Average Error: 14.6 → 0.0
Time: 2.4s
Precision: binary64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \left(\sqrt{b} \cdot \left|\frac{\sqrt{b}}{a}\right|\right) \cdot \left(\sqrt{b} \cdot \left|\frac{\sqrt{b}}{a}\right|\right)\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|1 - \left(\sqrt{b} \cdot \left|\frac{\sqrt{b}}{a}\right|\right) \cdot \left(\sqrt{b} \cdot \left|\frac{\sqrt{b}}{a}\right|\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) (((double) (((double) sqrt(b)) * ((double) fabs(((double) (((double) sqrt(b)) / a)))))) * ((double) (((double) sqrt(b)) * ((double) fabs(((double) (((double) sqrt(b)) / 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 14.6

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

  2. Simplified14.6

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

  4. Applied add-sqr-sqrt14.6

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

  5. Applied add-sqr-sqrt14.6

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

  6. Applied unswap-sqr14.6

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

  7. Simplified14.6

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2020181 
(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)))))