Average Error: 13.7 → 0.0
Time: 3.9s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\log \left(e^{\sqrt{\left|\frac{\frac{a + b}{a}}{\sqrt{a}} \cdot \frac{a - b}{\sqrt{a}}\right|}}\right)\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\log \left(e^{\sqrt{\left|\frac{\frac{a + b}{a}}{\sqrt{a}} \cdot \frac{a - b}{\sqrt{a}}\right|}}\right)
double code(double a, double b) {
	return sqrt(fabs((((a * a) - (b * b)) / (a * a))));
}
↓
double code(double a, double b) {
	return log(exp(sqrt(fabs(((((a + b) / a) / sqrt(a)) * ((a - b) / sqrt(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 13.7

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

  3. Applied difference-of-squares13.7

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

  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-log-exp0.0

    \[\leadsto \color{blue}{\log \left(e^{\sqrt{\left|\frac{a + b}{a} \cdot \frac{a - b}{a}\right|}}\right)}\]
  7. Using strategy rm

  8. Applied add-sqr-sqrt0.0

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

  9. Applied *-un-lft-identity0.0

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

  10. Applied times-frac0.0

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

  11. Applied associate-*r*0.0

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

  12. Simplified0.0

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

  13. Final simplification0.0

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

Reproduce

herbie shell --seed 2020060 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))