Average Error: 14.9 → 0.0
Time: 2.6s
Precision: binary64
\[\left(0 \leq b \land b \leq a\right) \land a \leq 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
↓
\[\sqrt{\left|\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a + b}{a} \cdot \frac{a - b}{a}\right)\right)\right|} \]
(FPCore (a b)
 :precision binary64
 (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))
↓
(FPCore (a b)
 :precision binary64
 (sqrt (fabs (expm1 (log1p (* (/ (+ a b) a) (/ (- a b) a)))))))
double code(double a, double b) {
	return sqrt(fabs((((a * a) - (b * b)) / (a * a))));
}
↓
double code(double a, double b) {
	return sqrt(fabs(expm1(log1p((((a + b) / a) * ((a - b) / a))))));
}

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(Math.expm1(Math.log1p((((a + b) / a) * ((a - b) / a))))));
}

def code(a, b):
	return math.sqrt(math.fabs((((a * a) - (b * b)) / (a * a))))
↓
def code(a, b):
	return math.sqrt(math.fabs(math.expm1(math.log1p((((a + b) / a) * ((a - b) / a))))))

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(expm1(log1p(Float64(Float64(Float64(a + b) / a) * Float64(Float64(a - b) / a))))))
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[(Exp[N[Log[1 + N[(N[(N[(a + b), $MachinePrecision] / a), $MachinePrecision] * N[(N[(a - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a + b}{a} \cdot \frac{a - b}{a}\right)\right)\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 14.9

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

  2. Applied egg-rr0.0

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

  3. Applied egg-rr0.0

    \[\leadsto \sqrt{\left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a + b}{a} \cdot \frac{a - b}{a}\right)\right)}\right|} \]

  4. Final simplification0.0

    \[\leadsto \sqrt{\left|\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a + b}{a} \cdot \frac{a - b}{a}\right)\right)\right|} \]

Reproduce

herbie shell --seed 2022133 
(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)))))