Average Error: 23.3 → 11.5
Time: 10.9s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.755371486194466 \cdot 10^{248}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{\sqrt[3]{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}}}{\sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2} \cdot \sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}, \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{8}{{\alpha}^{3}} + \left(\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]
\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 1.755371486194466 \cdot 10^{248}:\\
\;\;\;\;\frac{\sqrt[3]{{\left(\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{\sqrt[3]{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}}}{\sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2} \cdot \sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}, \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)\right)}^{3}}}{2}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r165323 = alpha;
        double r165324 = beta;
        double r165325 = r165323 + r165324;
        double r165326 = r165324 - r165323;
        double r165327 = r165325 * r165326;
        double r165328 = 2.0;
        double r165329 = i;
        double r165330 = r165328 * r165329;
        double r165331 = r165325 + r165330;
        double r165332 = r165327 / r165331;
        double r165333 = r165331 + r165328;
        double r165334 = r165332 / r165333;
        double r165335 = 1.0;
        double r165336 = r165334 + r165335;
        double r165337 = r165336 / r165328;
        return r165337;
}
↓
double f(double alpha, double beta, double i) {
        double r165338 = alpha;
        double r165339 = 1.7553714861944655e+248;
        bool r165340 = r165338 <= r165339;
        double r165341 = beta;
        double r165342 = r165338 + r165341;
        double r165343 = 2.0;
        double r165344 = i;
        double r165345 = fma(r165343, r165344, r165342);
        double r165346 = r165345 + r165343;
        double r165347 = sqrt(r165346);
        double r165348 = cbrt(r165347);
        double r165349 = r165348 * r165348;
        double r165350 = r165342 / r165349;
        double r165351 = cbrt(r165346);
        double r165352 = r165351 * r165351;
        double r165353 = r165350 / r165352;
        double r165354 = r165341 - r165338;
        double r165355 = r165354 / r165345;
        double r165356 = 1.0;
        double r165357 = fma(r165353, r165355, r165356);
        double r165358 = 3.0;
        double r165359 = pow(r165357, r165358);
        double r165360 = cbrt(r165359);
        double r165361 = r165360 / r165343;
        double r165362 = 8.0;
        double r165363 = pow(r165338, r165358);
        double r165364 = r165362 / r165363;
        double r165365 = r165343 / r165338;
        double r165366 = 4.0;
        double r165367 = r165338 * r165338;
        double r165368 = r165366 / r165367;
        double r165369 = r165365 - r165368;
        double r165370 = r165364 + r165369;
        double r165371 = r165370 / r165343;
        double r165372 = r165340 ? r165361 : r165371;
        return r165372;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 1.7553714861944655e+248

    1. Initial program 20.6

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

    2. Simplified9.5

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

    4. Applied add-cube-cbrt9.6

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

    5. Applied *-un-lft-identity9.6

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

    6. Applied times-frac9.6

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

    8. Applied add-sqr-sqrt9.6

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

    9. Applied cbrt-prod9.6

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

    11. Applied add-cbrt-cube9.5

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

    12. Simplified9.6

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

    if 1.7553714861944655e+248 < 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. Simplified54.3

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

    3. Taylor expanded around inf 40.5

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

    4. Simplified40.5

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

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.755371486194466 \cdot 10^{248}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{\sqrt[3]{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}}}{\sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2} \cdot \sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}, \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{8}{{\alpha}^{3}} + \left(\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))