Average Error: 14.9 → 0.0
Time: 1.8s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{-\left(a + b\right)}{\left(-a\right) \cdot \frac{a}{a - b}}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{-\left(a + b\right)}{\left(-a\right) \cdot \frac{a}{a - b}}\right|}
double code(double a, double b) {
	return sqrt(fabs((((a * a) - (b * b)) / (a * a))));
}
↓
double code(double a, double b) {
	return sqrt(fabs((-(a + b) / (-a * (a / (a - b))))));
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.9

    \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
  2. Using strategy rm

  3. Applied difference-of-squares14.9

    \[\leadsto \sqrt{\left|\frac{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}}{a \cdot a}\right|}\]

  4. Applied times-frac0.0

    \[\leadsto \sqrt{\left|\color{blue}{\frac{a + b}{a} \cdot \frac{a - b}{a}}\right|}\]
  5. Using strategy rm

  6. Applied clear-num0.0

    \[\leadsto \sqrt{\left|\frac{a + b}{a} \cdot \color{blue}{\frac{1}{\frac{a}{a - b}}}\right|}\]

  7. Applied frac-2neg0.0

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

  8. Applied frac-times0.0

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

  9. Simplified0.0

    \[\leadsto \sqrt{\left|\frac{\color{blue}{-\left(a + b\right)}}{\left(-a\right) \cdot \frac{a}{a - b}}\right|}\]

  10. Final simplification0.0

    \[\leadsto \sqrt{\left|\frac{-\left(a + b\right)}{\left(-a\right) \cdot \frac{a}{a - b}}\right|}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))