Statistics.Sample:$skurtosis from math-functions-0.1.5.2

Average Error: 5.0 → 0.1

Time: 1.9s

Precision: binary64

Math

\[\frac{x}{y \cdot y} - 3\]

↓

\[\frac{1}{y \cdot \frac{y}{x}} - 3\]

C

double code(double x, double y) {
	return ((double) (((double) (x / ((double) (y * y)))) - 3.0));
}

↓

double code(double x, double y) {
	return ((double) (((double) (1.0 / ((double) (y * ((double) (y / x)))))) - 3.0));
}

Error

Bits error versus x

Bits error versus y

Target

Original	5.0
Target	0.1
Herbie	0.1

\[\frac{\frac{x}{y}}{y} - 3\]

Derivation

1. Initial program 5.0

   \[\frac{x}{y \cdot y} - 3\]
2. Using strategy rm

3. Applied associate-/r* 0.1

   \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} - 3\]
4. Using strategy rm

5. Applied clear-num 0.1

   \[\leadsto \color{blue}{\frac{1}{\frac{y}{\frac{x}{y}}}} - 3\]
6. Using strategy rm

7. Applied add-sqr-sqrt 32.1

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

8. Applied *-un-lft-identity 32.1

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

9. Applied times-frac 32.1

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

10. Applied add-sqr-sqrt 32.1

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

11. Applied times-frac 32.1

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

12. Simplified 32.1

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

13. Simplified 0.1

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

14. Final simplification 0.1

    \[\leadsto \frac{1}{y \cdot \frac{y}{x}} - 3\]

Reproduce

herbie shell --seed 2020174 
(FPCore (x y)
  :name "Statistics.Sample:$skurtosis from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (- (/ (/ x y) y) 3.0)

  (- (/ x (* y y)) 3.0))