Average Error: 13.9 → 0.0
Time: 4.1s
Precision: binary64
\[0 \leq b \land b \leq a \land a \leq 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \log \left(e^{{\left({\left(\frac{b}{a}\right)}^{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right)}\right)}^{\left(\sqrt[3]{2}\right)}}\right)\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|1 - \log \left(e^{{\left({\left(\frac{b}{a}\right)}^{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right)}\right)}^{\left(\sqrt[3]{2}\right)}}\right)\right|}
(FPCore (a b)
 :precision binary64
 (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))
↓
(FPCore (a b)
 :precision binary64
 (sqrt
  (fabs
   (-
    1.0
    (log (exp (pow (pow (/ b a) (* (cbrt 2.0) (cbrt 2.0))) (cbrt 2.0))))))))
double code(double a, double b) {
	return sqrt(fabs(((a * a) - (b * b)) / (a * a)));
}
↓
double code(double a, double b) {
	return sqrt(fabs(1.0 - log(exp(pow(pow((b / a), (cbrt(2.0) * cbrt(2.0))), cbrt(2.0))))));
}

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 13.9

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

  2. Simplified13.9

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

  4. Applied add-log-exp_binary64_182213.9

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

  5. Simplified0.0

    \[\leadsto \sqrt{\left|1 - \log \color{blue}{\left(e^{{\left(\frac{b}{a}\right)}^{2}}\right)}\right|}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt_binary64_18180.0

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

  8. Applied pow-unpow_binary64_18600.0

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

  9. Final simplification0.0

    \[\leadsto \sqrt{\left|1 - \log \left(e^{{\left({\left(\frac{b}{a}\right)}^{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right)}\right)}^{\left(\sqrt[3]{2}\right)}}\right)\right|}\]

Reproduce

herbie shell --seed 2021064 
(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)))))