Eccentricity of an ellipse

Average Error: 14.2 → 14.3

Time: 1.8s

Precision: binary64

Cost: 13504

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

Math

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

↓

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

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (sqrt (/ (/ (fabs (- (* a a) (* b b))) a) 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(((fabs(((a * a) - (b * b))) / a) / 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(((abs(((a * a) - (b * b))) / a) / 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(((Math.abs(((a * a) - (b * b))) / a) / a));
}

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(((a * a) - (b * b))) / a) / 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(Float64(abs(Float64(Float64(a * a) - Float64(b * b))) / a) / 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(((abs(((a * a) - (b * b))) / a) / 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[(N[(N[Abs[N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision] / a), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 14.2

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

2. Simplified 14.2

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

   [Start] 14.2	\[ \sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
   rational_best-simplify-103 [=>] 14.2	\[ \sqrt{\color{blue}{\frac{\left|a \cdot a - b \cdot b\right|}{\left|a \cdot a\right|}}} \]
   rational_best-simplify-47 [<=] 14.2	\[ \sqrt{\frac{\left|a \cdot a - b \cdot b\right|}{\color{blue}{a \cdot a}}} \]

3. Applied egg-rr 14.4

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

4. Applied egg-rr 14.3

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

5. Final simplification 14.3

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

Alternatives

Alternative 1
Error	14.2
Cost	13504

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

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)))))