Eccentricity of an ellipse

Average Error: 14.9 → 0.0

Time: 8.4s

Precision: binary64

Cost: 7168

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

Math

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

↓

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

FPCore

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

↓

(FPCore (a b) :precision binary64 (pow (* a (/ (/ a (+ a b)) (- a b))) -0.5))

C

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

↓

double code(double a, double b) {
	return pow((a * ((a / (a + b)) / (a - b))), -0.5);
}

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 = (a * ((a / (a + b)) / (a - b))) ** (-0.5d0)
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.pow((a * ((a / (a + b)) / (a - b))), -0.5);
}

Python

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

↓

def code(a, b):
	return math.pow((a * ((a / (a + b)) / (a - b))), -0.5)

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 Float64(a * Float64(Float64(a / Float64(a + b)) / Float64(a - b))) ^ -0.5
end

MATLAB

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

↓

function tmp = code(a, b)
	tmp = (a * ((a / (a + b)) / (a - b))) ^ -0.5;
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[Power[N[(a * N[(N[(a / N[(a + b), $MachinePrecision]), $MachinePrecision] / N[(a - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]

Derivation

1. Initial program 14.9

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

2. Simplified 14.9

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

   [Start] 14.9	\[ \sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
   difference-of-squares [=>] 14.9	\[ \sqrt{\left|\frac{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}}{a \cdot a}\right|} \]

3. Applied egg-rr 14.9

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

4. Applied egg-rr 15.5

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

5. Simplified 0.0

   \[\leadsto \color{blue}{{\left(a \cdot \frac{\frac{a}{a + b}}{a - b}\right)}^{-0.5}} \]
   Proof

   [Start] 15.5	\[ {\left(\frac{\frac{a \cdot a}{a + b}}{a - b}\right)}^{-0.5} \]
   associate-/r* [<=] 14.9	\[ {\color{blue}{\left(\frac{a \cdot a}{\left(a + b\right) \cdot \left(a - b\right)}\right)}}^{-0.5} \]
   difference-of-squares [<=] 14.9	\[ {\left(\frac{a \cdot a}{\color{blue}{a \cdot a - b \cdot b}}\right)}^{-0.5} \]
   associate-*r/ [<=] 15.0	\[ {\color{blue}{\left(a \cdot \frac{a}{a \cdot a - b \cdot b}\right)}}^{-0.5} \]
   difference-of-squares [=>] 15.0	\[ {\left(a \cdot \frac{a}{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}}\right)}^{-0.5} \]
   associate-/r* [=>] 0.0	\[ {\left(a \cdot \color{blue}{\frac{\frac{a}{a + b}}{a - b}}\right)}^{-0.5} \]

6. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	6976

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

Alternative 2
Error	0.6
Cost	704

\[1 + \frac{b \cdot -0.5}{a} \cdot \frac{b}{a} \]

Alternative 3
Error	1.3
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023066 
(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)))))