Average Error: 0.4 → 0.4
Time: 28.1s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]
↓
\[\frac{\frac{\frac{\alpha + \left(\left(\beta + \beta \cdot \alpha\right) + 1.0\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
\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{\alpha + \left(\left(\beta + \beta \cdot \alpha\right) + 1.0\right)}{\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 r2884950 = alpha;
        double r2884951 = beta;
        double r2884952 = r2884950 + r2884951;
        double r2884953 = r2884951 * r2884950;
        double r2884954 = r2884952 + r2884953;
        double r2884955 = 1.0;
        double r2884956 = /* ERROR: no posit support in C */;
        double r2884957 = r2884954 + r2884956;
        double r2884958 = 2.0;
        double r2884959 = /* ERROR: no posit support in C */;
        double r2884960 = 1.0;
        double r2884961 = /* ERROR: no posit support in C */;
        double r2884962 = r2884959 * r2884961;
        double r2884963 = r2884952 + r2884962;
        double r2884964 = r2884957 / r2884963;
        double r2884965 = r2884964 / r2884963;
        double r2884966 = r2884963 + r2884956;
        double r2884967 = r2884965 / r2884966;
        return r2884967;
}
↓
double f(double alpha, double beta) {
        double r2884968 = alpha;
        double r2884969 = beta;
        double r2884970 = r2884969 * r2884968;
        double r2884971 = r2884969 + r2884970;
        double r2884972 = 1.0;
        double r2884973 = r2884971 + r2884972;
        double r2884974 = r2884968 + r2884973;
        double r2884975 = r2884968 + r2884969;
        double r2884976 = 2.0;
        double r2884977 = 1.0;
        double r2884978 = r2884976 * r2884977;
        double r2884979 = r2884975 + r2884978;
        double r2884980 = r2884974 / r2884979;
        double r2884981 = r2884980 / r2884979;
        double r2884982 = r2884979 + r2884972;
        double r2884983 = r2884981 / r2884982;
        return r2884983;
}

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-+l+0.4

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

  7. Applied associate-+r+0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019134 
(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))))