Average Error: 14.1 → 0.0
Time: 2.8s
Precision: binary64
\[\left(0 \leq b \land b \leq a\right) \land a \leq 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
↓
\[\left|\sqrt{\left|\mathsf{fma}\left(b, \frac{\frac{b}{a}}{a}, -1\right)\right|}\right| \]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\left|\sqrt{\left|\mathsf{fma}\left(b, \frac{\frac{b}{a}}{a}, -1\right)\right|}\right|

(FPCore (a b)
 :precision binary64
 (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))
↓
(FPCore (a b)
 :precision binary64
 (fabs (sqrt (fabs (fma b (/ (/ b a) a) -1.0)))))
double code(double a, double b) {
	return sqrt(fabs(((a * a) - (b * b)) / (a * a)));
}
↓
double code(double a, double b) {
	return fabs(sqrt(fabs(fma(b, ((b / a) / a), -1.0))));
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.1

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

  2. Simplified14.1

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

  3. Applied associate-/r*_binary640.0

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

  4. Applied add-sqr-sqrt_binary640.0

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

  5. Applied rem-sqrt-square_binary640.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2022088 
(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)))))