Average Error: 32.6 → 0.0
Time: 4.3s
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 r135799 = x;
        double r135800 = r135799 / r135799;
        double r135801 = 1.0;
        double r135802 = r135801 / r135799;
        double r135803 = r135799 * r135799;
        double r135804 = sqrt(r135803);
        double r135805 = r135802 * r135804;
        double r135806 = r135800 - r135805;
        return r135806;
}
↓
double f(double x) {
        double r135807 = 1.0;
        double r135808 = 1.0;
        double r135809 = x;
        double r135810 = r135808 / r135809;
        double r135811 = fabs(r135809);
        double r135812 = r135810 * r135811;
        double r135813 = log1p(r135812);
        double r135814 = expm1(r135813);
        double r135815 = r135807 - r135814;
        return r135815;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original32.6
Target0
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0.0\\ \end{array}\]

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 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 2020047 +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)))))