Average Error: 14.3 → 0.0
Time: 3.2s
Precision: binary64
\[0 \leq b \land b \leq a \land a \leq 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - {\left(\frac{b}{a}\right)}^{2}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|1 - {\left(\frac{b}{a}\right)}^{2}\right|}
(FPCore (a b)
 :precision binary64
 (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))
↓
(FPCore (a b) :precision binary64 (sqrt (fabs (- 1.0 (pow (/ b a) 2.0)))))
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 - pow((b / a), 2.0)));
}

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 14.3

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

  3. Applied associate-/r*_binary64_70414.5

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

  4. Simplified0.0

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

  6. Applied div-sub_binary64_7650.0

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

  7. Simplified0.0

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

  8. Taylor expanded around 0 14.3

    \[\leadsto \sqrt{\left|1 - \color{blue}{\frac{{b}^{2}}{{a}^{2}}}\right|}\]

  9. Simplified0.0

    \[\leadsto \sqrt{\left|1 - \color{blue}{{\left(\frac{b}{a}\right)}^{2}}\right|}\]

  10. Final simplification0.0

    \[\leadsto \sqrt{\left|1 - {\left(\frac{b}{a}\right)}^{2}\right|}\]

Reproduce

herbie shell --seed 2021032 
(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)))))