Average Error: 14.7 → 0.0
Time: 1.6s
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 14.7

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

  2. Simplified14.7

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

  4. Applied add-exp-log_binary6414.7

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

  5. Applied add-exp-log_binary6414.7

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

  6. Applied prod-exp_binary6414.7

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

  7. Applied add-exp-log_binary6414.7

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

  8. Applied add-exp-log_binary6414.7

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

  9. Applied prod-exp_binary6414.7

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

  10. Applied div-exp_binary640.0

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

  11. Simplified0.0

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

  12. Taylor expanded around 0 14.7

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

  13. Simplified0.0

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

  14. Final simplification0.0

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

Reproduce

herbie shell --seed 2021204 
(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)))))