Average Error: 1.0 → 1.0
Time: 18.1s
Precision: 64
\[\frac{\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(2\right)}{x}\right)\right)}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\]
↓
\[\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)\]
\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{2}{x} - \frac{1}{x - 1}\right)
double f(double x) {
        double r1548109 = 1.0;
        double r1548110 = /* ERROR: no posit support in C */;
        double r1548111 = x;
        double r1548112 = r1548111 + r1548110;
        double r1548113 = r1548110 / r1548112;
        double r1548114 = 2.0;
        double r1548115 = /* ERROR: no posit support in C */;
        double r1548116 = r1548115 / r1548111;
        double r1548117 = r1548113 - r1548116;
        double r1548118 = r1548111 - r1548110;
        double r1548119 = r1548110 / r1548118;
        double r1548120 = r1548117 + r1548119;
        return r1548120;
}
↓
double f(double x) {
        double r1548121 = 1.0;
        double r1548122 = x;
        double r1548123 = r1548122 + r1548121;
        double r1548124 = r1548121 / r1548123;
        double r1548125 = 2.0;
        double r1548126 = r1548125 / r1548122;
        double r1548127 = r1548122 - r1548121;
        double r1548128 = r1548121 / r1548127;
        double r1548129 = r1548126 - r1548128;
        double r1548130 = r1548124 - r1548129;
        return r1548130;
}

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 associate-+l-1.0

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

  4. Final simplification1.0

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

Reproduce

herbie shell --seed 2019107 +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)))))