Average Error: 0.6 → 0.6
Time: 19.0s
Precision: 64
\[\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(1\right)}{x}\right)\]
↓
\[\frac{\frac{1}{x + 1} - \frac{1}{x}}{\frac{1}{x} + \frac{1}{x + 1}} \cdot \frac{\frac{1}{x} + \frac{1}{x + 1}}{1.0}\]
\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{1}{x} + \frac{1}{x + 1}} \cdot \frac{\frac{1}{x} + \frac{1}{x + 1}}{1.0}
double f(double x) {
        double r1946120 = 1.0;
        double r1946121 = /* ERROR: no posit support in C */;
        double r1946122 = x;
        double r1946123 = r1946122 + r1946121;
        double r1946124 = r1946121 / r1946123;
        double r1946125 = r1946121 / r1946122;
        double r1946126 = r1946124 - r1946125;
        return r1946126;
}
↓
double f(double x) {
        double r1946127 = 1.0;
        double r1946128 = x;
        double r1946129 = r1946128 + r1946127;
        double r1946130 = r1946127 / r1946129;
        double r1946131 = r1946127 / r1946128;
        double r1946132 = r1946130 - r1946131;
        double r1946133 = r1946131 + r1946130;
        double r1946134 = r1946132 / r1946133;
        double r1946135 = 1.0;
        double r1946136 = r1946133 / r1946135;
        double r1946137 = r1946134 * r1946136;
        return r1946137;
}

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. Simplified1.0

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

  5. Simplified1.0

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

  7. Applied *p16-rgt-identity-expand1.0

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

  8. Applied p16-times-frac0.6

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

  9. Final simplification0.6

    \[\leadsto \frac{\frac{1}{x + 1} - \frac{1}{x}}{\frac{1}{x} + \frac{1}{x + 1}} \cdot \frac{\frac{1}{x} + \frac{1}{x + 1}}{1.0}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (-.p16 (/.p16 (real->posit16 1) (+.p16 x (real->posit16 1))) (/.p16 (real->posit16 1) x)))