Average Error: 32.2 → 0
Time: 6.0s
Precision: 64
\[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}\]
↓
\[1 - \frac{1 \cdot \left|x\right|}{x}\]
\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}
↓
1 - \frac{1 \cdot \left|x\right|}{x}
double f(double x) {
        double r104810 = x;
        double r104811 = r104810 / r104810;
        double r104812 = 1.0;
        double r104813 = r104812 / r104810;
        double r104814 = r104810 * r104810;
        double r104815 = sqrt(r104814);
        double r104816 = r104813 * r104815;
        double r104817 = r104811 - r104816;
        return r104817;
}
↓
double f(double x) {
        double r104818 = 1.0;
        double r104819 = 1.0;
        double r104820 = x;
        double r104821 = fabs(r104820);
        double r104822 = r104819 * r104821;
        double r104823 = r104822 / r104820;
        double r104824 = r104818 - r104823;
        return r104824;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 32.2

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

  2. Simplified4.7

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

  4. Applied *-un-lft-identity4.7

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

  5. Applied *-un-lft-identity4.7

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

  6. Applied times-frac4.7

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

  7. Applied associate-*l*4.7

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

  8. Simplified0

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

  9. Final simplification0

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

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x)
  :name "sqrt sqr"

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

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