Average Error: 0.4 → 0.4
Time: 19.1s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]
↓
\[\frac{\frac{\left(\beta + 1.0\right) \cdot \left(\alpha + 1.0\right)}{\left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \left(2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
\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{\left(\beta + 1.0\right) \cdot \left(\alpha + 1.0\right)}{\left(\alpha + \beta\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + \left(2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}
double f(double alpha, double beta) {
        double r1895863 = alpha;
        double r1895864 = beta;
        double r1895865 = r1895863 + r1895864;
        double r1895866 = r1895864 * r1895863;
        double r1895867 = r1895865 + r1895866;
        double r1895868 = 1.0;
        double r1895869 = /* ERROR: no posit support in C */;
        double r1895870 = r1895867 + r1895869;
        double r1895871 = 2.0;
        double r1895872 = /* ERROR: no posit support in C */;
        double r1895873 = 1.0;
        double r1895874 = /* ERROR: no posit support in C */;
        double r1895875 = r1895872 * r1895874;
        double r1895876 = r1895865 + r1895875;
        double r1895877 = r1895870 / r1895876;
        double r1895878 = r1895877 / r1895876;
        double r1895879 = r1895876 + r1895869;
        double r1895880 = r1895878 / r1895879;
        return r1895880;
}
↓
double f(double alpha, double beta) {
        double r1895881 = beta;
        double r1895882 = 1.0;
        double r1895883 = r1895881 + r1895882;
        double r1895884 = alpha;
        double r1895885 = r1895884 + r1895882;
        double r1895886 = r1895883 * r1895885;
        double r1895887 = r1895884 + r1895881;
        double r1895888 = 2.0;
        double r1895889 = 1.0;
        double r1895890 = r1895888 * r1895889;
        double r1895891 = r1895887 + r1895890;
        double r1895892 = r1895887 * r1895891;
        double r1895893 = r1895890 * r1895891;
        double r1895894 = r1895892 + r1895893;
        double r1895895 = r1895886 / r1895894;
        double r1895896 = r1895891 + r1895882;
        double r1895897 = r1895895 / r1895896;
        return r1895897;
}

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

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

  5. Applied associate-/l/0.4

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

  6. Simplified0.4

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

  8. Applied distribute-rgt-in0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(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))))