Average Error: 32.4 → 0.0
Time: 9.6s
Precision: 64
\[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}\]
↓
\[1 - \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x} \cdot \left|x\right|\right)\right)\]
\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}
↓
1 - \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x} \cdot \left|x\right|\right)\right)
double f(double x) {
        double r96358 = x;
        double r96359 = r96358 / r96358;
        double r96360 = 1.0;
        double r96361 = r96360 / r96358;
        double r96362 = r96358 * r96358;
        double r96363 = sqrt(r96362);
        double r96364 = r96361 * r96363;
        double r96365 = r96359 - r96364;
        return r96365;
}
↓
double f(double x) {
        double r96366 = 1.0;
        double r96367 = 1.0;
        double r96368 = x;
        double r96369 = r96367 / r96368;
        double r96370 = fabs(r96368);
        double r96371 = r96369 * r96370;
        double r96372 = log1p(r96371);
        double r96373 = expm1(r96372);
        double r96374 = r96366 - r96373;
        return r96374;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original32.4
Target0
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0.0\\ \end{array}\]

Derivation

  1. Initial program 32.4

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

  2. Simplified4.7

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

  4. Applied expm1-log1p-u0.0

    \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x} \cdot \left|x\right|\right)\right)}\]

  5. Final simplification0.0

    \[\leadsto 1 - \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x} \cdot \left|x\right|\right)\right)\]

Reproduce

herbie shell --seed 2019323 +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)))))