Average Error: 32.1 → 0
Time: 3.8s
Precision: 64
\[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}\]
↓
\[1 - \frac{\left|x\right|}{x}\]
double f(double x) {
        double r29323135 = x;
        double r29323136 = r29323135 / r29323135;
        double r29323137 = 1.0;
        double r29323138 = r29323137 / r29323135;
        double r29323139 = r29323135 * r29323135;
        double r29323140 = sqrt(r29323139);
        double r29323141 = r29323138 * r29323140;
        double r29323142 = r29323136 - r29323141;
        return r29323142;
}
↓
double f(double x) {
        double r29323143 = 1.0;
        double r29323144 = x;
        double r29323145 = fabs(r29323144);
        double r29323146 = r29323145 / r29323144;
        double r29323147 = r29323143 - r29323146;
        return r29323147;
}


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

Error

Bits error versus x

Target

Original32.1
Target0
Herbie0
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Derivation

  1. Initial program 32.1

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

  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 2019102 +o rules:numerics
(FPCore (x)
  :name "sqrt sqr"

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

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