Average Error: 14.1 → 0.0
Time: 2.7s
Precision: binary64
\[0 \leq b \land b \leq a \land a \leq 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
↓
\[\sqrt{\left|\mathsf{fma}\left(b, \frac{1}{\frac{a}{\frac{b}{a}}}, -1\right)\right|} \]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\mathsf{fma}\left(b, \frac{1}{\frac{a}{\frac{b}{a}}}, -1\right)\right|}

(FPCore (a b)
 :precision binary64
 (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))
↓
(FPCore (a b)
 :precision binary64
 (sqrt (fabs (fma b (/ 1.0 (/ a (/ b 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 sqrt(fabs(fma(b, (1.0 / (a / (b / 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. Using strategy rm

  4. Applied clear-num_binary6414.1

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2021210 
(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)))))