Average Error: 1.0 → 1.1
Time: 48.7s
Precision: 64
\[\frac{\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(2\right)}{x}\right)\right)}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\]
↓
\[\left(\frac{1}{x + 1} + \frac{1}{x - 1}\right) - \frac{2}{x}\]
\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(\frac{1}{x + 1} + \frac{1}{x - 1}\right) - \frac{2}{x}
double f(double x) {
        double r7666944 = 1.0;
        double r7666945 = /* ERROR: no posit support in C */;
        double r7666946 = x;
        double r7666947 = r7666946 + r7666945;
        double r7666948 = r7666945 / r7666947;
        double r7666949 = 2.0;
        double r7666950 = /* ERROR: no posit support in C */;
        double r7666951 = r7666950 / r7666946;
        double r7666952 = r7666948 - r7666951;
        double r7666953 = r7666946 - r7666945;
        double r7666954 = r7666945 / r7666953;
        double r7666955 = r7666952 + r7666954;
        return r7666955;
}
↓
double f(double x) {
        double r7666956 = 1.0;
        double r7666957 = x;
        double r7666958 = r7666957 + r7666956;
        double r7666959 = r7666956 / r7666958;
        double r7666960 = r7666957 - r7666956;
        double r7666961 = r7666956 / r7666960;
        double r7666962 = r7666959 + r7666961;
        double r7666963 = 2.0;
        double r7666964 = r7666963 / r7666957;
        double r7666965 = r7666962 - r7666964;
        return r7666965;
}

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 sub-neg1.0

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

  4. Applied associate-+l+1.0

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

  5. Simplified1.0

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

  7. Applied associate-+r-1.1

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

  8. Final simplification1.1

    \[\leadsto \left(\frac{1}{x + 1} + \frac{1}{x - 1}\right) - \frac{2}{x}\]

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(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)))))