Average Error: 32.1 → 0
Time: 3.6s
Precision: 64
\[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}\]
↓
\[1 - 1 \cdot \frac{\left|x\right|}{x}\]
\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}
↓
1 - 1 \cdot \frac{\left|x\right|}{x}
double f(double x) {
        double r110853 = x;
        double r110854 = r110853 / r110853;
        double r110855 = 1.0;
        double r110856 = r110855 / r110853;
        double r110857 = r110853 * r110853;
        double r110858 = sqrt(r110857);
        double r110859 = r110856 * r110858;
        double r110860 = r110854 - r110859;
        return r110860;
}
↓
double f(double x) {
        double r110861 = 1.0;
        double r110862 = 1.0;
        double r110863 = x;
        double r110864 = fabs(r110863);
        double r110865 = r110864 / r110863;
        double r110866 = r110862 * r110865;
        double r110867 = r110861 - r110866;
        return r110867;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original32.1
Target0
Herbie0
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0.0\\ \end{array}\]

Derivation

  1. Initial program 32.1

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

  2. Simplified4.9

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

  4. Applied div-inv4.9

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

  5. Applied associate-*l*4.9

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

  6. Simplified0

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

  7. Final simplification0

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

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (x)
  :name "sqrt sqr"
  :precision binary64

  :herbie-target
  (if (< x 0.0) 2 0.0)

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