Average Error: 32.4 → 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 r132417 = x;
        double r132418 = r132417 / r132417;
        double r132419 = 1.0;
        double r132420 = r132419 / r132417;
        double r132421 = r132417 * r132417;
        double r132422 = sqrt(r132421);
        double r132423 = r132420 * r132422;
        double r132424 = r132418 - r132423;
        return r132424;
}
↓
double f(double x) {
        double r132425 = 1.0;
        double r132426 = x;
        double r132427 = r132425 / r132426;
        double r132428 = -r132427;
        double r132429 = fabs(r132426);
        double r132430 = r132428 * r132429;
        double r132431 = 3.0;
        double r132432 = pow(r132430, r132431);
        double r132433 = cbrt(r132432);
        double r132434 = 1.0;
        double r132435 = r132433 + r132434;
        return r132435;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original32.4
Target0
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0.0\\ \end{array}\]

Derivation

  1. Initial program 32.4

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

  2. Simplified30.4

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

  4. Applied fma-udef4.9

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

  6. Applied add-cbrt-cube45.7

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

  7. Applied add-cbrt-cube49.6

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

  8. Applied cbrt-unprod44.1

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

  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 2020001 +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)))))