Average Error: 14.9 → 0.0
Time: 3.1s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{a + b}{a} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a - b}{a}\right)\right)\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{a + b}{a} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a - b}{a}\right)\right)\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) * expm1(log1p(((a - b) / 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 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 expm1-log1p-u0.0

    \[\leadsto \sqrt{\left|\frac{a + b}{a} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a - b}{a}\right)\right)}\right|}\]

  7. Final simplification0.0

    \[\leadsto \sqrt{\left|\frac{a + b}{a} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a - b}{a}\right)\right)\right|}\]

Reproduce

herbie shell --seed 2020079 +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)))))