Average Error: 1.0 → 1.0
Time: 1.0m
Precision: 64
\[\frac{\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(2\right)}{x}\right)\right)}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\]
↓
\[\frac{1}{x + 1} + \left(\frac{1}{x - 1} - \frac{2}{x}\right)\]
\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)}
↓
\frac{1}{x + 1} + \left(\frac{1}{x - 1} - \frac{2}{x}\right)
double f(double x) {
        double r1824114 = 1.0;
        double r1824115 = /* ERROR: no posit support in C */;
        double r1824116 = x;
        double r1824117 = r1824116 + r1824115;
        double r1824118 = r1824115 / r1824117;
        double r1824119 = 2.0;
        double r1824120 = /* ERROR: no posit support in C */;
        double r1824121 = r1824120 / r1824116;
        double r1824122 = r1824118 - r1824121;
        double r1824123 = r1824116 - r1824115;
        double r1824124 = r1824115 / r1824123;
        double r1824125 = r1824122 + r1824124;
        return r1824125;
}
↓
double f(double x) {
        double r1824126 = 1.0;
        double r1824127 = x;
        double r1824128 = r1824127 + r1824126;
        double r1824129 = r1824126 / r1824128;
        double r1824130 = r1824127 - r1824126;
        double r1824131 = r1824126 / r1824130;
        double r1824132 = 2.0;
        double r1824133 = r1824132 / r1824127;
        double r1824134 = r1824131 - r1824133;
        double r1824135 = r1824129 + r1824134;
        return r1824135;
}

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 +p16-rgt-identity-expand1.0

    \[\leadsto \frac{\left(\color{blue}{\left(\frac{\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right)}{\left(0.0\right)}\right)} - \left(\frac{\left(2\right)}{x}\right)\right)}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\]

  4. Applied associate--l+1.0

    \[\leadsto \frac{\color{blue}{\left(\frac{\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right)}{\left(\left(0.0\right) - \left(\frac{\left(2\right)}{x}\right)\right)}\right)}}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\]

  5. Applied associate-+l+1.0

    \[\leadsto \color{blue}{\frac{\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right)}{\left(\frac{\left(\left(0.0\right) - \left(\frac{\left(2\right)}{x}\right)\right)}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\right)}}\]

  6. Simplified1.0

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

  7. Final simplification1.0

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

Reproduce

herbie shell --seed 2019153 +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)))))