Average Error: 14.9 → 0.0
Time: 6.6s
Precision: binary64
\[\left(0 \leq b \land b \leq a\right) \land a \leq 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
↓
\[\sqrt{\left|\sqrt[3]{{\left(\frac{b}{a}\right)}^{6}} + -1\right|} \]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\sqrt[3]{{\left(\frac{b}{a}\right)}^{6}} + -1\right|}

(FPCore (a b)
 :precision binary64
 (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))
↓
(FPCore (a b)
 :precision binary64
 (sqrt (fabs (+ (cbrt (pow (/ b a) 6.0)) -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(cbrt(pow((b / a), 6.0)) + -1.0));
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.9

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

  2. Simplified14.4

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

  3. Applied clear-num_binary6414.4

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

  4. Simplified0.0

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

  5. Applied fma-udef_binary640.0

    \[\leadsto \sqrt{\left|\color{blue}{b \cdot \frac{1}{\frac{a}{\frac{b}{a}}} + -1}\right|} \]

  6. Applied add-cbrt-cube_binary640.0

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

  7. Simplified0.0

    \[\leadsto \sqrt{\left|\sqrt[3]{\color{blue}{{\left(\frac{b}{a}\right)}^{6}}} + -1\right|} \]

  8. Final simplification0.0

    \[\leadsto \sqrt{\left|\sqrt[3]{{\left(\frac{b}{a}\right)}^{6}} + -1\right|} \]

Reproduce

herbie shell --seed 2022068 
(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)))))