Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, A

Average Error: 0.6 → 0

Time: 1.3s

Precision: binary64

Math

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

↓

\[{x}^{-2} \cdot 1\]

C

double code(double x) {
	return ((double) (1.0 / ((double) (x * x))));
}

↓

double code(double x) {
	return ((double) (((double) pow(x, -2.0)) * 1.0));
}

Error

Bits error versus x

Target

Original	0.6
Target	0.2
Herbie	0

\[\frac{\frac{1}{x}}{x}\]

Derivation

1. Initial program 0.6

   \[\frac{1}{x \cdot x}\]
2. Using strategy rm

3. Applied associate-/r* 0.2

   \[\leadsto \color{blue}{\frac{\frac{1}{x}}{x}}\]
4. Using strategy rm

5. Applied add-cube-cbrt 0.9

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

6. Applied *-un-lft-identity 0.9

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

7. Applied times-frac 0.9

   \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{1}{\sqrt[3]{x}}}}{x}\]

8. Applied associate-/l* 0.9

   \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\frac{x}{\frac{1}{\sqrt[3]{x}}}}}\]

9. Simplified 0.9

   \[\leadsto \frac{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\color{blue}{\sqrt[3]{x} \cdot \frac{x}{1}}}\]
10. Using strategy rm

11. Applied cbrt-unprod 1.0

    \[\leadsto \frac{\frac{1}{\color{blue}{\sqrt[3]{x \cdot x}}}}{\sqrt[3]{x} \cdot \frac{x}{1}}\]
12. Using strategy rm

13. Applied associate-*r/ 1.0

    \[\leadsto \frac{\frac{1}{\sqrt[3]{x \cdot x}}}{\color{blue}{\frac{\sqrt[3]{x} \cdot x}{1}}}\]

14. Applied associate-/r/ 1.0

    \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt[3]{x \cdot x}}}{\sqrt[3]{x} \cdot x} \cdot 1}\]

15. Simplified 0

    \[\leadsto \color{blue}{{x}^{-2}} \cdot 1\]

16. Final simplification 0

    \[\leadsto {x}^{-2} \cdot 1\]

Reproduce

herbie shell --seed 2020179 
(FPCore (x)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (/ (/ 1.0 x) x)

  (/ 1.0 (* x x)))