Average Error: 1.0 → 1.0
Time: 5.7m
Precision: 64
\[\frac{\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(2\right)}{x}\right)\right)}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\]
↓
\[\frac{\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right)}{\left(\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right) - \left(\frac{\left(2\right)}{x}\right)\right)}\]
\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{\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right)}{\left(\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right) - \left(\frac{\left(2\right)}{x}\right)\right)}
double f(double x) {
        double r8251548 = 1.0;
        double r8251549 = /* ERROR: no posit support in C */;
        double r8251550 = x;
        double r8251551 = r8251550 + r8251549;
        double r8251552 = r8251549 / r8251551;
        double r8251553 = 2.0;
        double r8251554 = /* ERROR: no posit support in C */;
        double r8251555 = r8251554 / r8251550;
        double r8251556 = r8251552 - r8251555;
        double r8251557 = r8251550 - r8251549;
        double r8251558 = r8251549 / r8251557;
        double r8251559 = r8251556 + r8251558;
        return r8251559;
}
↓
double f(double x) {
        double r8251560 = 1.0;
        double r8251561 = /* ERROR: no posit support in C */;
        double r8251562 = x;
        double r8251563 = r8251562 + r8251561;
        double r8251564 = r8251561 / r8251563;
        double r8251565 = r8251562 - r8251561;
        double r8251566 = r8251561 / r8251565;
        double r8251567 = 2.0;
        double r8251568 = /* ERROR: no posit support in C */;
        double r8251569 = r8251568 / r8251562;
        double r8251570 = r8251566 - r8251569;
        double r8251571 = r8251564 + r8251570;
        return r8251571;
}

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

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

Reproduce

herbie shell --seed 2019168 +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)))))