Average Error: 32.0 → 0
Time: 18.0s
Precision: 64
\[\frac{x}{x} - \frac{1.0}{x} \cdot \sqrt{x \cdot x}\]
↓
\[1 - \left(\left(\frac{1.0}{x} \cdot \left|x\right|\right)\right)\]
\frac{x}{x} - \frac{1.0}{x} \cdot \sqrt{x \cdot x}
↓
1 - \left(\left(\frac{1.0}{x} \cdot \left|x\right|\right)\right)
double f(double x) {
        double r5327766 = x;
        double r5327767 = r5327766 / r5327766;
        double r5327768 = 1.0;
        double r5327769 = r5327768 / r5327766;
        double r5327770 = r5327766 * r5327766;
        double r5327771 = sqrt(r5327770);
        double r5327772 = r5327769 * r5327771;
        double r5327773 = r5327767 - r5327772;
        return r5327773;
}
↓
double f(double x) {
        double r5327774 = 1.0;
        double r5327775 = 1.0;
        double r5327776 = x;
        double r5327777 = r5327775 / r5327776;
        double r5327778 = fabs(r5327776);
        double r5327779 = r5327777 * r5327778;
        double r5327780 = /* ERROR: no posit support in C */;
        double r5327781 = /* ERROR: no posit support in C */;
        double r5327782 = r5327774 - r5327781;
        return r5327782;
}

Error

Bits error versus x

Target

Original32.0
Target0
Herbie0
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;2.0\\ \mathbf{else}:\\ \;\;\;\;0.0\\ \end{array}\]

Derivation

  1. Initial program 32.0

    \[\frac{x}{x} - \frac{1.0}{x} \cdot \sqrt{x \cdot x}\]

  2. Simplified4.6

    \[\leadsto \color{blue}{1 - \frac{1.0}{x} \cdot \left|x\right|}\]
  3. Using strategy rm

  4. Applied insert-posit160

    \[\leadsto 1 - \color{blue}{\left(\left(\frac{1.0}{x} \cdot \left|x\right|\right)\right)}\]

  5. Final simplification0

    \[\leadsto 1 - \left(\left(\frac{1.0}{x} \cdot \left|x\right|\right)\right)\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x)
  :name "sqrt sqr"

  :herbie-target
  (if (< x 0.0) 2.0 0.0)

  (- (/ x x) (* (/ 1.0 x) (sqrt (* x x)))))