Average Error: 32.3 → 0.0
Time: 5.8s
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 r153477 = x;
        double r153478 = r153477 / r153477;
        double r153479 = 1.0;
        double r153480 = r153479 / r153477;
        double r153481 = r153477 * r153477;
        double r153482 = sqrt(r153481);
        double r153483 = r153480 * r153482;
        double r153484 = r153478 - r153483;
        return r153484;
}
↓
double f(double x) {
        double r153485 = 1.0;
        double r153486 = 1.0;
        double r153487 = x;
        double r153488 = r153486 / r153487;
        double r153489 = fabs(r153487);
        double r153490 = r153488 * r153489;
        double r153491 = log1p(r153490);
        double r153492 = expm1(r153491);
        double r153493 = r153485 - r153492;
        return r153493;
}

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. Simplified5.0

    \[\leadsto \color{blue}{1 - \frac{1}{x} \cdot \left|x\right|}\]
  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 2020043 +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)))))