Average Error: 14.5 → 0.0
Time: 2.5s
Precision: binary64
\[\left(0 \leq b \land b \leq a\right) \land a \leq 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
↓
\[e^{\mathsf{log1p}\left(-{\left(\frac{b}{a}\right)}^{2}\right) \cdot 0.5} \]
(FPCore (a b)
 :precision binary64
 (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))
↓
(FPCore (a b) :precision binary64 (exp (* (log1p (- (pow (/ b a) 2.0))) 0.5)))
double code(double a, double b) {
	return sqrt(fabs((((a * a) - (b * b)) / (a * a))));
}
↓
double code(double a, double b) {
	return exp((log1p(-pow((b / a), 2.0)) * 0.5));
}

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.exp((Math.log1p(-Math.pow((b / a), 2.0)) * 0.5));
}

def code(a, b):
	return math.sqrt(math.fabs((((a * a) - (b * b)) / (a * a))))
↓
def code(a, b):
	return math.exp((math.log1p(-math.pow((b / a), 2.0)) * 0.5))

function code(a, b)
	return sqrt(abs(Float64(Float64(Float64(a * a) - Float64(b * b)) / Float64(a * a))))
end
↓
function code(a, b)
	return exp(Float64(log1p(Float64(-(Float64(b / a) ^ 2.0))) * 0.5))
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[Exp[N[(N[Log[1 + (-N[Power[N[(b / a), $MachinePrecision], 2.0], $MachinePrecision])], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]

\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
e^{\mathsf{log1p}\left(-{\left(\frac{b}{a}\right)}^{2}\right) \cdot 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 14.5

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

  2. Taylor expanded in a around 0 14.5

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

  3. Simplified0.0

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

  4. Applied egg-rr0.0

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-{\left(\frac{b}{a}\right)}^{2}\right) \cdot 0.5}} \]

  5. Final simplification0.0

    \[\leadsto e^{\mathsf{log1p}\left(-{\left(\frac{b}{a}\right)}^{2}\right) \cdot 0.5} \]

Reproduce

herbie shell --seed 2022153 
(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)))))