Eccentricity of an ellipse

Average Error: 14.3 → 0.0

Time: 10.0s

Precision: binary64

Cost: 13440

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

Math

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

↓

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

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) (/ 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(fabs((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(abs((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(Math.abs((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(math.fabs((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(abs(Float64(1.0 + Float64(-Float64(Float64(b / a) * Float64(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(abs((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[Abs[N[(1.0 + (-N[(N[(b / a), $MachinePrecision] * N[(b / a), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 14.3

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

2. Taylor expanded in a around inf 15.9

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

3. Simplified 14.3

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

4. Applied egg-rr 0.0

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

Alternatives

Alternative 1
Error	1.3
Cost	12864

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

Reproduce

herbie shell --seed 2023010 
(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)))))