Average Error: 32.1 → 0
Time: 6.2s
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 r167653 = x;
        double r167654 = r167653 / r167653;
        double r167655 = 1.0;
        double r167656 = r167655 / r167653;
        double r167657 = r167653 * r167653;
        double r167658 = sqrt(r167657);
        double r167659 = r167656 * r167658;
        double r167660 = r167654 - r167659;
        return r167660;
}
↓
double f(double x) {
        double r167661 = 1.0;
        double r167662 = x;
        double r167663 = fabs(r167662);
        double r167664 = 1.0;
        double r167665 = r167664 / r167662;
        double r167666 = r167663 * r167665;
        double r167667 = 3.0;
        double r167668 = pow(r167666, r167667);
        double r167669 = cbrt(r167668);
        double r167670 = r167661 - r167669;
        return r167670;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original32.1
Target0
Herbie0
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0.0\\ \end{array}\]

Derivation

  1. Initial program 32.1

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

  2. Simplified4.9

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

  4. Applied add-cbrt-cube45.1

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

  5. Applied add-cbrt-cube43.1

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

  6. Applied add-cbrt-cube43.1

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

  7. Applied cbrt-undiv49.0

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

  8. Applied cbrt-unprod42.0

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

  9. Simplified0

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

  10. Final simplification0

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

Reproduce

herbie shell --seed 2020045 +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)))))