Average Error: 0.4 → 0.3
Time: 28.4s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]
↓
\[\frac{\frac{\frac{1.0}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(\alpha + \beta\right)\right), \beta, \alpha\right)\right), 1.0, 1.0\right)\right)}}}{\beta + \left(2 \cdot 1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
\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{\frac{\frac{1.0}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(\alpha + \beta\right)\right), \beta, \alpha\right)\right), 1.0, 1.0\right)\right)}}}{\beta + \left(2 \cdot 1 + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}
double f(double alpha, double beta) {
        double r1130316 = alpha;
        double r1130317 = beta;
        double r1130318 = r1130316 + r1130317;
        double r1130319 = r1130317 * r1130316;
        double r1130320 = r1130318 + r1130319;
        double r1130321 = 1.0;
        double r1130322 = /* ERROR: no posit support in C */;
        double r1130323 = r1130320 + r1130322;
        double r1130324 = 2.0;
        double r1130325 = /* ERROR: no posit support in C */;
        double r1130326 = 1.0;
        double r1130327 = /* ERROR: no posit support in C */;
        double r1130328 = r1130325 * r1130327;
        double r1130329 = r1130318 + r1130328;
        double r1130330 = r1130323 / r1130329;
        double r1130331 = r1130330 / r1130329;
        double r1130332 = r1130329 + r1130322;
        double r1130333 = r1130331 / r1130332;
        return r1130333;
}
↓
double f(double alpha, double beta) {
        double r1130334 = 1.0;
        double r1130335 = alpha;
        double r1130336 = beta;
        double r1130337 = r1130335 + r1130336;
        double r1130338 = 2.0;
        double r1130339 = 1.0;
        double r1130340 = r1130338 * r1130339;
        double r1130341 = r1130337 + r1130340;
        double r1130342 = /*Error: no posit support in C */;
        double r1130343 = /*Error: no posit support in C */;
        double r1130344 = /*Error: no posit support in C */;
        double r1130345 = /*Error: no posit support in C */;
        double r1130346 = r1130341 / r1130345;
        double r1130347 = r1130334 / r1130346;
        double r1130348 = r1130340 + r1130335;
        double r1130349 = r1130336 + r1130348;
        double r1130350 = r1130347 / r1130349;
        double r1130351 = r1130341 + r1130334;
        double r1130352 = r1130350 / r1130351;
        return r1130352;
}

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 introduce-quire0.4

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

  4. Applied insert-quire-fdp-add0.4

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

  5. Applied insert-quire-add0.3

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

  7. Applied p16-*-un-lft-identity0.3

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

  8. Applied associate-/l*0.3

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

  10. Applied /p16-rgt-identity-expand0.3

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

  11. Applied associate-/r/0.3

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

  12. Applied p16-*-un-lft-identity0.3

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

  13. Applied p16-times-frac0.3

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

  14. Applied associate-/l*0.3

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

  15. Simplified0.3

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

  16. Final simplification0.3

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

Reproduce

herbie shell --seed 2019156 +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))))