Average Error: 1.0 → 0.6
Time: 53.2s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right) \land i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\left(\frac{\alpha}{\beta}\right) \cdot \left(\beta - \alpha\right)\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\left(\frac{\left(\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\beta}{\left(\frac{\left(\frac{\alpha}{\left(2.0\right)}\right)}{\left(i \cdot \left(2\right)\right)}\right)}\right)}\right) \cdot \left(\frac{\left(\frac{\beta}{\alpha}\right)}{\left(\frac{\alpha}{\left(\frac{\beta}{\left(i \cdot \left(2\right)\right)}\right)}\right)}\right)\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
\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(\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{\beta}{\alpha}\right)}{\left(\frac{\alpha}{\left(\frac{\beta}{\left(i \cdot \left(2\right)\right)}\right)}\right)}\right)\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}
double f(double alpha, double beta, double i) {
        double r3422318 = alpha;
        double r3422319 = beta;
        double r3422320 = r3422318 + r3422319;
        double r3422321 = r3422319 - r3422318;
        double r3422322 = r3422320 * r3422321;
        double r3422323 = 2.0;
        double r3422324 = /* ERROR: no posit support in C */;
        double r3422325 = i;
        double r3422326 = r3422324 * r3422325;
        double r3422327 = r3422320 + r3422326;
        double r3422328 = r3422322 / r3422327;
        double r3422329 = 2.0;
        double r3422330 = /* ERROR: no posit support in C */;
        double r3422331 = r3422327 + r3422330;
        double r3422332 = r3422328 / r3422331;
        double r3422333 = 1.0;
        double r3422334 = /* ERROR: no posit support in C */;
        double r3422335 = r3422332 + r3422334;
        double r3422336 = r3422335 / r3422330;
        return r3422336;
}
↓
double f(double alpha, double beta, double i) {
        double r3422337 = beta;
        double r3422338 = alpha;
        double r3422339 = r3422337 - r3422338;
        double r3422340 = 2.0;
        double r3422341 = /* ERROR: no posit support in C */;
        double r3422342 = r3422338 + r3422341;
        double r3422343 = i;
        double r3422344 = 2.0;
        double r3422345 = /* ERROR: no posit support in C */;
        double r3422346 = r3422343 * r3422345;
        double r3422347 = r3422342 + r3422346;
        double r3422348 = r3422337 + r3422347;
        double r3422349 = r3422339 / r3422348;
        double r3422350 = r3422337 + r3422338;
        double r3422351 = r3422337 + r3422346;
        double r3422352 = r3422338 + r3422351;
        double r3422353 = r3422350 / r3422352;
        double r3422354 = r3422349 * r3422353;
        double r3422355 = 1.0;
        double r3422356 = /* ERROR: no posit support in C */;
        double r3422357 = r3422354 + r3422356;
        double r3422358 = r3422357 / r3422341;
        return r3422358;
}

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-expand1.0

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

  4. Applied p16-times-frac0.6

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

  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. Simplified0.6

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

  8. Applied /p16-rgt-identity-expand0.6

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

  9. Applied associate-/r/0.6

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

  10. Applied p16-*-un-lft-identity0.6

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

  11. Applied p16-times-frac0.6

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

  12. Simplified0.6

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

  13. Simplified0.6

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

  15. Applied *p16-rgt-identity-expand0.6

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

  16. Applied associate-/l*0.6

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

  17. Simplified0.6

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

  18. Final simplification0.6

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

Reproduce

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