Average Error: 31.1 → 0
Time: 4.0s
Precision: 64
\[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}\]
↓
\[1 - \frac{\left|x\right|}{x}\]
double f(double x) {
        double r12800613 = x;
        double r12800614 = r12800613 / r12800613;
        double r12800615 = 1.0;
        double r12800616 = r12800615 / r12800613;
        double r12800617 = r12800613 * r12800613;
        double r12800618 = sqrt(r12800617);
        double r12800619 = r12800616 * r12800618;
        double r12800620 = r12800614 - r12800619;
        return r12800620;
}
↓
double f(double x) {
        double r12800621 = 1.0;
        double r12800622 = x;
        double r12800623 = fabs(r12800622);
        double r12800624 = r12800623 / r12800622;
        double r12800625 = r12800621 - r12800624;
        return r12800625;
}


\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}
↓
1 - \frac{\left|x\right|}{x}

Error

Bits error versus x

Target

Original31.1
Target0
Herbie0
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Derivation

  1. Initial program 31.1

    \[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}\]

  2. Simplified0

    \[\leadsto \color{blue}{1 - \frac{\left|x\right|}{x}}\]

  3. Final simplification0

    \[\leadsto 1 - \frac{\left|x\right|}{x}\]

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (x)
  :name "sqrt sqr"

  :herbie-target
  (if (< x 0) 2 0)

  (- (/ x x) (* (/ 1 x) (sqrt (* x x)))))