Average Error: 15.1 → 0.0
Time: 9.8s
Precision: binary64
\[\left(0 \leq b \land b \leq a\right) \land a \leq 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\frac{b}{a}}\\ \sqrt{\left|\mathsf{fma}\left(b, \frac{t_0 \cdot t_0}{\sqrt{a}} \cdot \frac{t_0}{\sqrt{a}}, -1\right)\right|} \end{array} \]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\begin{array}{l}
t_0 := \sqrt[3]{\frac{b}{a}}\\
\sqrt{\left|\mathsf{fma}\left(b, \frac{t_0 \cdot t_0}{\sqrt{a}} \cdot \frac{t_0}{\sqrt{a}}, -1\right)\right|}
\end{array}

(FPCore (a b)
 :precision binary64
 (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))
↓
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (cbrt (/ b a))))
   (sqrt (fabs (fma b (* (/ (* t_0 t_0) (sqrt a)) (/ t_0 (sqrt a))) -1.0)))))
double code(double a, double b) {
	return sqrt(fabs(((a * a) - (b * b)) / (a * a)));
}
↓
double code(double a, double b) {
	double t_0 = cbrt(b / a);
	return sqrt(fabs(fma(b, (((t_0 * t_0) / sqrt(a)) * (t_0 / sqrt(a))), -1.0)));
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.1

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

  2. Simplified15.1

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

  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

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

  5. Applied add-cube-cbrt_binary640.0

    \[\leadsto \sqrt{\left|\mathsf{fma}\left(b, \frac{\color{blue}{\left(\sqrt[3]{\frac{b}{a}} \cdot \sqrt[3]{\frac{b}{a}}\right) \cdot \sqrt[3]{\frac{b}{a}}}}{\sqrt{a} \cdot \sqrt{a}}, -1\right)\right|} \]

  6. Applied times-frac_binary640.0

    \[\leadsto \sqrt{\left|\mathsf{fma}\left(b, \color{blue}{\frac{\sqrt[3]{\frac{b}{a}} \cdot \sqrt[3]{\frac{b}{a}}}{\sqrt{a}} \cdot \frac{\sqrt[3]{\frac{b}{a}}}{\sqrt{a}}}, -1\right)\right|} \]

  7. Final simplification0.0

    \[\leadsto \sqrt{\left|\mathsf{fma}\left(b, \frac{\sqrt[3]{\frac{b}{a}} \cdot \sqrt[3]{\frac{b}{a}}}{\sqrt{a}} \cdot \frac{\sqrt[3]{\frac{b}{a}}}{\sqrt{a}}, -1\right)\right|} \]

Reproduce

herbie shell --seed 2021275 
(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)))))