Average Error: 16.3 → 3.1
Time: 25.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999999999973689:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2.0} \cdot {\left(\left(\alpha + \beta\right) + 2.0\right)}^{\frac{1}{3}}}, \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2.0}}, -\left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right)\right)\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}{2.0}\\ \end{array}\]
\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 -0.9999999999973689:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2.0} \cdot {\left(\left(\alpha + \beta\right) + 2.0\right)}^{\frac{1}{3}}}, \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2.0}}, -\left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right)\right)\right)}{2.0}\\

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

\end{array}
double f(double alpha, double beta) {
        double r4016074 = beta;
        double r4016075 = alpha;
        double r4016076 = r4016074 - r4016075;
        double r4016077 = r4016075 + r4016074;
        double r4016078 = 2.0;
        double r4016079 = r4016077 + r4016078;
        double r4016080 = r4016076 / r4016079;
        double r4016081 = 1.0;
        double r4016082 = r4016080 + r4016081;
        double r4016083 = r4016082 / r4016078;
        return r4016083;
}
↓
double f(double alpha, double beta) {
        double r4016084 = beta;
        double r4016085 = alpha;
        double r4016086 = r4016084 - r4016085;
        double r4016087 = r4016085 + r4016084;
        double r4016088 = 2.0;
        double r4016089 = r4016087 + r4016088;
        double r4016090 = r4016086 / r4016089;
        double r4016091 = -0.9999999999973689;
        bool r4016092 = r4016090 <= r4016091;
        double r4016093 = cbrt(r4016084);
        double r4016094 = r4016093 * r4016093;
        double r4016095 = cbrt(r4016089);
        double r4016096 = 0.3333333333333333;
        double r4016097 = pow(r4016089, r4016096);
        double r4016098 = r4016095 * r4016097;
        double r4016099 = r4016094 / r4016098;
        double r4016100 = r4016093 / r4016095;
        double r4016101 = 4.0;
        double r4016102 = r4016085 * r4016085;
        double r4016103 = r4016101 / r4016102;
        double r4016104 = r4016088 / r4016085;
        double r4016105 = 8.0;
        double r4016106 = r4016105 / r4016085;
        double r4016107 = r4016106 / r4016102;
        double r4016108 = r4016104 + r4016107;
        double r4016109 = r4016103 - r4016108;
        double r4016110 = -r4016109;
        double r4016111 = fma(r4016099, r4016100, r4016110);
        double r4016112 = r4016111 / r4016088;
        double r4016113 = 1.0;
        double r4016114 = r4016113 + r4016090;
        double r4016115 = r4016114 / r4016088;
        double r4016116 = r4016092 ? r4016112 : r4016115;
        return r4016116;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (/ (- beta alpha) (+ (+ alpha beta) 2.0)) < -0.9999999999973689

    1. Initial program 60.3

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

    3. Applied div-sub60.3

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

    4. Applied associate-+l-58.5

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

    6. Applied add-cube-cbrt58.5

      \[\leadsto \frac{\frac{\beta}{\color{blue}{\left(\sqrt[3]{\left(\alpha + \beta\right) + 2.0} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2.0}\right) \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2.0}}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]

    7. Applied add-cube-cbrt58.5

      \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}\right) \cdot \sqrt[3]{\beta}}}{\left(\sqrt[3]{\left(\alpha + \beta\right) + 2.0} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2.0}\right) \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2.0}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]

    8. Applied times-frac58.5

      \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2.0} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2.0}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2.0}}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]

    9. Applied fma-neg58.5

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2.0} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2.0}}, \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2.0}}, -\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right)}}{2.0}\]
    10. Using strategy rm

    11. Applied pow1/358.5

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\color{blue}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{\frac{1}{3}}} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2.0}}, \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2.0}}, -\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right)}{2.0}\]

    12. Taylor expanded around inf 10.9

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

    13. Simplified10.9

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

    if -0.9999999999973689 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

    1. Initial program 0.3

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999999999973689:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2.0} \cdot {\left(\left(\alpha + \beta\right) + 2.0\right)}^{\frac{1}{3}}}, \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2.0}}, -\left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right)\right)\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +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))