Average Error: 31.8 → 0
Time: 6.9s
Precision: 64
\[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}\]
↓
\[\mathsf{fma}\left(1, -\frac{\left|x\right|}{x}, 1\right)\]
\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}
↓
\mathsf{fma}\left(1, -\frac{\left|x\right|}{x}, 1\right)
double f(double x) {
        double r110984 = x;
        double r110985 = r110984 / r110984;
        double r110986 = 1.0;
        double r110987 = r110986 / r110984;
        double r110988 = r110984 * r110984;
        double r110989 = sqrt(r110988);
        double r110990 = r110987 * r110989;
        double r110991 = r110985 - r110990;
        return r110991;
}
↓
double f(double x) {
        double r110992 = 1.0;
        double r110993 = x;
        double r110994 = fabs(r110993);
        double r110995 = r110994 / r110993;
        double r110996 = -r110995;
        double r110997 = 1.0;
        double r110998 = fma(r110992, r110996, r110997);
        return r110998;
}

Error

Bits error versus x

Target

Original31.8
Target0
Herbie0
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0.0\\ \end{array}\]

Derivation

  1. Initial program 31.8

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

  2. Simplified4.5

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

  3. Taylor expanded around 0 0

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

  4. Simplified0

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

  5. Final simplification0

    \[\leadsto \mathsf{fma}\left(1, -\frac{\left|x\right|}{x}, 1\right)\]

Reproduce

herbie shell --seed 2019303 +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)))))