Average Error: 23.9 → 12.1
Time: 33.5s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\frac{\log \left(e^{\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}\right)}{2}\]
\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}
↓
\frac{\log \left(e^{\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}\right)}{2}
double f(double alpha, double beta, double i) {
        double r71872 = alpha;
        double r71873 = beta;
        double r71874 = r71872 + r71873;
        double r71875 = r71873 - r71872;
        double r71876 = r71874 * r71875;
        double r71877 = 2.0;
        double r71878 = i;
        double r71879 = r71877 * r71878;
        double r71880 = r71874 + r71879;
        double r71881 = r71876 / r71880;
        double r71882 = r71880 + r71877;
        double r71883 = r71881 / r71882;
        double r71884 = 1.0;
        double r71885 = r71883 + r71884;
        double r71886 = r71885 / r71877;
        return r71886;
}
↓
double f(double alpha, double beta, double i) {
        double r71887 = beta;
        double r71888 = alpha;
        double r71889 = r71887 - r71888;
        double r71890 = 2.0;
        double r71891 = i;
        double r71892 = r71888 + r71887;
        double r71893 = fma(r71890, r71891, r71892);
        double r71894 = r71893 + r71890;
        double r71895 = r71889 / r71894;
        double r71896 = r71892 / r71893;
        double r71897 = 1.0;
        double r71898 = fma(r71895, r71896, r71897);
        double r71899 = exp(r71898);
        double r71900 = log(r71899);
        double r71901 = r71900 / r71890;
        return r71901;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 23.9

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

  2. Simplified12.1

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}{2}}\]
  3. Using strategy rm

  4. Applied add-log-exp12.1

    \[\leadsto \frac{\color{blue}{\log \left(e^{\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}\right)}}{2}\]

  5. Final simplification12.1

    \[\leadsto \frac{\log \left(e^{\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}\right)}{2}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2)) 1) 2))