Average Error: 1.0 → 1.0
Time: 6.2m
Precision: 64
\[\frac{\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(2\right)}{x}\right)\right)}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\]
↓
\[\left(\mathsf{qma}\left(\left(\mathsf{qms}\left(\left(\left(\frac{1}{1 + x}\right)\right), \left(\frac{2}{x}\right), 1.0\right)\right), \left(\frac{1}{x - 1}\right), 1.0\right)\right)\]
\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{1}{1 + x}\right)\right), \left(\frac{2}{x}\right), 1.0\right)\right), \left(\frac{1}{x - 1}\right), 1.0\right)\right)
double f(double x) {
        double r4109442 = 1.0;
        double r4109443 = /* ERROR: no posit support in C */;
        double r4109444 = x;
        double r4109445 = r4109444 + r4109443;
        double r4109446 = r4109443 / r4109445;
        double r4109447 = 2.0;
        double r4109448 = /* ERROR: no posit support in C */;
        double r4109449 = r4109448 / r4109444;
        double r4109450 = r4109446 - r4109449;
        double r4109451 = r4109444 - r4109443;
        double r4109452 = r4109443 / r4109451;
        double r4109453 = r4109450 + r4109452;
        return r4109453;
}
↓
double f(double x) {
        double r4109454 = 1.0;
        double r4109455 = x;
        double r4109456 = r4109454 + r4109455;
        double r4109457 = r4109454 / r4109456;
        double r4109458 = /*Error: no posit support in C */;
        double r4109459 = 2.0;
        double r4109460 = r4109459 / r4109455;
        double r4109461 = 1.0;
        double r4109462 = /*Error: no posit support in C */;
        double r4109463 = r4109455 - r4109454;
        double r4109464 = r4109454 / r4109463;
        double r4109465 = /*Error: no posit support in C */;
        double r4109466 = /*Error: no posit support in C */;
        return r4109466;
}

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-rgt-identity-expand1.0

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

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

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

  5. Applied p16-times-frac1.0

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

  6. Applied introduce-quire1.0

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

  7. Applied insert-quire-sub1.0

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

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

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

  9. Simplified1.0

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

  10. Final simplification1.0

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(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)))))