Average Error: 0.6 → 1.0
Time: 55.1s
Precision: 64
\[\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(1\right)}{x}\right)\]
↓
\[\frac{\left(\frac{1}{x + 1} + \frac{1}{x}\right) \cdot \left(\frac{1}{x + 1} - \frac{1}{x}\right)}{\frac{1}{x + 1} + \frac{1}{x}}\]
\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(1\right)}{x}\right)
↓
\frac{\left(\frac{1}{x + 1} + \frac{1}{x}\right) \cdot \left(\frac{1}{x + 1} - \frac{1}{x}\right)}{\frac{1}{x + 1} + \frac{1}{x}}
double f(double x) {
        double r9053113 = 1.0;
        double r9053114 = /* ERROR: no posit support in C */;
        double r9053115 = x;
        double r9053116 = r9053115 + r9053114;
        double r9053117 = r9053114 / r9053116;
        double r9053118 = r9053114 / r9053115;
        double r9053119 = r9053117 - r9053118;
        return r9053119;
}
↓
double f(double x) {
        double r9053120 = 1.0;
        double r9053121 = x;
        double r9053122 = r9053121 + r9053120;
        double r9053123 = r9053120 / r9053122;
        double r9053124 = r9053120 / r9053121;
        double r9053125 = r9053123 + r9053124;
        double r9053126 = r9053123 - r9053124;
        double r9053127 = r9053125 * r9053126;
        double r9053128 = r9053127 / r9053125;
        return r9053128;
}

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. Final simplification1.0

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

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (-.p16 (/.p16 (real->posit16 1) (+.p16 x (real->posit16 1))) (/.p16 (real->posit16 1) x)))