Average Error: 32.3 → 0
Time: 4.8s
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 r12099 = x;
        double r12100 = r12099 / r12099;
        double r12101 = 1.0;
        double r12102 = r12101 / r12099;
        double r12103 = r12099 * r12099;
        double r12104 = sqrt(r12103);
        double r12105 = r12102 * r12104;
        double r12106 = r12100 - r12105;
        return r12106;
}
↓
double f(double x) {
        double r12107 = 1.0;
        double r12108 = x;
        double r12109 = fabs(r12108);
        double r12110 = r12109 / r12108;
        double r12111 = -r12110;
        double r12112 = 1.0;
        double r12113 = fma(r12107, r12111, r12112);
        return r12113;
}

Error

Bits error versus x

Target

Original32.3
Target0
Herbie0
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0.0\\ \end{array}\]

Derivation

  1. Initial program 32.3

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

  2. Simplified4.6

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

  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 2019315 +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)))))