Average Error: 1.0 → 1.0
Time: 5.5m
Precision: 64
\[\frac{\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(2\right)}{x}\right)\right)}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\]
↓
\[\left(\mathsf{qma}\left(\left(\mathsf{qms}\left(\left(\left(\frac{1}{x + 1}\right)\right), 1.0, \left(\frac{2}{x}\right)\right)\right), \left(\frac{1}{x - 1}\right), 1.0\right)\right)\]
\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}{x + 1}\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 r6127920 = 1.0;
        double r6127921 = /* ERROR: no posit support in C */;
        double r6127922 = x;
        double r6127923 = r6127922 + r6127921;
        double r6127924 = r6127921 / r6127923;
        double r6127925 = 2.0;
        double r6127926 = /* ERROR: no posit support in C */;
        double r6127927 = r6127926 / r6127922;
        double r6127928 = r6127924 - r6127927;
        double r6127929 = r6127922 - r6127921;
        double r6127930 = r6127921 / r6127929;
        double r6127931 = r6127928 + r6127930;
        return r6127931;
}
↓
double f(double x) {
        double r6127932 = 1.0;
        double r6127933 = x;
        double r6127934 = r6127933 + r6127932;
        double r6127935 = r6127932 / r6127934;
        double r6127936 = /*Error: no posit support in C */;
        double r6127937 = 1.0;
        double r6127938 = 2.0;
        double r6127939 = r6127938 / r6127933;
        double r6127940 = /*Error: no posit support in C */;
        double r6127941 = r6127933 - r6127932;
        double r6127942 = r6127932 / r6127941;
        double r6127943 = /*Error: no posit support in C */;
        double r6127944 = /*Error: no posit support in C */;
        return r6127944;
}

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

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

  4. Applied introduce-quire1.0

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

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

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

  6. Applied insert-quire-add1.0

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

  7. Final simplification1.0

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

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)))))