Octave 3.8, jcobi/1

Average Error: 0.7 → 0.6

Time: 30.7s

Precision: 64

\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]

Math

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

↓

\[\frac{\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2.0}\right)\right), \left(\frac{-\alpha}{\left(\beta + \alpha\right) + 2.0}\right), 1.0\right)\right), 1.0, 1.0\right)\right)}{2.0}\]

C

double f(double alpha, double beta) {
        double r667953 = beta;
        double r667954 = alpha;
        double r667955 = r667953 - r667954;
        double r667956 = r667954 + r667953;
        double r667957 = 2.0;
        double r667958 = /* ERROR: no posit support in C */;
        double r667959 = r667956 + r667958;
        double r667960 = r667955 / r667959;
        double r667961 = 1.0;
        double r667962 = /* ERROR: no posit support in C */;
        double r667963 = r667960 + r667962;
        double r667964 = r667963 / r667958;
        return r667964;
}

↓

double f(double alpha, double beta) {
        double r667965 = beta;
        double r667966 = alpha;
        double r667967 = r667965 + r667966;
        double r667968 = 2.0;
        double r667969 = r667967 + r667968;
        double r667970 = r667965 / r667969;
        double r667971 = /*Error: no posit support in C */;
        double r667972 = -r667966;
        double r667973 = r667972 / r667969;
        double r667974 = 1.0;
        double r667975 = /*Error: no posit support in C */;
        double r667976 = /*Error: no posit support in C */;
        double r667977 = /*Error: no posit support in C */;
        double r667978 = r667977 / r667968;
        return r667978;
}

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-identity 0.7

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

4. Applied associate-/l* 0.8

   \[\leadsto \frac{\left(\frac{\color{blue}{\left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}{\left(\beta - \alpha\right)}\right)}\right)}}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
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-neg 0.8

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

9. Applied distribute-lft-in 0.7

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

11. Applied introduce-quire 0.7

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

12. Applied insert-quire-add 0.7

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

13. Applied insert-quire-add 0.7

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

14. Simplified 0.6

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

15. Final simplification 0.6

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

Reproduce

herbie shell --seed 2019156 
(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)))