Average Error: 32.3 → 0.0
Time: 1.3s
Precision: 64
\[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}\]
↓
\[\left(1 - \frac{\left|x\right| \cdot 1}{x}\right) + \left|x\right| \cdot \left(\left(-\frac{1}{x}\right) + \frac{1}{x}\right)\]
\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}
↓
\left(1 - \frac{\left|x\right| \cdot 1}{x}\right) + \left|x\right| \cdot \left(\left(-\frac{1}{x}\right) + \frac{1}{x}\right)
double f(double x) {
        double r112824 = x;
        double r112825 = r112824 / r112824;
        double r112826 = 1.0;
        double r112827 = r112826 / r112824;
        double r112828 = r112824 * r112824;
        double r112829 = sqrt(r112828);
        double r112830 = r112827 * r112829;
        double r112831 = r112825 - r112830;
        return r112831;
}
↓
double f(double x) {
        double r112832 = 1.0;
        double r112833 = x;
        double r112834 = fabs(r112833);
        double r112835 = 1.0;
        double r112836 = r112834 * r112835;
        double r112837 = r112836 / r112833;
        double r112838 = r112832 - r112837;
        double r112839 = r112835 / r112833;
        double r112840 = -r112839;
        double r112841 = r112840 + r112839;
        double r112842 = r112834 * r112841;
        double r112843 = r112838 + r112842;
        return r112843;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original32.3
Target0
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0.0\\ \end{array}\]

Derivation

  1. Initial program 32.3

    \[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt32.3

    \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{x}{x}} \cdot \sqrt[3]{\frac{x}{x}}\right) \cdot \sqrt[3]{\frac{x}{x}}} - \frac{1}{x} \cdot \sqrt{x \cdot x}\]

  4. Applied prod-diff45.6

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{x}{x}} \cdot \sqrt[3]{\frac{x}{x}}, \sqrt[3]{\frac{x}{x}}, -\sqrt{x \cdot x} \cdot \frac{1}{x}\right) + \mathsf{fma}\left(-\sqrt{x \cdot x}, \frac{1}{x}, \sqrt{x \cdot x} \cdot \frac{1}{x}\right)}\]

  5. Simplified32.0

    \[\leadsto \color{blue}{\left(1 - \frac{\left|x\right| \cdot 1}{x}\right)} + \mathsf{fma}\left(-\sqrt{x \cdot x}, \frac{1}{x}, \sqrt{x \cdot x} \cdot \frac{1}{x}\right)\]

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020003 +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)))))