Average Error: 13.8 → 0.0
Time: 3.0s
Precision: binary64
\[\left(0 \leq b \land b \leq a\right) \land a \leq 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
↓
\[\sqrt{\left|\log \left(e^{\frac{a + b}{\frac{a}{\frac{a - b}{a}}}}\right)\right|} \]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\log \left(e^{\frac{a + b}{\frac{a}{\frac{a - b}{a}}}}\right)\right|}

(FPCore (a b)
 :precision binary64
 (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))
↓
(FPCore (a b)
 :precision binary64
 (sqrt (fabs (log (exp (/ (+ a b) (/ a (/ (- a b) a))))))))
double code(double a, double b) {
	return sqrt(fabs((((a * a) - (b * b)) / (a * a))));
}
↓
double code(double a, double b) {
	return sqrt(fabs(log(exp(((a + b) / (a / ((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 13.8

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

  2. Applied egg-rr0.0

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

  3. Applied egg-rr0.0

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

  4. Applied egg-rr0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2022129 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (and (and (<= 0.0 b) (<= b a)) (<= a 1.0))
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))