Average Error: 14.3 → 0.0
Time: 1.8s
Precision: binary64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \log \left(e^{{\left(\frac{b}{a}\right)}^{2}}\right)\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|1 - \log \left(e^{{\left(\frac{b}{a}\right)}^{2}}\right)\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) (1.0 - ((double) log(((double) exp(((double) pow(((double) (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. Simplified14.2

    \[\leadsto \color{blue}{\sqrt{\left|1 - b \cdot \frac{b}{a \cdot a}\right|}}\]
  3. Using strategy rm

  4. Applied add-cbrt-cube27.1

    \[\leadsto \sqrt{\left|1 - b \cdot \frac{b}{a \cdot \color{blue}{\sqrt[3]{\left(a \cdot a\right) \cdot a}}}\right|}\]

  5. Applied add-cbrt-cube27.1

    \[\leadsto \sqrt{\left|1 - b \cdot \frac{b}{\color{blue}{\sqrt[3]{\left(a \cdot a\right) \cdot a}} \cdot \sqrt[3]{\left(a \cdot a\right) \cdot a}}\right|}\]

  6. Applied cbrt-unprod44.1

    \[\leadsto \sqrt{\left|1 - b \cdot \frac{b}{\color{blue}{\sqrt[3]{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)}}}\right|}\]

  7. Applied add-cbrt-cube44.1

    \[\leadsto \sqrt{\left|1 - b \cdot \frac{\color{blue}{\sqrt[3]{\left(b \cdot b\right) \cdot b}}}{\sqrt[3]{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)}}\right|}\]

  8. Applied cbrt-undiv44.1

    \[\leadsto \sqrt{\left|1 - b \cdot \color{blue}{\sqrt[3]{\frac{\left(b \cdot b\right) \cdot b}{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)}}}\right|}\]

  9. Applied add-cbrt-cube44.1

    \[\leadsto \sqrt{\left|1 - \color{blue}{\sqrt[3]{\left(b \cdot b\right) \cdot b}} \cdot \sqrt[3]{\frac{\left(b \cdot b\right) \cdot b}{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)}}\right|}\]

  10. Applied cbrt-unprod44.1

    \[\leadsto \sqrt{\left|1 - \color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot b\right) \cdot \frac{\left(b \cdot b\right) \cdot b}{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)}}}\right|}\]

  11. Simplified0.0

    \[\leadsto \sqrt{\left|1 - \sqrt[3]{\color{blue}{{\left(\frac{b}{a}\right)}^{6}}}\right|}\]
  12. Using strategy rm

  13. Applied add-log-exp0.0

    \[\leadsto \sqrt{\left|1 - \color{blue}{\log \left(e^{\sqrt[3]{{\left(\frac{b}{a}\right)}^{6}}}\right)}\right|}\]

  14. Simplified0.0

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

  15. Final simplification0.0

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

Reproduce

herbie shell --seed 2020185 
(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)))))