Average Error: 32.5 → 0.0
Time: 2.0s
Precision: 64
\[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}\]
↓
\[\sqrt[3]{{\left(\left(-\frac{1}{x}\right) \cdot \left|x\right|\right)}^{3}} + 1\]
\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}
↓
\sqrt[3]{{\left(\left(-\frac{1}{x}\right) \cdot \left|x\right|\right)}^{3}} + 1
double f(double x) {
        double r93558 = x;
        double r93559 = r93558 / r93558;
        double r93560 = 1.0;
        double r93561 = r93560 / r93558;
        double r93562 = r93558 * r93558;
        double r93563 = sqrt(r93562);
        double r93564 = r93561 * r93563;
        double r93565 = r93559 - r93564;
        return r93565;
}
↓
double f(double x) {
        double r93566 = 1.0;
        double r93567 = x;
        double r93568 = r93566 / r93567;
        double r93569 = -r93568;
        double r93570 = fabs(r93567);
        double r93571 = r93569 * r93570;
        double r93572 = 3.0;
        double r93573 = pow(r93571, r93572);
        double r93574 = cbrt(r93573);
        double r93575 = 1.0;
        double r93576 = r93574 + r93575;
        return r93576;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original32.5
Target0
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0.0\\ \end{array}\]

Derivation

  1. Initial program 32.5

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

  2. Simplified30.5

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

  4. Applied fma-udef4.7

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

  6. Applied add-cbrt-cube46.0

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

  7. Applied add-cbrt-cube50.1

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

  8. Applied cbrt-unprod44.2

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020024 +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)))))