Average Error: 0.4 → 0.4
Time: 49.8s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]
↓
\[\frac{\frac{1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\frac{\left(1.0 + \left(\alpha + \beta\right)\right) + 1 \cdot 2}{\alpha + \left(\left(\alpha \cdot \beta + \beta\right) + 1.0\right)} \cdot \left(\left(1 \cdot 2 + \alpha\right) + \beta\right)}\]
\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{1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\frac{\left(1.0 + \left(\alpha + \beta\right)\right) + 1 \cdot 2}{\alpha + \left(\left(\alpha \cdot \beta + \beta\right) + 1.0\right)} \cdot \left(\left(1 \cdot 2 + \alpha\right) + \beta\right)}
double f(double alpha, double beta) {
        double r3775920 = alpha;
        double r3775921 = beta;
        double r3775922 = r3775920 + r3775921;
        double r3775923 = r3775921 * r3775920;
        double r3775924 = r3775922 + r3775923;
        double r3775925 = 1.0;
        double r3775926 = /* ERROR: no posit support in C */;
        double r3775927 = r3775924 + r3775926;
        double r3775928 = 2.0;
        double r3775929 = /* ERROR: no posit support in C */;
        double r3775930 = 1.0;
        double r3775931 = /* ERROR: no posit support in C */;
        double r3775932 = r3775929 * r3775931;
        double r3775933 = r3775922 + r3775932;
        double r3775934 = r3775927 / r3775933;
        double r3775935 = r3775934 / r3775933;
        double r3775936 = r3775933 + r3775926;
        double r3775937 = r3775935 / r3775936;
        return r3775937;
}
↓
double f(double alpha, double beta) {
        double r3775938 = 1.0;
        double r3775939 = alpha;
        double r3775940 = beta;
        double r3775941 = r3775939 + r3775940;
        double r3775942 = 2.0;
        double r3775943 = 1.0;
        double r3775944 = r3775942 * r3775943;
        double r3775945 = r3775941 + r3775944;
        double r3775946 = r3775938 / r3775945;
        double r3775947 = r3775938 + r3775941;
        double r3775948 = r3775943 * r3775942;
        double r3775949 = r3775947 + r3775948;
        double r3775950 = r3775939 * r3775940;
        double r3775951 = r3775950 + r3775940;
        double r3775952 = r3775951 + r3775938;
        double r3775953 = r3775939 + r3775952;
        double r3775954 = r3775949 / r3775953;
        double r3775955 = r3775948 + r3775939;
        double r3775956 = r3775955 + r3775940;
        double r3775957 = r3775954 * r3775956;
        double r3775958 = r3775946 / r3775957;
        return r3775958;
}

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 *p16-rgt-identity-expand0.4

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

  4. Applied p16-*-un-lft-identity0.4

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

  5. Applied p16-times-frac0.4

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

  6. Applied associate-/l*0.4

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

  7. Simplified0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019165 
(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))))