Average Error: 32.6 → 0
Time: 3.1s
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 r130553 = x;
        double r130554 = r130553 / r130553;
        double r130555 = 1.0;
        double r130556 = r130555 / r130553;
        double r130557 = r130553 * r130553;
        double r130558 = sqrt(r130557);
        double r130559 = r130556 * r130558;
        double r130560 = r130554 - r130559;
        return r130560;
}
↓
double f(double x) {
        double r130561 = 1.0;
        double r130562 = 1.0;
        double r130563 = x;
        double r130564 = fabs(r130563);
        double r130565 = r130564 / r130563;
        double r130566 = r130562 * r130565;
        double r130567 = r130561 - r130566;
        return r130567;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original32.6
Target0
Herbie0
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0.0\\ \end{array}\]

Derivation

  1. Initial program 32.6

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

  2. Simplified4.9

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

  4. Applied *-un-lft-identity4.9

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

  5. Applied *-un-lft-identity4.9

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

  6. Applied times-frac4.9

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

  7. Applied associate-*l*4.9

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

  8. Simplified0

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

  9. Final simplification0

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

Reproduce

herbie shell --seed 2020047 
(FPCore (x)
  :name "sqrt sqr"
  :precision binary64

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

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