Eccentricity of an ellipse

Average Error: 22.2% → 0.02%

Time: 5.3s

Precision: binary64

Cost: 6976

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

Math

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

↓

\[\sqrt{1 - \frac{\frac{b}{\frac{a}{b}}}{a}} \]

FPCore

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

↓

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

C

double code(double a, double b) {
	return sqrt(fabs((((a * a) - (b * b)) / (a * a))));
}

↓

double code(double a, double b) {
	return sqrt((1.0 - ((b / (a / b)) / a)));
}

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((1.0d0 - ((b / (a / b)) / a)))
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((1.0 - ((b / (a / b)) / a)));
}

Python

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

↓

def code(a, b):
	return math.sqrt((1.0 - ((b / (a / b)) / a)))

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

MATLAB

function tmp = code(a, b)
	tmp = sqrt(abs((((a * a) - (b * b)) / (a * a))));
end

↓

function tmp = code(a, b)
	tmp = sqrt((1.0 - ((b / (a / b)) / a)));
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[(1.0 - N[(N[(b / N[(a / b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 22.2

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

2. Taylor expanded in a around 0 22.2

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

3. Simplified 0.02

   \[\leadsto \color{blue}{\sqrt{1 - \frac{b}{a} \cdot \frac{b}{a}}} \]
   Proof

   [Start] 22.2	\[ \sqrt{\left|\frac{{a}^{2} - {b}^{2}}{{a}^{2}}\right|} \]
   fabs-div [=>] 22.2	\[ \sqrt{\color{blue}{\frac{\left|{a}^{2} - {b}^{2}\right|}{\left|{a}^{2}\right|}}} \]
   unpow2 [=>] 22.2	\[ \sqrt{\frac{\left|\color{blue}{a \cdot a} - {b}^{2}\right|}{\left|{a}^{2}\right|}} \]
   unpow2 [=>] 22.2	\[ \sqrt{\frac{\left|a \cdot a - \color{blue}{b \cdot b}\right|}{\left|{a}^{2}\right|}} \]
   rem-square-sqrt [<=] 22.21	\[ \sqrt{\frac{\left|\color{blue}{\sqrt{a \cdot a - b \cdot b} \cdot \sqrt{a \cdot a - b \cdot b}}\right|}{\left|{a}^{2}\right|}} \]
   fabs-sqr [=>] 22.21	\[ \sqrt{\frac{\color{blue}{\sqrt{a \cdot a - b \cdot b} \cdot \sqrt{a \cdot a - b \cdot b}}}{\left|{a}^{2}\right|}} \]
   rem-square-sqrt [=>] 22.2	\[ \sqrt{\frac{\color{blue}{a \cdot a - b \cdot b}}{\left|{a}^{2}\right|}} \]
   unpow2 [=>] 22.2	\[ \sqrt{\frac{a \cdot a - b \cdot b}{\left|\color{blue}{a \cdot a}\right|}} \]
   fabs-sqr [=>] 22.2	\[ \sqrt{\frac{a \cdot a - b \cdot b}{\color{blue}{a \cdot a}}} \]
   div-sub [=>] 22.2	\[ \sqrt{\color{blue}{\frac{a \cdot a}{a \cdot a} - \frac{b \cdot b}{a \cdot a}}} \]
   *-inverses [=>] 22.2	\[ \sqrt{\color{blue}{1} - \frac{b \cdot b}{a \cdot a}} \]
   times-frac [=>] 0.02	\[ \sqrt{1 - \color{blue}{\frac{b}{a} \cdot \frac{b}{a}}} \]

4. Applied egg-rr 0.02

   \[\leadsto \sqrt{1 - \color{blue}{\frac{\frac{b}{\frac{a}{b}}}{a}}} \]

5. Final simplification 0.02

   \[\leadsto \sqrt{1 - \frac{\frac{b}{\frac{a}{b}}}{a}} \]

Alternatives

Alternative 1
Error	0.02%
Cost	6976

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

Alternative 2
Error	0.91%
Cost	704

\[1 + \left(\frac{b}{a} \cdot \frac{b}{a}\right) \cdot -0.5 \]

Alternative 3
Error	2.09%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023104 
(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)))))