Average Error: 14.1 → 0.0
Time: 2.8s
Precision: binary64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\left(\sqrt[3]{\frac{a + b}{a} \cdot \frac{a - b}{a}} \cdot \sqrt[3]{\frac{a + b}{a} \cdot \frac{a - b}{a}}\right) \cdot \sqrt[3]{\frac{a + b}{a} \cdot \frac{a - b}{a}}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\left(\sqrt[3]{\frac{a + b}{a} \cdot \frac{a - b}{a}} \cdot \sqrt[3]{\frac{a + b}{a} \cdot \frac{a - b}{a}}\right) \cdot \sqrt[3]{\frac{a + b}{a} \cdot \frac{a - b}{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) (((double) (((double) cbrt(((double) (((double) (((double) (a + b)) / a)) * ((double) (((double) (a - b)) / a)))))) * ((double) cbrt(((double) (((double) (((double) (a + b)) / a)) * ((double) (((double) (a - b)) / a)))))))) * ((double) cbrt(((double) (((double) (((double) (a + b)) / a)) * ((double) (((double) (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 14.1

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

  3. Applied difference-of-squares14.1

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

  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 add-cube-cbrt0.0

    \[\leadsto \sqrt{\left|\color{blue}{\left(\sqrt[3]{\frac{a + b}{a} \cdot \frac{a - b}{a}} \cdot \sqrt[3]{\frac{a + b}{a} \cdot \frac{a - b}{a}}\right) \cdot \sqrt[3]{\frac{a + b}{a} \cdot \frac{a - b}{a}}}\right|}\]

  7. Final simplification0.0

    \[\leadsto \sqrt{\left|\left(\sqrt[3]{\frac{a + b}{a} \cdot \frac{a - b}{a}} \cdot \sqrt[3]{\frac{a + b}{a} \cdot \frac{a - b}{a}}\right) \cdot \sqrt[3]{\frac{a + b}{a} \cdot \frac{a - b}{a}}\right|}\]

Reproduce

herbie shell --seed 2020161 
(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)))))