Average Error: 14.0 → 0.0
Time: 3.4s
Precision: binary64
\[0 \leq b \land b \leq a \land a \leq 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{1 - {\left(\frac{b}{a}\right)}^{4}}{1 + {\left(\frac{b}{a}\right)}^{2}}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{1 - {\left(\frac{b}{a}\right)}^{4}}{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) 4.0)) (+ 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), 4.0)) / (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 14.0

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

  2. Simplified14.0

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

  4. Applied add-exp-log_binary64_79814.1

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

  5. Applied add-exp-log_binary64_79814.1

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

  6. Applied prod-exp_binary64_80914.1

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

  7. Applied add-exp-log_binary64_79814.1

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

  8. Applied add-exp-log_binary64_79814.1

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

  9. Applied prod-exp_binary64_80914.1

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

  10. Applied div-exp_binary64_8110.0

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

  11. Simplified0.0

    \[\leadsto \sqrt{\left|1 - e^{\color{blue}{2 \cdot \log \left(\frac{b}{a}\right)}}\right|}\]
  12. Using strategy rm

  13. Applied flip--_binary64_7350.0

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

  14. Simplified0.0

    \[\leadsto \sqrt{\left|\frac{\color{blue}{1 - {\left(\frac{b}{a}\right)}^{4}}}{1 + e^{2 \cdot \log \left(\frac{b}{a}\right)}}\right|}\]

  15. Simplified0.0

    \[\leadsto \sqrt{\left|\frac{1 - {\left(\frac{b}{a}\right)}^{4}}{\color{blue}{1 + {\left(\frac{b}{a}\right)}^{2}}}\right|}\]

  16. Final simplification0.0

    \[\leadsto \sqrt{\left|\frac{1 - {\left(\frac{b}{a}\right)}^{4}}{1 + {\left(\frac{b}{a}\right)}^{2}}\right|}\]

Reproduce

herbie shell --seed 2020354 
(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)))))