Average Error: 0.7 → 0.7
Time: 29.7s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\left(\frac{1.0}{\left(\alpha + \beta\right) + 2.0} \cdot \beta + \frac{1.0}{\left(\alpha + \beta\right) + 2.0} \cdot \left(-\alpha\right)\right) + 1.0}{2.0}\]
\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}
↓
\frac{\left(\frac{1.0}{\left(\alpha + \beta\right) + 2.0} \cdot \beta + \frac{1.0}{\left(\alpha + \beta\right) + 2.0} \cdot \left(-\alpha\right)\right) + 1.0}{2.0}
double f(double alpha, double beta) {
        double r971955 = beta;
        double r971956 = alpha;
        double r971957 = r971955 - r971956;
        double r971958 = r971956 + r971955;
        double r971959 = 2.0;
        double r971960 = /* ERROR: no posit support in C */;
        double r971961 = r971958 + r971960;
        double r971962 = r971957 / r971961;
        double r971963 = 1.0;
        double r971964 = /* ERROR: no posit support in C */;
        double r971965 = r971962 + r971964;
        double r971966 = r971965 / r971960;
        return r971966;
}
↓
double f(double alpha, double beta) {
        double r971967 = 1.0;
        double r971968 = alpha;
        double r971969 = beta;
        double r971970 = r971968 + r971969;
        double r971971 = 2.0;
        double r971972 = r971970 + r971971;
        double r971973 = r971967 / r971972;
        double r971974 = r971973 * r971969;
        double r971975 = -r971968;
        double r971976 = r971973 * r971975;
        double r971977 = r971974 + r971976;
        double r971978 = r971977 + r971967;
        double r971979 = r971978 / r971971;
        return r971979;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Initial program 0.7

    \[\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
  2. Using strategy rm

  3. Applied p16-*-un-lft-identity0.7

    \[\leadsto \frac{\left(\frac{\left(\frac{\color{blue}{\left(\left(1.0\right) \cdot \left(\beta - \alpha\right)\right)}}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

  4. Applied associate-/l*0.7

    \[\leadsto \frac{\left(\frac{\color{blue}{\left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}{\left(\beta - \alpha\right)}\right)}\right)}}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
  5. Using strategy rm

  6. Applied associate-/r/0.8

    \[\leadsto \frac{\left(\frac{\color{blue}{\left(\left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right) \cdot \left(\beta - \alpha\right)\right)}}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
  7. Using strategy rm

  8. Applied sub-neg0.8

    \[\leadsto \frac{\left(\frac{\left(\left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right) \cdot \color{blue}{\left(\frac{\beta}{\left(-\alpha\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

  9. Applied distribute-lft-in0.7

    \[\leadsto \frac{\left(\frac{\color{blue}{\left(\frac{\left(\left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right) \cdot \beta\right)}{\left(\left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right) \cdot \left(-\alpha\right)\right)}\right)}}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

  10. Final simplification0.7

    \[\leadsto \frac{\left(\frac{1.0}{\left(\alpha + \beta\right) + 2.0} \cdot \beta + \frac{1.0}{\left(\alpha + \beta\right) + 2.0} \cdot \left(-\alpha\right)\right) + 1.0}{2.0}\]

Reproduce

herbie shell --seed 2019154 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (>.p16 alpha (real->posit16 -1)) (>.p16 beta (real->posit16 -1)))
  (/.p16 (+.p16 (/.p16 (-.p16 beta alpha) (+.p16 (+.p16 alpha beta) (real->posit16 2.0))) (real->posit16 1.0)) (real->posit16 2.0)))