Average Error: 14.4 → 0.0
Time: 6.9s
Precision: binary64
\[0 \leq b \land b \leq a \land a \leq 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{1}{\frac{a}{a + b}} \cdot \frac{a - b}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{1}{\frac{a}{a + b}} \cdot \frac{a - b}{a}\right|}
(FPCore (a b)
 :precision binary64
 (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))
↓
(FPCore (a b)
 :precision binary64
 (sqrt (fabs (* (/ 1.0 (/ a (+ a b))) (/ (- a b) a)))))
double code(double a, double b) {
	return sqrt(fabs(((a * a) - (b * b)) / (a * a)));
}
↓
double code(double a, double b) {
	return sqrt(fabs((1.0 / (a / (a + b))) * ((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 14.4

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

  3. Applied difference-of-squares_binary6414.4

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

  4. Applied times-frac_binary640.0

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

  6. Applied *-un-lft-identity_binary640.0

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

  7. Applied *-un-lft-identity_binary640.0

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

  8. Applied distribute-lft-out_binary640.0

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

  9. Applied associate-/l*_binary640.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020277 
(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)))))