Average Error: 1.0 → 1.0
Time: 4.6m
Precision: 64
\[\frac{\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(2\right)}{x}\right)\right)}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\]
↓
\[\left(\mathsf{qma}\left(\left(\mathsf{qms}\left(\left(\left(\frac{1}{x + 1}\right)\right), 1.0, \left(\frac{2}{x}\right)\right)\right), 1, \left(\frac{1.0}{x - 1}\right)\right)\right)\]
\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{1}{x + 1}\right)\right), 1.0, \left(\frac{2}{x}\right)\right)\right), 1, \left(\frac{1.0}{x - 1}\right)\right)\right)
double f(double x) {
        double r3226161 = 1.0;
        double r3226162 = /* ERROR: no posit support in C */;
        double r3226163 = x;
        double r3226164 = r3226163 + r3226162;
        double r3226165 = r3226162 / r3226164;
        double r3226166 = 2.0;
        double r3226167 = /* ERROR: no posit support in C */;
        double r3226168 = r3226167 / r3226163;
        double r3226169 = r3226165 - r3226168;
        double r3226170 = r3226163 - r3226162;
        double r3226171 = r3226162 / r3226170;
        double r3226172 = r3226169 + r3226171;
        return r3226172;
}
↓
double f(double x) {
        double r3226173 = 1.0;
        double r3226174 = x;
        double r3226175 = r3226174 + r3226173;
        double r3226176 = r3226173 / r3226175;
        double r3226177 = /*Error: no posit support in C */;
        double r3226178 = 1.0;
        double r3226179 = 2.0;
        double r3226180 = r3226179 / r3226174;
        double r3226181 = /*Error: no posit support in C */;
        double r3226182 = r3226174 - r3226173;
        double r3226183 = r3226178 / r3226182;
        double r3226184 = /*Error: no posit support in C */;
        double r3226185 = /*Error: no posit support in C */;
        return r3226185;
}

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

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

  11. Final simplification1.0

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

Reproduce

herbie shell --seed 2019163 +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)))))