Average Error: 1.0 → 1.0
Time: 5.5m
Precision: 64
\[\frac{\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(2\right)}{x}\right)\right)}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\]
↓
\[\left(\mathsf{qma}\left(\left(\mathsf{qms}\left(\left(\left(\frac{1}{x + 1}\right)\right), \left(\frac{2}{x}\right), 1.0\right)\right), \left(\frac{1.0}{x - 1}\right), 1\right)\right)\]
\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), \left(\frac{2}{x}\right), 1.0\right)\right), \left(\frac{1.0}{x - 1}\right), 1\right)\right)
double f(double x) {
        double r4854933 = 1.0;
        double r4854934 = /* ERROR: no posit support in C */;
        double r4854935 = x;
        double r4854936 = r4854935 + r4854934;
        double r4854937 = r4854934 / r4854936;
        double r4854938 = 2.0;
        double r4854939 = /* ERROR: no posit support in C */;
        double r4854940 = r4854939 / r4854935;
        double r4854941 = r4854937 - r4854940;
        double r4854942 = r4854935 - r4854934;
        double r4854943 = r4854934 / r4854942;
        double r4854944 = r4854941 + r4854943;
        return r4854944;
}
↓
double f(double x) {
        double r4854945 = 1.0;
        double r4854946 = x;
        double r4854947 = r4854946 + r4854945;
        double r4854948 = r4854945 / r4854947;
        double r4854949 = /*Error: no posit support in C */;
        double r4854950 = 2.0;
        double r4854951 = r4854950 / r4854946;
        double r4854952 = 1.0;
        double r4854953 = /*Error: no posit support in C */;
        double r4854954 = r4854946 - r4854945;
        double r4854955 = r4854952 / r4854954;
        double r4854956 = /*Error: no posit support in C */;
        double r4854957 = /*Error: no posit support in C */;
        return r4854957;
}

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-*-un-lft-identity1.0

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

  5. Applied p16-times-frac1.0

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

  6. Applied introduce-quire1.0

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

  7. Applied insert-quire-sub1.0

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

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

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

  9. Simplified1.0

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

  10. Final simplification1.0

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

Reproduce

herbie shell --seed 2019163 
(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)))))