Average Error: 1.0 → 1.0
Time: 5.9m
Precision: 64
\[\frac{\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(2\right)}{x}\right)\right)}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\]
↓
\[\left(\mathsf{qma}\left(\left(\mathsf{qms}\left(\left(\left(\frac{\left(1\right)}{\left(\frac{\left(1\right)}{x}\right)}\right)\right), \left(1.0\right), \left(\frac{\left(2\right)}{x}\right)\right)\right), \left(1\right), \left(\frac{\left(1.0\right)}{\left(x - \left(1\right)\right)}\right)\right)\right)\]
\frac{\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(2\right)}{x}\right)\right)}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}
↓
\left(\mathsf{qma}\left(\left(\mathsf{qms}\left(\left(\left(\frac{\left(1\right)}{\left(\frac{\left(1\right)}{x}\right)}\right)\right), \left(1.0\right), \left(\frac{\left(2\right)}{x}\right)\right)\right), \left(1\right), \left(\frac{\left(1.0\right)}{\left(x - \left(1\right)\right)}\right)\right)\right)
double f(double x) {
        double r5609327 = 1.0;
        double r5609328 = /* ERROR: no posit support in C */;
        double r5609329 = x;
        double r5609330 = r5609329 + r5609328;
        double r5609331 = r5609328 / r5609330;
        double r5609332 = 2.0;
        double r5609333 = /* ERROR: no posit support in C */;
        double r5609334 = r5609333 / r5609329;
        double r5609335 = r5609331 - r5609334;
        double r5609336 = r5609329 - r5609328;
        double r5609337 = r5609328 / r5609336;
        double r5609338 = r5609335 + r5609337;
        return r5609338;
}
↓
double f(double x) {
        double r5609339 = 1.0;
        double r5609340 = /* ERROR: no posit support in C */;
        double r5609341 = x;
        double r5609342 = r5609340 + r5609341;
        double r5609343 = r5609340 / r5609342;
        double r5609344 = /*Error: no posit support in C */;
        double r5609345 = 1.0;
        double r5609346 = /* ERROR: no posit support in C */;
        double r5609347 = 2.0;
        double r5609348 = /* ERROR: no posit support in C */;
        double r5609349 = r5609348 / r5609341;
        double r5609350 = /*Error: no posit support in C */;
        double r5609351 = r5609341 - r5609340;
        double r5609352 = r5609346 / r5609351;
        double r5609353 = /*Error: no posit support in C */;
        double r5609354 = /*Error: no posit support in C */;
        return r5609354;
}

Error

Bits error versus x

Derivation

  1. Initial program 1.0

    \[\frac{\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(2\right)}{x}\right)\right)}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\]
  2. Using strategy rm

  3. Applied p16-*-un-lft-identity1.0

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

  4. Applied *p16-rgt-identity-expand1.0

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

  5. Applied p16-times-frac1.0

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

  6. Applied p16-*-un-lft-identity1.0

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

  7. Applied introduce-quire1.0

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

  8. Applied insert-quire-fdp-sub1.0

    \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{qms}\left(\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right)\right), \left(1.0\right), \left(\frac{\left(2\right)}{x}\right)\right)\right)\right)}}{\left(\left(\frac{\left(1\right)}{\left(1.0\right)}\right) \cdot \left(\frac{\left(1.0\right)}{\left(x - \left(1\right)\right)}\right)\right)}\]

  9. Applied insert-quire-fdp-add1.0

    \[\leadsto \color{blue}{\left(\mathsf{qma}\left(\left(\mathsf{qms}\left(\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right)\right), \left(1.0\right), \left(\frac{\left(2\right)}{x}\right)\right)\right), \left(\frac{\left(1\right)}{\left(1.0\right)}\right), \left(\frac{\left(1.0\right)}{\left(x - \left(1\right)\right)}\right)\right)\right)}\]

  10. Simplified1.0

    \[\leadsto \color{blue}{\left(\mathsf{qma}\left(\left(\mathsf{qms}\left(\left(\left(\frac{\left(1\right)}{\left(\frac{\left(1\right)}{x}\right)}\right)\right), \left(1.0\right), \left(\frac{\left(2\right)}{x}\right)\right)\right), \left(1\right), \left(\frac{\left(1.0\right)}{\left(x - \left(1\right)\right)}\right)\right)\right)}\]

  11. Final simplification1.0

    \[\leadsto \left(\mathsf{qma}\left(\left(\mathsf{qms}\left(\left(\left(\frac{\left(1\right)}{\left(\frac{\left(1\right)}{x}\right)}\right)\right), \left(1.0\right), \left(\frac{\left(2\right)}{x}\right)\right)\right), \left(1\right), \left(\frac{\left(1.0\right)}{\left(x - \left(1\right)\right)}\right)\right)\right)\]

Reproduce

herbie shell --seed 2019164 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  (+.p16 (-.p16 (/.p16 (real->posit16 1) (+.p16 x (real->posit16 1))) (/.p16 (real->posit16 2) x)) (/.p16 (real->posit16 1) (-.p16 x (real->posit16 1)))))