Octave 3.8, jcobi/1

Average Error: 0.7 → 0.6

Time: 32.3s

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 r653088 = beta;
        double r653089 = alpha;
        double r653090 = r653088 - r653089;
        double r653091 = r653089 + r653088;
        double r653092 = 2.0;
        double r653093 = /* ERROR: no posit support in C */;
        double r653094 = r653091 + r653093;
        double r653095 = r653090 / r653094;
        double r653096 = 1.0;
        double r653097 = /* ERROR: no posit support in C */;
        double r653098 = r653095 + r653097;
        double r653099 = r653098 / r653093;
        return r653099;
}

↓

double f(double alpha, double beta) {
        double r653100 = beta;
        double r653101 = alpha;
        double r653102 = r653100 + r653101;
        double r653103 = 2.0;
        double r653104 = r653102 + r653103;
        double r653105 = r653100 / r653104;
        double r653106 = /*Error: no posit support in C */;
        double r653107 = -r653101;
        double r653108 = r653107 / r653104;
        double r653109 = 1.0;
        double r653110 = /*Error: no posit support in C */;
        double r653111 = /*Error: no posit support in C */;
        double r653112 = /*Error: no posit support in C */;
        double r653113 = r653112 / r653103;
        return r653113;
}

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-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 +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)))