Octave 3.8, jcobi/3

Average Error: 0.4 → 0.3

Time: 44.4s

Precision: 64

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

Math

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

↓

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

C

double f(double alpha, double beta) {
        double r3125162 = alpha;
        double r3125163 = beta;
        double r3125164 = r3125162 + r3125163;
        double r3125165 = r3125163 * r3125162;
        double r3125166 = r3125164 + r3125165;
        double r3125167 = 1.0;
        double r3125168 = /* ERROR: no posit support in C */;
        double r3125169 = r3125166 + r3125168;
        double r3125170 = 2.0;
        double r3125171 = /* ERROR: no posit support in C */;
        double r3125172 = 1.0;
        double r3125173 = /* ERROR: no posit support in C */;
        double r3125174 = r3125171 * r3125173;
        double r3125175 = r3125164 + r3125174;
        double r3125176 = r3125169 / r3125175;
        double r3125177 = r3125176 / r3125175;
        double r3125178 = r3125175 + r3125168;
        double r3125179 = r3125177 / r3125178;
        return r3125179;
}

↓

double f(double alpha, double beta) {
        double r3125180 = 1.0;
        double r3125181 = /* ERROR: no posit support in C */;
        double r3125182 = alpha;
        double r3125183 = r3125181 + r3125182;
        double r3125184 = beta;
        double r3125185 = r3125184 + r3125181;
        double r3125186 = 2.0;
        double r3125187 = /* ERROR: no posit support in C */;
        double r3125188 = 1.0;
        double r3125189 = /* ERROR: no posit support in C */;
        double r3125190 = r3125187 * r3125189;
        double r3125191 = r3125184 + r3125190;
        double r3125192 = r3125191 + r3125182;
        double r3125193 = r3125185 / r3125192;
        double r3125194 = r3125183 * r3125193;
        double r3125195 = /*Error: no posit support in C */;
        double r3125196 = /*Error: no posit support in C */;
        double r3125197 = r3125189 * r3125187;
        double r3125198 = r3125184 + r3125197;
        double r3125199 = r3125198 + r3125182;
        double r3125200 = r3125181 / r3125199;
        double r3125201 = r3125196 * r3125200;
        double r3125202 = r3125182 + r3125184;
        double r3125203 = r3125202 + r3125190;
        double r3125204 = r3125203 + r3125181;
        double r3125205 = r3125201 / r3125204;
        return r3125205;
}

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 *p16-rgt-identity-expand 0.4

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

4. Applied p16-*-un-lft-identity 0.4

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

5. Applied *p16-rgt-identity-expand 0.4

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

6. Applied distribute-lft1-in 0.4

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

7. Applied p16-times-frac 0.4

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

8. Applied p16-times-frac 0.4

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

9. Simplified 0.4

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

11. Applied *p16-rgt-identity-expand 0.4

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

12. Applied p16-*-un-lft-identity 0.4

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

13. Applied p16-times-frac 0.4

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

14. Simplified 0.3

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

16. Applied introduce-quire 0.3

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

17. Simplified 0.3

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

18. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(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))))