Eccentricity of an ellipse

Average Error: 14.3 → 0.0

Time: 4.3s

Precision: binary64

Cost: 7232

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

Math

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

↓

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

FPCore

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

↓

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

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

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

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 - ((1.0 + ((b / a) / (a / b))) + -1.0)))

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(1.0 + Float64(Float64(b / a) / Float64(a / b))) + -1.0)))
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 - ((1.0 + ((b / a) / (a / b))) + -1.0)));
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[(1.0 + N[(N[(b / a), $MachinePrecision] / N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 14.3

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

2. Simplified 0.0

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

   [Start] 14.3	\[ \sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
   div-sub [=>] 14.3	\[ \sqrt{\left|\color{blue}{\frac{a \cdot a}{a \cdot a} - \frac{b \cdot b}{a \cdot a}}\right|} \]
   *-inverses [=>] 14.3	\[ \sqrt{\left|\color{blue}{1} - \frac{b \cdot b}{a \cdot a}\right|} \]
   times-frac [=>] 0.0	\[ \sqrt{\left|1 - \color{blue}{\frac{b}{a} \cdot \frac{b}{a}}\right|} \]

3. Applied egg-rr 0.0

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{1 - {\left(\frac{b}{a}\right)}^{2}}\right)} - 1} \]

4. Simplified 0.0

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

   [Start] 0.0	\[ e^{\mathsf{log1p}\left(\sqrt{1 - {\left(\frac{b}{a}\right)}^{2}}\right)} - 1 \]
   expm1-def [=>] 0.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - {\left(\frac{b}{a}\right)}^{2}}\right)\right)} \]
   expm1-log1p [=>] 0.0	\[ \color{blue}{\sqrt{1 - {\left(\frac{b}{a}\right)}^{2}}} \]

5. Applied egg-rr 0.0

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

6. Applied egg-rr 0.0

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

7. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	6976

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

Alternative 2
Error	0.6
Cost	704

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

Alternative 3
Error	1.3
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023039 
(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)))))