * r156248;
        double r156250 = r156244 + r156249;
        double r156251 = r156246 / r156250;
        double r156252 = r156250 + r156247;
        double r156253 = r156251 / r156252;
        double r156254 = 1.0;
        double r156255 = r156253 + r156254;
        double r156256 = r156255 / r156247;
        return r156256;
}
↓
double f(double alpha, double beta, double i) {
        double r156257 = alpha;
        double r156258 = 1.0395626917288666e+182;
        bool r156259 = r156257 <= r156258;
        double r156260 = beta;
        double r156261 = r156257 + r156260;
        double r156262 = 2.0;
        double r156263 = i;
        double r156264 = r156262 * r156263;
        double r156265 = r156261 + r156264;
        double r156266 = r156265 + r156262;
        double r156267 = sqrt(r156266);
        double r156268 = r156261 / r156267;
        double r156269 = r156260 - r156257;
        double r156270 = cbrt(r156269);
        double r156271 = cbrt(r156266);
        double r156272 = fabs(r156271);
        double r156273 = r156272 / r156270;
        double r156274 = r156270 / r156273;
        double r156275 = sqrt(r156271);
        double r156276 = r156270 / r156275;
        double r156277 = r156274 * r156276;
        double r156278 = fma(r156263, r156262, r156261);
        double r156279 = r156277 / r156278;
        double r156280 = 1.0;
        double r156281 = fma(r156268, r156279, r156280);
        double r156282 = 3.0;
        double r156283 = pow(r156281, r156282);
        double r156284 = cbrt(r156283);
        double r156285 = r156284 / r156262;
        double r156286 = 1.0;
        double r156287 = r156286 / r156257;
        double r156288 = 8.0;
        double r156289 = pow(r156257, r156282);
        double r156290 = r156286 / r156289;
        double r156291 = r156288 * r156290;
        double r156292 = 4.0;
        double r156293 = 2.0;
        double r156294 = pow(r156257, r156293);
        double r156295 = r156286 / r156294;
        double r156296 = r156292 * r156295;
        double r156297 = r156291 - r156296;
        double r156298 = fma(r156262, r156287, r156297);
        double r156299 = r156298 / r156262;
        double r156300 = r156259 ? r156285 : r156299;
        return r156300;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 1.0395626917288666e+182

    1. Initial program 17.7

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

    3. Applied add-sqr-sqrt17.7

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

    4. Applied *-un-lft-identity17.7

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

    5. Applied times-frac6.3

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

    6. Applied times-frac6.3

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

    7. Simplified6.3

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

    8. Simplified6.3

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

    10. Applied add-cube-cbrt6.3

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

    11. Applied sqrt-prod6.3

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

    12. Applied add-cube-cbrt6.4

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

    13. Applied times-frac6.4

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

    14. Simplified6.4

      \[\leadsto \frac{\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\color{blue}{\frac{\sqrt[3]{\beta - \alpha}}{\frac{\left|\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right|}{\sqrt[3]{\beta - \alpha}}}} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} + 1}{2}\]
    15. Using strategy rm

    16. Applied add-cbrt-cube6.3

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

    17. Simplified6.3

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

    if 1.0395626917288666e+182 < 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

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

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.0395626917288666 \cdot 10^{182}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\mathsf{fma}\left(\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}, \frac{\frac{\sqrt[3]{\beta - \alpha}}{\frac{\left|\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right|}{\sqrt[3]{\beta - \alpha}}} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}, 1\right)\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(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))