Average Error: 16.6 → 6.9
Time: 20.0s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 9249179198057380398250710513657839616:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(\alpha, \frac{1}{\left(\alpha + \beta\right) + 2}, -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}} \cdot \sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}}}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 9249179198057380398250710513657839616:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(\alpha, \frac{1}{\left(\alpha + \beta\right) + 2}, -1\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{1}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}} \cdot \sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}}}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r67201 = beta;
        double r67202 = alpha;
        double r67203 = r67201 - r67202;
        double r67204 = r67202 + r67201;
        double r67205 = 2.0;
        double r67206 = r67204 + r67205;
        double r67207 = r67203 / r67206;
        double r67208 = 1.0;
        double r67209 = r67207 + r67208;
        double r67210 = r67209 / r67205;
        return r67210;
}
↓
double f(double alpha, double beta) {
        double r67211 = alpha;
        double r67212 = 9.24917919805738e+36;
        bool r67213 = r67211 <= r67212;
        double r67214 = beta;
        double r67215 = r67211 + r67214;
        double r67216 = 2.0;
        double r67217 = r67215 + r67216;
        double r67218 = r67214 / r67217;
        double r67219 = 1.0;
        double r67220 = r67219 / r67217;
        double r67221 = 1.0;
        double r67222 = -r67221;
        double r67223 = fma(r67211, r67220, r67222);
        double r67224 = r67218 - r67223;
        double r67225 = r67224 / r67216;
        double r67226 = r67217 / r67214;
        double r67227 = cbrt(r67226);
        double r67228 = r67227 * r67227;
        double r67229 = r67219 / r67228;
        double r67230 = r67229 / r67227;
        double r67231 = 4.0;
        double r67232 = 2.0;
        double r67233 = pow(r67211, r67232);
        double r67234 = r67231 / r67233;
        double r67235 = r67216 / r67211;
        double r67236 = r67234 - r67235;
        double r67237 = 8.0;
        double r67238 = 3.0;
        double r67239 = pow(r67211, r67238);
        double r67240 = r67237 / r67239;
        double r67241 = r67236 - r67240;
        double r67242 = r67230 - r67241;
        double r67243 = r67242 / r67216;
        double r67244 = r67213 ? r67225 : r67243;
        return r67244;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 9.24917919805738e+36

    1. Initial program 2.0

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

    3. Applied div-sub2.0

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

    4. Applied associate-+l-2.0

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

    6. Applied div-inv2.0

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

    7. Applied fma-neg2.0

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

    if 9.24917919805738e+36 < alpha

    1. Initial program 50.5

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

    3. Applied div-sub50.5

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

    4. Applied associate-+l-49.0

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

    6. Applied clear-num49.0

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

    8. Applied add-cube-cbrt49.0

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

    9. Applied associate-/r*49.0

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

    10. Taylor expanded around inf 18.3

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

    11. Simplified18.3

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

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 9249179198057380398250710513657839616:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(\alpha, \frac{1}{\left(\alpha + \beta\right) + 2}, -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}} \cdot \sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}}}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))