Average Error: 1.0 → 1.0
Time: 48.3s
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 r7935133 = 1.0;
        double r7935134 = /* ERROR: no posit support in C */;
        double r7935135 = x;
        double r7935136 = r7935135 + r7935134;
        double r7935137 = r7935134 / r7935136;
        double r7935138 = 2.0;
        double r7935139 = /* ERROR: no posit support in C */;
        double r7935140 = r7935139 / r7935135;
        double r7935141 = r7935137 - r7935140;
        double r7935142 = r7935135 - r7935134;
        double r7935143 = r7935134 / r7935142;
        double r7935144 = r7935141 + r7935143;
        return r7935144;
}
↓
double f(double x) {
        double r7935145 = 1.0;
        double r7935146 = x;
        double r7935147 = r7935146 + r7935145;
        double r7935148 = r7935145 / r7935147;
        double r7935149 = r7935146 - r7935145;
        double r7935150 = r7935145 / r7935149;
        double r7935151 = r7935148 + r7935150;
        double r7935152 = 2.0;
        double r7935153 = r7935152 / r7935146;
        double r7935154 = r7935151 - r7935153;
        return r7935154;
}

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-1.0

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

  8. Final simplification1.0

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

Reproduce

herbie shell --seed 2019133 +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)))))