Average Error: 14.4 → 0.0
Time: 3.0s
Precision: binary64
\[0 \leq b \land b \leq a \land a \leq 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
↓
\[{\left(\left|1 - \sqrt[3]{{\left(\frac{b}{a}\right)}^{6}}\right|\right)}^{0.5} \]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
{\left(\left|1 - \sqrt[3]{{\left(\frac{b}{a}\right)}^{6}}\right|\right)}^{0.5}

(FPCore (a b)
 :precision binary64
 (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))
↓
(FPCore (a b)
 :precision binary64
 (pow (fabs (- 1.0 (cbrt (pow (/ b a) 6.0)))) 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(1.0 - cbrt(pow((b / a), 6.0))), 0.5);
}

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 14.4

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

  2. Simplified14.4

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

  4. Applied add-cbrt-cube_binary6414.4

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

  5. Simplified0.0

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

  7. Applied pow1_binary640.0

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

  8. Applied sqrt-pow1_binary640.0

    \[\leadsto \color{blue}{{\left(\left|1 - \sqrt[3]{{\left(\frac{b}{a}\right)}^{6}}\right|\right)}^{\left(\frac{1}{2}\right)}} \]

  9. Final simplification0.0

    \[\leadsto {\left(\left|1 - \sqrt[3]{{\left(\frac{b}{a}\right)}^{6}}\right|\right)}^{0.5} \]

Reproduce

herbie shell --seed 2021198 
(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)))))