Average Error: 14.1 → 0.0
Time: 3.0s
Precision: binary64
\[0 \leq b \land b \leq a \land a \leq 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}
(FPCore (a b)
 :precision binary64
 (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))
↓
(FPCore (a b) :precision binary64 (sqrt (fabs (- 1.0 (* (/ b a) (/ b a))))))
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) (1.0 - ((double) ((b / a) * (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 Error: 14.1 bits

    \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
  2. Using strategy rm

  3. Applied difference-of-squaresError: 14.1 bits

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

  4. Applied associate-/l*Error: 14.7 bits

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

  5. SimplifiedError: 0.0 bits

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

  6. Taylor expanded around 0 Error: 14.1 bits

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

  7. SimplifiedError: 0.0 bits

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

  8. Final simplificationError: 0.0 bits

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

Reproduce

herbie shell --seed 2020205 
(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)))))