Eccentricity of an ellipse

Average Error: 13.7 → 0.0

Time: 1.6min

Precision: binary64

Cost: 13376

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

Math

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

↓

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

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

Derivation

1. Initial program 13.7

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

2. Simplified 0.0

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

3. Applied egg-rr 0.0

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

Alternatives

Alternative 1
Error	1.2
Cost	12864

\[\sqrt{\left|1\right|} \]

Reproduce

herbie shell --seed 2023033 
(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)))))