Average Error: 16.4 → 3.5
Time: 5.6s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \le -1:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) - \left(-1\right)}{2}\\ \end{array}\]
\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 r87063 = beta;
        double r87064 = alpha;
        double r87065 = r87063 - r87064;
        double r87066 = r87064 + r87063;
        double r87067 = 2.0;
        double r87068 = r87066 + r87067;
        double r87069 = r87065 / r87068;
        double r87070 = 1.0;
        double r87071 = r87069 + r87070;
        double r87072 = r87071 / r87067;
        return r87072;
}
↓
double f(double alpha, double beta) {
        double r87073 = beta;
        double r87074 = alpha;
        double r87075 = r87073 - r87074;
        double r87076 = r87074 + r87073;
        double r87077 = 2.0;
        double r87078 = r87076 + r87077;
        double r87079 = r87075 / r87078;
        double r87080 = -1.0;
        bool r87081 = r87079 <= r87080;
        double r87082 = r87073 / r87078;
        double r87083 = 4.0;
        double r87084 = 1.0;
        double r87085 = 2.0;
        double r87086 = pow(r87074, r87085);
        double r87087 = r87084 / r87086;
        double r87088 = r87084 / r87074;
        double r87089 = 8.0;
        double r87090 = 3.0;
        double r87091 = pow(r87074, r87090);
        double r87092 = r87084 / r87091;
        double r87093 = r87089 * r87092;
        double r87094 = fma(r87077, r87088, r87093);
        double r87095 = -r87094;
        double r87096 = fma(r87083, r87087, r87095);
        double r87097 = r87082 - r87096;
        double r87098 = r87097 / r87077;
        double r87099 = r87074 / r87078;
        double r87100 = r87082 - r87099;
        double r87101 = 1.0;
        double r87102 = -r87101;
        double r87103 = r87100 - r87102;
        double r87104 = r87103 / r87077;
        double r87105 = r87081 ? r87098 : r87104;
        return r87105;
}

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))