Octave 3.8, jcobi/1

Average Error: 0.7 → 0.6

Time: 1.1m

Precision: 64

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

Math

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

↓

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

C

double f(double alpha, double beta) {
        double r3753101 = beta;
        double r3753102 = alpha;
        double r3753103 = r3753101 - r3753102;
        double r3753104 = r3753102 + r3753101;
        double r3753105 = 2.0;
        double r3753106 = /* ERROR: no posit support in C */;
        double r3753107 = r3753104 + r3753106;
        double r3753108 = r3753103 / r3753107;
        double r3753109 = 1.0;
        double r3753110 = /* ERROR: no posit support in C */;
        double r3753111 = r3753108 + r3753110;
        double r3753112 = r3753111 / r3753106;
        return r3753112;
}

↓

double f(double alpha, double beta) {
        double r3753113 = beta;
        double r3753114 = 2.0;
        double r3753115 = alpha;
        double r3753116 = r3753114 + r3753115;
        double r3753117 = r3753113 + r3753116;
        double r3753118 = r3753113 / r3753117;
        double r3753119 = /*Error: no posit support in C */;
        double r3753120 = -r3753115;
        double r3753121 = r3753120 / r3753117;
        double r3753122 = 1.0;
        double r3753123 = /*Error: no posit support in C */;
        double r3753124 = /*Error: no posit support in C */;
        double r3753125 = /*Error: no posit support in C */;
        double r3753126 = r3753125 / r3753114;
        return r3753126;
}

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-rgt-identity-expand 0.7

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

4. Applied p16-*-un-lft-identity 0.7

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

5. Applied p16-times-frac 0.8

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

6. Simplified 0.8

   \[\leadsto \frac{\left(\frac{\left(\left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right) \cdot \color{blue}{\left(\beta - \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
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-rgt-in 0.7

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

11. Applied introduce-quire 0.7

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

12. Applied insert-quire-add 0.7

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

13. Applied insert-quire-add 0.7

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

14. Simplified 0.6

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

15. Final simplification 0.6

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(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)))