Average Error: 16.2 → 14.6
Time: 20.8s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 429068.6689692988:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{{2.0}^{3} + {\left(\beta + \alpha\right)}^{3}}, \left(2.0 \cdot 2.0 - 2.0 \cdot \left(\beta + \alpha\right)\right) + \left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right), 1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2.0} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 429068.6689692988:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{{2.0}^{3} + {\left(\beta + \alpha\right)}^{3}}, \left(2.0 \cdot 2.0 - 2.0 \cdot \left(\beta + \alpha\right)\right) + \left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right), 1.0\right)}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2.0} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right)}{2.0}\\

\end{array}
double f(double alpha, double beta) {
        double r2719046 = beta;
        double r2719047 = alpha;
        double r2719048 = r2719046 - r2719047;
        double r2719049 = r2719047 + r2719046;
        double r2719050 = 2.0;
        double r2719051 = r2719049 + r2719050;
        double r2719052 = r2719048 / r2719051;
        double r2719053 = 1.0;
        double r2719054 = r2719052 + r2719053;
        double r2719055 = r2719054 / r2719050;
        return r2719055;
}
↓
double f(double alpha, double beta) {
        double r2719056 = alpha;
        double r2719057 = 429068.6689692988;
        bool r2719058 = r2719056 <= r2719057;
        double r2719059 = beta;
        double r2719060 = r2719059 - r2719056;
        double r2719061 = 2.0;
        double r2719062 = 3.0;
        double r2719063 = pow(r2719061, r2719062);
        double r2719064 = r2719059 + r2719056;
        double r2719065 = pow(r2719064, r2719062);
        double r2719066 = r2719063 + r2719065;
        double r2719067 = r2719060 / r2719066;
        double r2719068 = r2719061 * r2719061;
        double r2719069 = r2719061 * r2719064;
        double r2719070 = r2719068 - r2719069;
        double r2719071 = r2719064 * r2719064;
        double r2719072 = r2719070 + r2719071;
        double r2719073 = 1.0;
        double r2719074 = fma(r2719067, r2719072, r2719073);
        double r2719075 = r2719074 / r2719061;
        double r2719076 = r2719064 + r2719061;
        double r2719077 = r2719059 / r2719076;
        double r2719078 = 4.0;
        double r2719079 = r2719056 * r2719056;
        double r2719080 = r2719078 / r2719079;
        double r2719081 = r2719061 / r2719056;
        double r2719082 = r2719080 - r2719081;
        double r2719083 = 8.0;
        double r2719084 = r2719079 * r2719056;
        double r2719085 = r2719083 / r2719084;
        double r2719086 = r2719082 - r2719085;
        double r2719087 = r2719077 - r2719086;
        double r2719088 = r2719087 / r2719061;
        double r2719089 = r2719058 ? r2719075 : r2719088;
        return r2719089;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 429068.6689692988

    1. Initial program 0.0

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

    3. Applied flip3-+13.1

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

    4. Applied associate-/r/13.1

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

    5. Applied fma-def13.1

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

    if 429068.6689692988 < alpha

    1. Initial program 49.5

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

    3. Applied div-sub49.5

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

    4. Applied associate-+l-48.0

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

    5. Taylor expanded around -inf 17.8

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

    6. Simplified17.8

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right)}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification14.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 429068.6689692988:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{{2.0}^{3} + {\left(\beta + \alpha\right)}^{3}}, \left(2.0 \cdot 2.0 - 2.0 \cdot \left(\beta + \alpha\right)\right) + \left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right), 1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2.0} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 +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))