Average Error: 32.4 → 0
Time: 5.9s
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 r87390 = x;
        double r87391 = r87390 / r87390;
        double r87392 = 1.0;
        double r87393 = r87392 / r87390;
        double r87394 = r87390 * r87390;
        double r87395 = sqrt(r87394);
        double r87396 = r87393 * r87395;
        double r87397 = r87391 - r87396;
        return r87397;
}
↓
double f(double x) {
        double r87398 = 1.0;
        double r87399 = 1.0;
        double r87400 = x;
        double r87401 = fabs(r87400);
        double r87402 = r87399 * r87401;
        double r87403 = r87402 / r87400;
        double r87404 = r87398 - r87403;
        return r87404;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original32.4
Target0
Herbie0
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0.0\\ \end{array}\]

Derivation

  1. Initial program 32.4

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

  2. Simplified4.5

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

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

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

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

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

  6. Applied times-frac4.5

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

  7. Applied associate-*l*4.5

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

  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 2019208 +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)))))