Average Error: 14.0 → 0.0
Time: 5.0s
Precision: binary64
Cost: 13440
\[\left(0 \leq b \land b \leq a\right) \land a \leq 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
↓
\[e^{\mathsf{log1p}\left(\frac{\frac{-b}{a}}{\frac{a}{b}}\right) \cdot 0.5} \]
(FPCore (a b)
 :precision binary64
 (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))
↓
(FPCore (a b)
 :precision binary64
 (exp (* (log1p (/ (/ (- b) a) (/ a b))) 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(((-b / a) / (a / b))) * 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(((-b / a) / (a / b))) * 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(((-b / a) / (a / b))) * 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(Float64(-b) / a) / Float64(a / b))) * 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[(N[((-b) / a), $MachinePrecision] / N[(a / b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]

\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
e^{\mathsf{log1p}\left(\frac{\frac{-b}{a}}{\frac{a}{b}}\right) \cdot 0.5}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.0

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

  2. Applied egg-rr0.0

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

  3. Applied egg-rr0.0

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

  4. Final simplification0.0

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

Alternatives

Alternative 1
Error0.0
Cost7040
\[{\left(1 - \frac{\frac{b}{a}}{\frac{a}{b}}\right)}^{0.5} \]
Alternative 2
Error0.6
Cost6976
\[\mathsf{fma}\left(-0.5, \frac{b}{a} \cdot \frac{b}{a}, 1\right) \]
Alternative 3
Error1.3
Cost64
\[1 \]

Error

Reproduce

herbie shell --seed 2022293 
(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)))))