Average Error: 13.9 → 0.0
Time: 2.1s
Precision: binary64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - b \cdot \frac{\frac{b}{a}}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|1 - b \cdot \frac{\frac{b}{a}}{a}\right|}
double code(double a, double b) {
	return ((double) sqrt(((double) fabs(((double) (((double) (((double) (a * a)) - ((double) (b * b)))) / ((double) (a * a))))))));
}
↓
double code(double a, double b) {
	return ((double) sqrt(((double) fabs(((double) (1.0 - ((double) (b * ((double) (((double) (b / a)) / a))))))))));
}

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 13.9

    \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]

  2. Simplified13.9

    \[\leadsto \color{blue}{\sqrt{\left|1 - b \cdot \frac{b}{a \cdot a}\right|}}\]
  3. Using strategy rm

  4. Applied associate-/r*0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020184 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1.0)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))