Average Error: 32.3 → 0.0
Time: 2.0s
Precision: 64
\[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}\]
↓
\[2 \cdot \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right) + \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right)\]
\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}
↓
2 \cdot \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right) + \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right)
double f(double x) {
        double r210278 = x;
        double r210279 = r210278 / r210278;
        double r210280 = 1.0;
        double r210281 = r210280 / r210278;
        double r210282 = r210278 * r210278;
        double r210283 = sqrt(r210282);
        double r210284 = r210281 * r210283;
        double r210285 = r210279 - r210284;
        return r210285;
}
↓
double f(double x) {
        double r210286 = 2.0;
        double r210287 = 1.0;
        double r210288 = x;
        double r210289 = r210287 / r210288;
        double r210290 = -r210289;
        double r210291 = fabs(r210288);
        double r210292 = 1.0;
        double r210293 = fma(r210290, r210291, r210292);
        double r210294 = exp(r210293);
        double r210295 = cbrt(r210294);
        double r210296 = log(r210295);
        double r210297 = r210286 * r210296;
        double r210298 = r210297 + r210296;
        return r210298;
}

Error

Bits error versus x

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. Simplified30.9

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

  4. Applied add-log-exp4.7

    \[\leadsto \color{blue}{\log \left(e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}\right)}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt0.0

    \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right)}\]

  7. Applied log-prod0.0

    \[\leadsto \color{blue}{\log \left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right) + \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right)}\]

  8. Simplified0.0

    \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right)} + \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right)\]

  9. Final simplification0.0

    \[\leadsto 2 \cdot \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right) + \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right)\]

Reproduce

herbie shell --seed 2019362 +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)))))