Average Error: 0.6 → 0.9
Time: 12.2s
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 r2734867 = 1.0;
        double r2734868 = /* ERROR: no posit support in C */;
        double r2734869 = x;
        double r2734870 = r2734869 + r2734868;
        double r2734871 = r2734868 / r2734870;
        double r2734872 = r2734868 / r2734869;
        double r2734873 = r2734871 - r2734872;
        return r2734873;
}
↓
double f(double x) {
        double r2734874 = 1.0;
        double r2734875 = x;
        double r2734876 = r2734875 + r2734874;
        double r2734877 = r2734874 / r2734876;
        double r2734878 = r2734874 / r2734875;
        double r2734879 = r2734877 + r2734878;
        double r2734880 = r2734877 - r2734878;
        double r2734881 = r2734879 / r2734880;
        double r2734882 = r2734879 / r2734881;
        return r2734882;
}

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 2019128 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (-.p16 (/.p16 (real->posit16 1) (+.p16 x (real->posit16 1))) (/.p16 (real->posit16 1) x)))