Average Error: 0.4 → 0.4
Time: 19.3s
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 r1372911 = alpha;
        double r1372912 = beta;
        double r1372913 = r1372911 + r1372912;
        double r1372914 = r1372912 * r1372911;
        double r1372915 = r1372913 + r1372914;
        double r1372916 = 1.0;
        double r1372917 = /* ERROR: no posit support in C */;
        double r1372918 = r1372915 + r1372917;
        double r1372919 = 2.0;
        double r1372920 = /* ERROR: no posit support in C */;
        double r1372921 = 1.0;
        double r1372922 = /* ERROR: no posit support in C */;
        double r1372923 = r1372920 * r1372922;
        double r1372924 = r1372913 + r1372923;
        double r1372925 = r1372918 / r1372924;
        double r1372926 = r1372925 / r1372924;
        double r1372927 = r1372924 + r1372917;
        double r1372928 = r1372926 / r1372927;
        return r1372928;
}
↓
double f(double alpha, double beta) {
        double r1372929 = beta;
        double r1372930 = 1.0;
        double r1372931 = r1372929 + r1372930;
        double r1372932 = alpha;
        double r1372933 = r1372932 + r1372930;
        double r1372934 = r1372931 * r1372933;
        double r1372935 = r1372932 + r1372929;
        double r1372936 = 2.0;
        double r1372937 = 1.0;
        double r1372938 = r1372936 * r1372937;
        double r1372939 = r1372935 + r1372938;
        double r1372940 = r1372935 * r1372939;
        double r1372941 = r1372938 * r1372939;
        double r1372942 = r1372940 + r1372941;
        double r1372943 = r1372934 / r1372942;
        double r1372944 = r1372939 + r1372930;
        double r1372945 = r1372943 / r1372944;
        return r1372945;
}

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 
(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))))