Average Error: 32.8 → 0.0
Time: 12.5s
Precision: 64
\[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}\]
↓
\[1 - \sqrt[3]{{\left(\left|x\right| \cdot \frac{1}{x}\right)}^{3}}\]
\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}
↓
1 - \sqrt[3]{{\left(\left|x\right| \cdot \frac{1}{x}\right)}^{3}}
double f(double x) {
        double r94851 = x;
        double r94852 = r94851 / r94851;
        double r94853 = 1.0;
        double r94854 = r94853 / r94851;
        double r94855 = r94851 * r94851;
        double r94856 = sqrt(r94855);
        double r94857 = r94854 * r94856;
        double r94858 = r94852 - r94857;
        return r94858;
}
↓
double f(double x) {
        double r94859 = 1.0;
        double r94860 = x;
        double r94861 = fabs(r94860);
        double r94862 = 1.0;
        double r94863 = r94862 / r94860;
        double r94864 = r94861 * r94863;
        double r94865 = 3.0;
        double r94866 = pow(r94864, r94865);
        double r94867 = cbrt(r94866);
        double r94868 = r94859 - r94867;
        return r94868;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original32.8
Target0
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0.0\\ \end{array}\]

Derivation

  1. Initial program 32.8

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

  2. Simplified4.5

    \[\leadsto \color{blue}{1 - \frac{1}{x} \cdot \left|x\right|}\]
  3. Using strategy rm

  4. Applied add-cbrt-cube45.5

    \[\leadsto 1 - \frac{1}{x} \cdot \color{blue}{\sqrt[3]{\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|}}\]

  5. Applied add-cbrt-cube43.8

    \[\leadsto 1 - \frac{1}{\color{blue}{\sqrt[3]{\left(x \cdot x\right) \cdot x}}} \cdot \sqrt[3]{\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|}\]

  6. Applied add-cbrt-cube43.8

    \[\leadsto 1 - \frac{\color{blue}{\sqrt[3]{\left(1 \cdot 1\right) \cdot 1}}}{\sqrt[3]{\left(x \cdot x\right) \cdot x}} \cdot \sqrt[3]{\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|}\]

  7. Applied cbrt-undiv49.5

    \[\leadsto 1 - \color{blue}{\sqrt[3]{\frac{\left(1 \cdot 1\right) \cdot 1}{\left(x \cdot x\right) \cdot x}}} \cdot \sqrt[3]{\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|}\]

  8. Applied cbrt-unprod42.9

    \[\leadsto 1 - \color{blue}{\sqrt[3]{\frac{\left(1 \cdot 1\right) \cdot 1}{\left(x \cdot x\right) \cdot x} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)}}\]

  9. Simplified0.0

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

  10. Final simplification0.0

    \[\leadsto 1 - \sqrt[3]{{\left(\left|x\right| \cdot \frac{1}{x}\right)}^{3}}\]

Reproduce

herbie shell --seed 2019325 +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)))))