Average Error: 14.5 → 0.0
Time: 2.6s
Precision: binary64
\[\left(0 \leq b \land b \leq a\right) \land a \leq 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
↓
\[\sqrt{\left|\frac{1 - {\left(\frac{b}{a}\right)}^{4}}{-1 - {\left(\frac{b}{a}\right)}^{2}}\right|} \]
(FPCore (a b)
 :precision binary64
 (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))
↓
(FPCore (a b)
 :precision binary64
 (sqrt (fabs (/ (- 1.0 (pow (/ b a) 4.0)) (- -1.0 (pow (/ b a) 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 - pow((b / a), 4.0)) / (-1.0 - pow((b / a), 2.0)))));
}

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = sqrt(abs((((a * a) - (b * b)) / (a * a))))
end function
↓
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = sqrt(abs(((1.0d0 - ((b / a) ** 4.0d0)) / ((-1.0d0) - ((b / a) ** 2.0d0)))))
end function

public static double code(double a, double b) {
	return Math.sqrt(Math.abs((((a * a) - (b * b)) / (a * a))));
}
↓
public static double code(double a, double b) {
	return Math.sqrt(Math.abs(((1.0 - Math.pow((b / a), 4.0)) / (-1.0 - Math.pow((b / a), 2.0)))));
}

def code(a, b):
	return math.sqrt(math.fabs((((a * a) - (b * b)) / (a * a))))
↓
def code(a, b):
	return math.sqrt(math.fabs(((1.0 - math.pow((b / a), 4.0)) / (-1.0 - math.pow((b / a), 2.0)))))

function code(a, b)
	return sqrt(abs(Float64(Float64(Float64(a * a) - Float64(b * b)) / Float64(a * a))))
end
↓
function code(a, b)
	return sqrt(abs(Float64(Float64(1.0 - (Float64(b / a) ^ 4.0)) / Float64(-1.0 - (Float64(b / a) ^ 2.0)))))
end

function tmp = code(a, b)
	tmp = sqrt(abs((((a * a) - (b * b)) / (a * a))));
end
↓
function tmp = code(a, b)
	tmp = sqrt(abs(((1.0 - ((b / a) ^ 4.0)) / (-1.0 - ((b / a) ^ 2.0)))));
end

code[a_, b_] := N[Sqrt[N[Abs[N[(N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]
↓
code[a_, b_] := N[Sqrt[N[Abs[N[(N[(1.0 - N[Power[N[(b / a), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[Power[N[(b / a), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{1 - {\left(\frac{b}{a}\right)}^{4}}{-1 - {\left(\frac{b}{a}\right)}^{2}}\right|}

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 14.5

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

  2. Simplified14.5

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

  3. Taylor expanded in b around 0 14.5

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

  4. Simplified0.0

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

  5. Applied egg-rr0.0

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

  6. Applied egg-rr0.0

    \[\leadsto \sqrt{\left|\color{blue}{\frac{1 - {\left(\frac{b}{a}\right)}^{4}}{-1 - {\left(\frac{b}{a}\right)}^{2}}}\right|} \]

  7. Final simplification0.0

    \[\leadsto \sqrt{\left|\frac{1 - {\left(\frac{b}{a}\right)}^{4}}{-1 - {\left(\frac{b}{a}\right)}^{2}}\right|} \]

Reproduce

herbie shell --seed 2022165 
(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)))))