Octave 3.8, jcobi/2

Average Error: 1.0 → 0.6

Time: 53.0s

Precision: 64

\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right) \land i \gt \left(0\right)\]

Math

\[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\left(\frac{\alpha}{\beta}\right) \cdot \left(\beta - \alpha\right)\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

↓

\[\frac{\left(\frac{\left(\left(\frac{\left(\frac{\beta}{\alpha}\right)}{\left(\frac{\beta}{\left(\frac{\left(\frac{\alpha}{\left(2.0\right)}\right)}{\left(i \cdot \left(2\right)\right)}\right)}\right)}\right) \cdot \left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\beta}{\alpha}\right)}{\left(i \cdot \left(2\right)\right)}\right)}\right)\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

C

double f(double alpha, double beta, double i) {
        double r3814248 = alpha;
        double r3814249 = beta;
        double r3814250 = r3814248 + r3814249;
        double r3814251 = r3814249 - r3814248;
        double r3814252 = r3814250 * r3814251;
        double r3814253 = 2.0;
        double r3814254 = /* ERROR: no posit support in C */;
        double r3814255 = i;
        double r3814256 = r3814254 * r3814255;
        double r3814257 = r3814250 + r3814256;
        double r3814258 = r3814252 / r3814257;
        double r3814259 = 2.0;
        double r3814260 = /* ERROR: no posit support in C */;
        double r3814261 = r3814257 + r3814260;
        double r3814262 = r3814258 / r3814261;
        double r3814263 = 1.0;
        double r3814264 = /* ERROR: no posit support in C */;
        double r3814265 = r3814262 + r3814264;
        double r3814266 = r3814265 / r3814260;
        return r3814266;
}

↓

double f(double alpha, double beta, double i) {
        double r3814267 = beta;
        double r3814268 = alpha;
        double r3814269 = r3814267 + r3814268;
        double r3814270 = 2.0;
        double r3814271 = /* ERROR: no posit support in C */;
        double r3814272 = r3814268 + r3814271;
        double r3814273 = i;
        double r3814274 = 2.0;
        double r3814275 = /* ERROR: no posit support in C */;
        double r3814276 = r3814273 * r3814275;
        double r3814277 = r3814272 + r3814276;
        double r3814278 = r3814267 + r3814277;
        double r3814279 = r3814269 / r3814278;
        double r3814280 = r3814267 - r3814268;
        double r3814281 = r3814269 + r3814276;
        double r3814282 = r3814280 / r3814281;
        double r3814283 = r3814279 * r3814282;
        double r3814284 = 1.0;
        double r3814285 = /* ERROR: no posit support in C */;
        double r3814286 = r3814283 + r3814285;
        double r3814287 = r3814286 / r3814271;
        return r3814287;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Initial program 1.0

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

3. Applied *p16-rgt-identity-expand 1.0

   \[\leadsto \frac{\left(\frac{\left(\frac{\left(\frac{\left(\left(\frac{\alpha}{\beta}\right) \cdot \left(\beta - \alpha\right)\right)}{\color{blue}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right) \cdot \left(1.0\right)\right)}}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

4. Applied p16-times-frac 0.6

   \[\leadsto \frac{\left(\frac{\left(\frac{\color{blue}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right) \cdot \left(\frac{\left(\beta - \alpha\right)}{\left(1.0\right)}\right)\right)}}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

5. Applied associate-/l* 0.6

   \[\leadsto \frac{\left(\frac{\color{blue}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}{\left(\frac{\left(\beta - \alpha\right)}{\left(1.0\right)}\right)}\right)}\right)}}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

6. Simplified 0.6

   \[\leadsto \frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\color{blue}{\left(\frac{\left(\frac{\left(i \cdot \left(2\right)\right)}{\left(\frac{\left(\frac{\beta}{\alpha}\right)}{\left(2.0\right)}\right)}\right)}{\left(\beta - \alpha\right)}\right)}}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
7. Using strategy rm

8. Applied p16-*-un-lft-identity 0.6

   \[\leadsto \frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(i \cdot \left(2\right)\right)}{\left(\frac{\left(\frac{\beta}{\alpha}\right)}{\left(2.0\right)}\right)}\right)}{\color{blue}{\left(\left(1.0\right) \cdot \left(\beta - \alpha\right)\right)}}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

9. Applied *p16-rgt-identity-expand 0.6

   \[\leadsto \frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\color{blue}{\left(\left(\frac{\left(i \cdot \left(2\right)\right)}{\left(\frac{\left(\frac{\beta}{\alpha}\right)}{\left(2.0\right)}\right)}\right) \cdot \left(1.0\right)\right)}}{\left(\left(1.0\right) \cdot \left(\beta - \alpha\right)\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

10. Applied p16-times-frac 0.7

    \[\leadsto \frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\color{blue}{\left(\left(\frac{\left(\frac{\left(i \cdot \left(2\right)\right)}{\left(\frac{\left(\frac{\beta}{\alpha}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right) \cdot \left(\frac{\left(1.0\right)}{\left(\beta - \alpha\right)}\right)\right)}}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

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

    \[\leadsto \frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\color{blue}{\left(\left(1.0\right) \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)\right)}}\right)}{\left(\left(\frac{\left(\frac{\left(i \cdot \left(2\right)\right)}{\left(\frac{\left(\frac{\beta}{\alpha}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right) \cdot \left(\frac{\left(1.0\right)}{\left(\beta - \alpha\right)}\right)\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

12. Applied *p16-rgt-identity-expand 0.7

    \[\leadsto \frac{\left(\frac{\left(\frac{\left(\frac{\color{blue}{\left(\left(\frac{\alpha}{\beta}\right) \cdot \left(1.0\right)\right)}}{\left(\left(1.0\right) \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)\right)}\right)}{\left(\left(\frac{\left(\frac{\left(i \cdot \left(2\right)\right)}{\left(\frac{\left(\frac{\beta}{\alpha}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right) \cdot \left(\frac{\left(1.0\right)}{\left(\beta - \alpha\right)}\right)\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

13. Applied p16-times-frac 0.7

    \[\leadsto \frac{\left(\frac{\left(\frac{\color{blue}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(1.0\right)}\right) \cdot \left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)\right)}}{\left(\left(\frac{\left(\frac{\left(i \cdot \left(2\right)\right)}{\left(\frac{\left(\frac{\beta}{\alpha}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right) \cdot \left(\frac{\left(1.0\right)}{\left(\beta - \alpha\right)}\right)\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

14. Applied p16-times-frac 0.7

    \[\leadsto \frac{\left(\frac{\color{blue}{\left(\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\left(i \cdot \left(2\right)\right)}{\left(\frac{\left(\frac{\beta}{\alpha}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}\right) \cdot \left(\frac{\left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\left(1.0\right)}{\left(\beta - \alpha\right)}\right)}\right)\right)}}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

15. Simplified 0.6

    \[\leadsto \frac{\left(\frac{\left(\color{blue}{\left(\frac{\left(\frac{\beta}{\alpha}\right)}{\left(\frac{\beta}{\left(\frac{\left(\frac{\alpha}{\left(2.0\right)}\right)}{\left(i \cdot \left(2\right)\right)}\right)}\right)}\right)} \cdot \left(\frac{\left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\left(1.0\right)}{\left(\beta - \alpha\right)}\right)}\right)\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

16. Simplified 0.6

    \[\leadsto \frac{\left(\frac{\left(\left(\frac{\left(\frac{\beta}{\alpha}\right)}{\left(\frac{\beta}{\left(\frac{\left(\frac{\alpha}{\left(2.0\right)}\right)}{\left(i \cdot \left(2\right)\right)}\right)}\right)}\right) \cdot \color{blue}{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\beta}{\alpha}\right)}{\left(i \cdot \left(2\right)\right)}\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

17. Final simplification 0.6

    \[\leadsto \frac{\left(\frac{\left(\left(\frac{\left(\frac{\beta}{\alpha}\right)}{\left(\frac{\beta}{\left(\frac{\left(\frac{\alpha}{\left(2.0\right)}\right)}{\left(i \cdot \left(2\right)\right)}\right)}\right)}\right) \cdot \left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\beta}{\alpha}\right)}{\left(i \cdot \left(2\right)\right)}\right)}\right)\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

Reproduce

herbie shell --seed 2019164 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (>.p16 alpha (real->posit16 -1)) (>.p16 beta (real->posit16 -1)) (>.p16 i (real->posit16 0)))
  (/.p16 (+.p16 (/.p16 (/.p16 (*.p16 (+.p16 alpha beta) (-.p16 beta alpha)) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i))) (+.p16 (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i)) (real->posit16 2.0))) (real->posit16 1.0)) (real->posit16 2.0)))