Average Error: 0.6 → 0.9
Time: 48.7s
Precision: 64
\[\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(1\right)}{x}\right)\]
↓
\[\frac{\frac{1}{x + 1} + \frac{1}{x}}{\frac{\frac{1}{x + 1} + \frac{1}{x}}{\frac{1}{x + 1} - \frac{1}{x}}}\]
\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(1\right)}{x}\right)
↓
\frac{\frac{1}{x + 1} + \frac{1}{x}}{\frac{\frac{1}{x + 1} + \frac{1}{x}}{\frac{1}{x + 1} - \frac{1}{x}}}
double f(double x) {
        double r11428607 = 1.0;
        double r11428608 = /* ERROR: no posit support in C */;
        double r11428609 = x;
        double r11428610 = r11428609 + r11428608;
        double r11428611 = r11428608 / r11428610;
        double r11428612 = r11428608 / r11428609;
        double r11428613 = r11428611 - r11428612;
        return r11428613;
}
↓
double f(double x) {
        double r11428614 = 1.0;
        double r11428615 = x;
        double r11428616 = r11428615 + r11428614;
        double r11428617 = r11428614 / r11428616;
        double r11428618 = r11428614 / r11428615;
        double r11428619 = r11428617 + r11428618;
        double r11428620 = r11428617 - r11428618;
        double r11428621 = r11428619 / r11428620;
        double r11428622 = r11428619 / r11428621;
        return r11428622;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.6

    \[\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(1\right)}{x}\right)\]
  2. Using strategy rm

  3. Applied p16-flip--1.3

    \[\leadsto \color{blue}{\frac{\left(\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) \cdot \left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right)\right) - \left(\left(\frac{\left(1\right)}{x}\right) \cdot \left(\frac{\left(1\right)}{x}\right)\right)\right)}{\left(\frac{\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right)}{\left(\frac{\left(1\right)}{x}\right)}\right)}}\]
  4. Using strategy rm

  5. Applied difference-of-squares1.0

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

  6. Applied associate-/l*0.9

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

  7. Final simplification0.9

    \[\leadsto \frac{\frac{1}{x + 1} + \frac{1}{x}}{\frac{\frac{1}{x + 1} + \frac{1}{x}}{\frac{1}{x + 1} - \frac{1}{x}}}\]

Reproduce

herbie shell --seed 2019135 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (-.p16 (/.p16 (real->posit16 1) (+.p16 x (real->posit16 1))) (/.p16 (real->posit16 1) x)))