Average Error: 0.4 → 0.4
Time: 20.5s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]
↓
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}
↓
\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}
double f(double alpha, double beta) {
        double r2223871 = alpha;
        double r2223872 = beta;
        double r2223873 = r2223871 + r2223872;
        double r2223874 = r2223872 * r2223871;
        double r2223875 = r2223873 + r2223874;
        double r2223876 = 1.0;
        double r2223877 = /* ERROR: no posit support in C */;
        double r2223878 = r2223875 + r2223877;
        double r2223879 = 2.0;
        double r2223880 = /* ERROR: no posit support in C */;
        double r2223881 = 1.0;
        double r2223882 = /* ERROR: no posit support in C */;
        double r2223883 = r2223880 * r2223882;
        double r2223884 = r2223873 + r2223883;
        double r2223885 = r2223878 / r2223884;
        double r2223886 = r2223885 / r2223884;
        double r2223887 = r2223884 + r2223877;
        double r2223888 = r2223886 / r2223887;
        return r2223888;
}
↓
double f(double alpha, double beta) {
        double r2223889 = alpha;
        double r2223890 = beta;
        double r2223891 = r2223889 + r2223890;
        double r2223892 = r2223890 * r2223889;
        double r2223893 = r2223891 + r2223892;
        double r2223894 = 1.0;
        double r2223895 = r2223893 + r2223894;
        double r2223896 = 2.0;
        double r2223897 = 1.0;
        double r2223898 = r2223896 * r2223897;
        double r2223899 = r2223891 + r2223898;
        double r2223900 = r2223895 / r2223899;
        double r2223901 = r2223900 / r2223899;
        double r2223902 = r2223899 + r2223894;
        double r2223903 = r2223901 / r2223902;
        return r2223903;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Initial program 0.4

    \[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]
  2. Using strategy rm

  3. Applied associate-+l+0.4

    \[\leadsto \frac{\left(\frac{\left(\frac{\color{blue}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\frac{\left(\beta \cdot \alpha\right)}{\left(1.0\right)}\right)}\right)}}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]
  4. Using strategy rm

  5. Applied associate-+r+0.4

    \[\leadsto \frac{\left(\frac{\left(\frac{\color{blue}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

  6. Final simplification0.4

    \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

Reproduce

herbie shell --seed 2019128 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (>.p16 alpha (real->posit16 -1)) (>.p16 beta (real->posit16 -1)))
  (/.p16 (/.p16 (/.p16 (+.p16 (+.p16 (+.p16 alpha beta) (*.p16 beta alpha)) (real->posit16 1.0)) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) (real->posit16 1)))) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) (real->posit16 1)))) (+.p16 (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) (real->posit16 1))) (real->posit16 1.0))))