Average Error: 15.0 → 0.0
Time: 4.6s
Precision: binary64
\[\left(0 \leq b \land b \leq a\right) \land a \leq 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
↓
\[{\left({\left(\frac{a - b \cdot \frac{b}{a}}{a}\right)}^{1.5}\right)}^{0.3333333333333333} \]
(FPCore (a b)
 :precision binary64
 (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))
↓
(FPCore (a b)
 :precision binary64
 (pow (pow (/ (- a (* b (/ b a))) a) 1.5) 0.3333333333333333))
double code(double a, double b) {
	return sqrt(fabs((((a * a) - (b * b)) / (a * a))));
}
↓
double code(double a, double b) {
	return pow(pow(((a - (b * (b / a))) / a), 1.5), 0.3333333333333333);
}

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 = (((a - (b * (b / a))) / a) ** 1.5d0) ** 0.3333333333333333d0
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.pow(Math.pow(((a - (b * (b / a))) / a), 1.5), 0.3333333333333333);
}

def code(a, b):
	return math.sqrt(math.fabs((((a * a) - (b * b)) / (a * a))))
↓
def code(a, b):
	return math.pow(math.pow(((a - (b * (b / a))) / a), 1.5), 0.3333333333333333)

function code(a, b)
	return sqrt(abs(Float64(Float64(Float64(a * a) - Float64(b * b)) / Float64(a * a))))
end
↓
function code(a, b)
	return (Float64(Float64(a - Float64(b * Float64(b / a))) / a) ^ 1.5) ^ 0.3333333333333333
end

function tmp = code(a, b)
	tmp = sqrt(abs((((a * a) - (b * b)) / (a * a))));
end
↓
function tmp = code(a, b)
	tmp = (((a - (b * (b / a))) / a) ^ 1.5) ^ 0.3333333333333333;
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[Power[N[Power[N[(N[(a - N[(b * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], 1.5], $MachinePrecision], 0.3333333333333333], $MachinePrecision]

\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
{\left({\left(\frac{a - b \cdot \frac{b}{a}}{a}\right)}^{1.5}\right)}^{0.3333333333333333}

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 15.0

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

  2. Simplified0.0

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

  3. Applied egg-rr0.0

    \[\leadsto \color{blue}{{\left({\left(\frac{a - b \cdot \frac{b}{a}}{a}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

  4. Final simplification0.0

    \[\leadsto {\left({\left(\frac{a - b \cdot \frac{b}{a}}{a}\right)}^{1.5}\right)}^{0.3333333333333333} \]

Reproduce

herbie shell --seed 2022155 
(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)))))