Average Error: 14.8 → 0.0
Time: 2.1s
Precision: binary64
\[0 \leq b \land b \leq a \land a \leq 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
↓
\[\begin{array}{l} t_0 := \frac{\sqrt{b}}{a}\\ \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left|\mathsf{fma}\left(b, t_0 \cdot t_0, -1\right)\right|}\right)\right) \end{array} \]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\begin{array}{l}
t_0 := \frac{\sqrt{b}}{a}\\
\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left|\mathsf{fma}\left(b, t_0 \cdot t_0, -1\right)\right|}\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 (/ (sqrt b) a)))
   (expm1 (log1p (sqrt (fabs (fma b (* t_0 t_0) -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 = sqrt(b) / a;
	return expm1(log1p(sqrt(fabs(fma(b, (t_0 * t_0), -1.0)))));
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.8

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

  2. Simplified14.8

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

  4. Applied add-sqr-sqrt_binary6414.8

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

  5. Simplified14.8

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

  6. Simplified0.0

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

  8. Applied expm1-log1p-u_binary640.0

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left|\mathsf{fma}\left(b, \left|\frac{\sqrt{b}}{a}\right| \cdot \left|\frac{\sqrt{b}}{a}\right|, -1\right)\right|}\right)\right)} \]

  9. Final simplification0.0

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left|\mathsf{fma}\left(b, \frac{\sqrt{b}}{a} \cdot \frac{\sqrt{b}}{a}, -1\right)\right|}\right)\right) \]

Reproduce

herbie shell --seed 2021211 
(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)))))