Average Error: 16.4 → 3.0
Time: 5.6s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \le -0.9999999998847824977943332669383380562067:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right) \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(\alpha, \frac{1}{\left(\alpha + \beta\right) + 2}, -1\right)}{2}\\ \end{array}\]
\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 -0.9999999998847824977943332669383380562067:\\
\;\;\;\;\frac{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right) \cdot \sqrt[3]{\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{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(\alpha, \frac{1}{\left(\alpha + \beta\right) + 2}, -1\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r113095 = beta;
        double r113096 = alpha;
        double r113097 = r113095 - r113096;
        double r113098 = r113096 + r113095;
        double r113099 = 2.0;
        double r113100 = r113098 + r113099;
        double r113101 = r113097 / r113100;
        double r113102 = 1.0;
        double r113103 = r113101 + r113102;
        double r113104 = r113103 / r113099;
        return r113104;
}
↓
double f(double alpha, double beta) {
        double r113105 = beta;
        double r113106 = alpha;
        double r113107 = r113105 - r113106;
        double r113108 = r113106 + r113105;
        double r113109 = 2.0;
        double r113110 = r113108 + r113109;
        double r113111 = r113107 / r113110;
        double r113112 = -0.9999999998847825;
        bool r113113 = r113111 <= r113112;
        double r113114 = r113105 / r113110;
        double r113115 = cbrt(r113114);
        double r113116 = r113115 * r113115;
        double r113117 = r113116 * r113115;
        double r113118 = 4.0;
        double r113119 = 1.0;
        double r113120 = 2.0;
        double r113121 = pow(r113106, r113120);
        double r113122 = r113119 / r113121;
        double r113123 = r113119 / r113106;
        double r113124 = 8.0;
        double r113125 = 3.0;
        double r113126 = pow(r113106, r113125);
        double r113127 = r113119 / r113126;
        double r113128 = r113124 * r113127;
        double r113129 = fma(r113109, r113123, r113128);
        double r113130 = -r113129;
        double r113131 = fma(r113118, r113122, r113130);
        double r113132 = r113117 - r113131;
        double r113133 = r113132 / r113109;
        double r113134 = r113119 / r113110;
        double r113135 = 1.0;
        double r113136 = -r113135;
        double r113137 = fma(r113106, r113134, r113136);
        double r113138 = r113114 - r113137;
        double r113139 = r113138 / r113109;
        double r113140 = r113113 ? r113133 : r113139;
        return r113140;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (/ (- beta alpha) (+ (+ alpha beta) 2.0)) < -0.9999999998847825

    1. Initial program 60.2

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

    3. Applied div-sub60.2

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

    4. Applied associate-+l-58.4

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

    6. Applied add-cube-cbrt58.4

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

    7. Taylor expanded around inf 10.5

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

    8. Simplified10.5

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

    if -0.9999999998847825 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

    1. Initial program 0.2

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

    3. Applied div-sub0.2

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

    4. Applied associate-+l-0.2

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

    6. Applied div-inv0.2

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

    7. Applied fma-neg0.2

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

  4. Final simplification3.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \le -0.9999999998847824977943332669383380562067:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right) \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(\alpha, \frac{1}{\left(\alpha + \beta\right) + 2}, -1\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +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))