Average Error: 14.5 → 0.0
Time: 3.1s
Precision: binary64
\[0 \leq b \land b \leq a \land a \leq 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
↓
\[{\left(\left|\mathsf{fma}\left(b, \frac{\frac{b}{a}}{a}, -1\right)\right|\right)}^{0.5} \]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
{\left(\left|\mathsf{fma}\left(b, \frac{\frac{b}{a}}{a}, -1\right)\right|\right)}^{0.5}

(FPCore (a b)
 :precision binary64
 (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))
↓
(FPCore (a b) :precision binary64 (pow (fabs (fma b (/ (/ b a) a) -1.0)) 0.5))
double code(double a, double b) {
	return sqrt(fabs(((a * a) - (b * b)) / (a * a)));
}
↓
double code(double a, double b) {
	return pow(fabs(fma(b, ((b / a) / a), -1.0)), 0.5);
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.5

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

  2. Simplified14.4

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

  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 pow1/2_binary640.0

    \[\leadsto \color{blue}{{\left(\left|\mathsf{fma}\left(b, \frac{\frac{b}{a}}{a}, -1\right)\right|\right)}^{0.5}} \]

  5. Final simplification0.0

    \[\leadsto {\left(\left|\mathsf{fma}\left(b, \frac{\frac{b}{a}}{a}, -1\right)\right|\right)}^{0.5} \]

Reproduce

herbie shell --seed 2021215 
(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)))))