Average Error: 32.4 → 0.0
Time: 2.8s
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 r161964 = x;
        double r161965 = r161964 / r161964;
        double r161966 = 1.0;
        double r161967 = r161966 / r161964;
        double r161968 = r161964 * r161964;
        double r161969 = sqrt(r161968);
        double r161970 = r161967 * r161969;
        double r161971 = r161965 - r161970;
        return r161971;
}
↓
double f(double x) {
        double r161972 = 1.0;
        double r161973 = x;
        double r161974 = r161972 / r161973;
        double r161975 = -r161974;
        double r161976 = fabs(r161973);
        double r161977 = r161975 * r161976;
        double r161978 = 3.0;
        double r161979 = pow(r161977, r161978);
        double r161980 = cbrt(r161979);
        double r161981 = 1.0;
        double r161982 = r161980 + r161981;
        return r161982;
}

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. Simplified31.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}\]
  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-cube49.5

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

  8. Applied cbrt-unprod44.4

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

  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 2020100 +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)))))