Average Error: 16.4 → 3.5
Time: 4.9s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \le -1:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) - \left(-1\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \le -1:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) - \left(-1\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r102186 = beta;
        double r102187 = alpha;
        double r102188 = r102186 - r102187;
        double r102189 = r102187 + r102186;
        double r102190 = 2.0;
        double r102191 = r102189 + r102190;
        double r102192 = r102188 / r102191;
        double r102193 = 1.0;
        double r102194 = r102192 + r102193;
        double r102195 = r102194 / r102190;
        return r102195;
}
↓
double f(double alpha, double beta) {
        double r102196 = beta;
        double r102197 = alpha;
        double r102198 = r102196 - r102197;
        double r102199 = r102197 + r102196;
        double r102200 = 2.0;
        double r102201 = r102199 + r102200;
        double r102202 = r102198 / r102201;
        double r102203 = -1.0;
        bool r102204 = r102202 <= r102203;
        double r102205 = r102196 / r102201;
        double r102206 = 4.0;
        double r102207 = 1.0;
        double r102208 = 2.0;
        double r102209 = pow(r102197, r102208);
        double r102210 = r102207 / r102209;
        double r102211 = r102207 / r102197;
        double r102212 = 8.0;
        double r102213 = 3.0;
        double r102214 = pow(r102197, r102213);
        double r102215 = r102207 / r102214;
        double r102216 = r102212 * r102215;
        double r102217 = fma(r102200, r102211, r102216);
        double r102218 = -r102217;
        double r102219 = fma(r102206, r102210, r102218);
        double r102220 = r102205 - r102219;
        double r102221 = r102220 / r102200;
        double r102222 = r102197 / r102201;
        double r102223 = r102205 - r102222;
        double r102224 = 1.0;
        double r102225 = -r102224;
        double r102226 = r102223 - r102225;
        double r102227 = r102226 / r102200;
        double r102228 = r102204 ? r102221 : r102227;
        return r102228;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (/ (- beta alpha) (+ (+ alpha beta) 2.0)) < -1.0

    1. Initial program 60.6

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

    3. Applied div-sub60.6

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

    4. Applied associate-+l-58.6

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

    5. Taylor expanded around inf 11.5

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

    6. Simplified11.5

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

    if -1.0 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

    1. Initial program 0.6

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

    3. Applied div-sub0.6

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

    4. Applied associate-+l-0.6

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

    6. Applied sub-neg0.6

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

    7. Applied associate--r+0.6

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

  4. Final simplification3.5

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

Reproduce

herbie shell --seed 2020056 +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))