Average Error: 1.0 → 1.0
Time: 1.0m
Precision: 64
\[\frac{\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(2\right)}{x}\right)\right)}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\]
↓
\[\left(\mathsf{qma}\left(\left(\mathsf{qms}\left(\left(\left(\frac{1}{1 + x}\right)\right), 1.0, \left(\frac{2}{x}\right)\right)\right), \left(\frac{1}{x - 1}\right), 1.0\right)\right)\]
\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), 1.0, \left(\frac{2}{x}\right)\right)\right), \left(\frac{1}{x - 1}\right), 1.0\right)\right)
double f(double x) {
        double r2138949 = 1.0;
        double r2138950 = /* ERROR: no posit support in C */;
        double r2138951 = x;
        double r2138952 = r2138951 + r2138950;
        double r2138953 = r2138950 / r2138952;
        double r2138954 = 2.0;
        double r2138955 = /* ERROR: no posit support in C */;
        double r2138956 = r2138955 / r2138951;
        double r2138957 = r2138953 - r2138956;
        double r2138958 = r2138951 - r2138950;
        double r2138959 = r2138950 / r2138958;
        double r2138960 = r2138957 + r2138959;
        return r2138960;
}
↓
double f(double x) {
        double r2138961 = 1.0;
        double r2138962 = x;
        double r2138963 = r2138961 + r2138962;
        double r2138964 = r2138961 / r2138963;
        double r2138965 = /*Error: no posit support in C */;
        double r2138966 = 1.0;
        double r2138967 = 2.0;
        double r2138968 = r2138967 / r2138962;
        double r2138969 = /*Error: no posit support in C */;
        double r2138970 = r2138962 - r2138961;
        double r2138971 = r2138961 / r2138970;
        double r2138972 = /*Error: no posit support in C */;
        double r2138973 = /*Error: no posit support in C */;
        return r2138973;
}

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 p16-*-un-lft-identity1.0

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

  7. Applied introduce-quire1.0

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

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

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

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

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

  10. Simplified1.0

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

  11. Final simplification1.0

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

Reproduce

herbie shell --seed 2019154 +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)))))