Average Error: 15.7 → 3.0
Time: 22.8s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999999999999637:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \mathsf{fma}\left(\alpha, \frac{1}{\left(\alpha + \beta\right) + 2.0}, -1.0\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999999999999637:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \mathsf{fma}\left(\alpha, \frac{1}{\left(\alpha + \beta\right) + 2.0}, -1.0\right)}{2.0}\\

\end{array}
double f(double alpha, double beta) {
        double r1597211 = beta;
        double r1597212 = alpha;
        double r1597213 = r1597211 - r1597212;
        double r1597214 = r1597212 + r1597211;
        double r1597215 = 2.0;
        double r1597216 = r1597214 + r1597215;
        double r1597217 = r1597213 / r1597216;
        double r1597218 = 1.0;
        double r1597219 = r1597217 + r1597218;
        double r1597220 = r1597219 / r1597215;
        return r1597220;
}
↓
double f(double alpha, double beta) {
        double r1597221 = beta;
        double r1597222 = alpha;
        double r1597223 = r1597221 - r1597222;
        double r1597224 = r1597222 + r1597221;
        double r1597225 = 2.0;
        double r1597226 = r1597224 + r1597225;
        double r1597227 = r1597223 / r1597226;
        double r1597228 = -0.9999999999999637;
        bool r1597229 = r1597227 <= r1597228;
        double r1597230 = r1597221 / r1597226;
        double r1597231 = 4.0;
        double r1597232 = r1597222 * r1597222;
        double r1597233 = r1597231 / r1597232;
        double r1597234 = 8.0;
        double r1597235 = r1597232 * r1597222;
        double r1597236 = r1597234 / r1597235;
        double r1597237 = r1597233 - r1597236;
        double r1597238 = r1597225 / r1597222;
        double r1597239 = r1597237 - r1597238;
        double r1597240 = r1597230 - r1597239;
        double r1597241 = r1597240 / r1597225;
        double r1597242 = 1.0;
        double r1597243 = r1597242 / r1597226;
        double r1597244 = 1.0;
        double r1597245 = -r1597244;
        double r1597246 = fma(r1597222, r1597243, r1597245);
        double r1597247 = r1597230 - r1597246;
        double r1597248 = r1597247 / r1597225;
        double r1597249 = r1597229 ? r1597241 : r1597248;
        return r1597249;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (/ (- beta alpha) (+ (+ alpha beta) 2.0)) < -0.9999999999999637

    1. Initial program 60.4

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

    3. Applied div-sub60.4

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

    4. Applied associate-+l-58.7

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]
    5. Using strategy rm

    6. Applied add-log-exp58.7

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

    7. Taylor expanded around inf 10.6

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

    8. Simplified10.6

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

    if -0.9999999999999637 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

    1. Initial program 0.4

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

    3. Applied div-sub0.4

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

    4. Applied associate-+l-0.4

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]
    5. Using strategy rm

    6. Applied div-inv0.4

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

    7. Applied fma-neg0.4

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\mathsf{fma}\left(\alpha, \frac{1}{\left(\alpha + \beta\right) + 2.0}, -1.0\right)}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999999999999637:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \mathsf{fma}\left(\alpha, \frac{1}{\left(\alpha + \beta\right) + 2.0}, -1.0\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))