Average Error: 1.0 → 1.0
Time: 4.6m
Precision: 64
\[\frac{\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(2\right)}{x}\right)\right)}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\]
↓
\[\left(\mathsf{qma}\left(\left(\mathsf{qms}\left(\left(\left(\frac{1}{x + 1}\right)\right), \left(\frac{2}{x}\right), 1.0\right)\right), 1.0, \left(\frac{1}{x - 1}\right)\right)\right)\]
\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), \left(\frac{2}{x}\right), 1.0\right)\right), 1.0, \left(\frac{1}{x - 1}\right)\right)\right)
double f(double x) {
        double r4611580 = 1.0;
        double r4611581 = /* ERROR: no posit support in C */;
        double r4611582 = x;
        double r4611583 = r4611582 + r4611581;
        double r4611584 = r4611581 / r4611583;
        double r4611585 = 2.0;
        double r4611586 = /* ERROR: no posit support in C */;
        double r4611587 = r4611586 / r4611582;
        double r4611588 = r4611584 - r4611587;
        double r4611589 = r4611582 - r4611581;
        double r4611590 = r4611581 / r4611589;
        double r4611591 = r4611588 + r4611590;
        return r4611591;
}
↓
double f(double x) {
        double r4611592 = 1.0;
        double r4611593 = x;
        double r4611594 = r4611593 + r4611592;
        double r4611595 = r4611592 / r4611594;
        double r4611596 = /*Error: no posit support in C */;
        double r4611597 = 2.0;
        double r4611598 = r4611597 / r4611593;
        double r4611599 = 1.0;
        double r4611600 = /*Error: no posit support in C */;
        double r4611601 = r4611593 - r4611592;
        double r4611602 = r4611592 / r4611601;
        double r4611603 = /*Error: no posit support in C */;
        double r4611604 = /*Error: no posit support in C */;
        return r4611604;
}

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

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

  4. Applied introduce-quire1.0

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

  5. Applied insert-quire-sub1.0

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

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

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

  7. Final simplification1.0

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

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)))))