Average Error: 32.4 → 0.0
Time: 2.0s
Precision: 64
\[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}\]
↓
\[2 \cdot \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right) + \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right)\]
\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}
↓
2 \cdot \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right) + \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right)
double code(double x) {
	return ((x / x) - ((1.0 / x) * sqrt((x * x))));
}
↓
double code(double x) {
	return ((2.0 * log(cbrt(exp(fma(-(1.0 / x), fabs(x), 1.0))))) + log(cbrt(exp(fma(-(1.0 / x), fabs(x), 1.0)))));
}

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. Simplified30.6

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

  4. Applied add-log-exp4.8

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

  6. Applied add-cube-cbrt0.0

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

  7. Applied log-prod0.0

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

  8. Simplified0.0

    \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right)} + \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right)\]

  9. Final simplification0.0

    \[\leadsto 2 \cdot \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right) + \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right)\]

Reproduce

herbie shell --seed 2020057 +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)))))