Eccentricity of an ellipse

Average Error: 14.7 → 0.0

Time: 6.9s

Precision: binary64

Cost: 13376

\[\left(0 \leq b \land b \leq a\right) \land a \leq 1\]

Math

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

↓

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

FPCore

(FPCore (a b)
 :precision binary64
 (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))

↓

(FPCore (a b) :precision binary64 (sqrt (fabs (- 1.0 (/ b (* a (/ a b)))))))

C

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 - (b / (a * (a / b))))));
}

Fortran

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 * (a / b))))))
end function

Java

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 - (b / (a * (a / b))))));
}

Python

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 - (b / (a * (a / b))))))

Julia

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(1.0 - Float64(b / Float64(a * Float64(a / b))))))
end

MATLAB

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 * (a / b))))));
end

Wolfram

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[(1.0 - N[(b / N[(a * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 14.7

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

2. Taylor expanded in a around 0 14.7

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

3. Simplified 0.0

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

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	53.6
Cost	1220

\[\begin{array}{l} t_0 := 1 + \left(1 - \frac{b}{a}\right)\\ \mathbf{if}\;a \leq 2.3164228816471113 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{b}{a} \cdot 2}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - b \cdot \frac{b}{a \cdot a}}{t_0}\\ \end{array} \]

Alternative 2
Error	52.0
Cost	1216

\[\frac{1 + \frac{b}{a} \cdot \left(2 - \frac{b}{a}\right)}{1 + \left(1 - \frac{b}{a}\right)} \]

Alternative 3
Error	59.8
Cost	832

\[\frac{\frac{b}{a} \cdot 2}{1 + \left(1 - \frac{b}{a}\right)} \]

Alternative 4
Error	59.8
Cost	192

\[\frac{b}{a} \]

Alternative 5
Error	1.3
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022241 
(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)))))