Average Error: 14.3 → 0.0
Time: 1.9s
Precision: binary64
\[0 \leq b \land b \leq a \land a \leq 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\sqrt[3]{{\left(1 - \frac{b}{a} \cdot \frac{b}{a}\right)}^{3}}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\sqrt[3]{{\left(1 - \frac{b}{a} \cdot \frac{b}{a}\right)}^{3}}\right|}
(FPCore (a b)
 :precision binary64
 (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))
↓
(FPCore (a b)
 :precision binary64
 (sqrt (fabs (cbrt (pow (- 1.0 (* (/ b a) (/ b a))) 3.0)))))
double code(double a, double b) {
	return ((double) sqrt(((double) fabs((((double) (((double) (a * a)) - ((double) (b * b)))) / ((double) (a * a)))))));
}
↓
double code(double a, double b) {
	return ((double) sqrt(((double) fabs(((double) cbrt(((double) pow(((double) (1.0 - ((double) ((b / a) * (b / a))))), 3.0))))))));
}

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. Simplified14.3

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

  4. Applied times-frac_binary640.0

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

  5. Applied add-sqr-sqrt_binary640.0

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

  6. Applied difference-of-squares_binary640.0

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

  7. Simplified0.0

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

  8. Simplified0.0

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

  10. Applied add-cbrt-cube_binary640.0

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

  11. Applied add-cbrt-cube_binary640.0

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

  12. Applied cbrt-unprod_binary640.0

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

  13. Simplified0.0

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

  14. Taylor expanded around 0 14.3

    \[\leadsto \sqrt{\left|\sqrt[3]{{\color{blue}{\left(1 - \frac{{b}^{2}}{{a}^{2}}\right)}}^{3}}\right|}\]

  15. Simplified0.0

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

  16. Final simplification0.0

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

Reproduce

herbie shell --seed 2020210 
(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)))))