Eccentricity of an ellipse

Percentage Accurate: 77.4% → 100.0%

Time: 3.4s

Precision: binary64

Cost: 13440

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

Math

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

↓

\[\begin{array}{l} \\ \sqrt{1 - \left(\left(1 + {\left(\frac{b}{a}\right)}^{2}\right) + -1\right)} \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (sqrt (- 1.0 (+ (+ 1.0 (pow (/ b a) 2.0)) -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 + pow((b / a), 2.0)) + -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) ** 2.0d0)) + (-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 + Math.pow((b / a), 2.0)) + -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 + math.pow((b / a), 2.0)) + -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(b / a) ^ 2.0)) + -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) ^ 2.0)) + -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[Power[N[(b / a), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 77.7%

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

2. Simplified 100.0%

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

   [Start] 77.7%	\[ \sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
   div-sub [=>] 77.7%	\[ \sqrt{\left|\color{blue}{\frac{a \cdot a}{a \cdot a} - \frac{b \cdot b}{a \cdot a}}\right|} \]
   *-inverses [=>] 77.7%	\[ \sqrt{\left|\color{blue}{1} - \frac{b \cdot b}{a \cdot a}\right|} \]
   times-frac [=>] 100.0%	\[ \sqrt{\left|1 - \color{blue}{\frac{b}{a} \cdot \frac{b}{a}}\right|} \]

3. Taylor expanded in b around 0 77.7%

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

4. Simplified 100.0%

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

   [Start] 77.7%	\[ \sqrt{\left|1 - \frac{{b}^{2}}{{a}^{2}}\right|} \]
   fabs-sub [=>] 77.7%	\[ \sqrt{\color{blue}{\left|\frac{{b}^{2}}{{a}^{2}} - 1\right|}} \]
   unpow2 [=>] 77.7%	\[ \sqrt{\left|\frac{\color{blue}{b \cdot b}}{{a}^{2}} - 1\right|} \]
   unpow2 [=>] 77.7%	\[ \sqrt{\left|\frac{b \cdot b}{\color{blue}{a \cdot a}} - 1\right|} \]
   times-frac [=>] 100.0%	\[ \sqrt{\left|\color{blue}{\frac{b}{a} \cdot \frac{b}{a}} - 1\right|} \]
   unpow2 [<=] 100.0%	\[ \sqrt{\left|\color{blue}{{\left(\frac{b}{a}\right)}^{2}} - 1\right|} \]
   fabs-sub [<=] 100.0%	\[ \sqrt{\color{blue}{\left|1 - {\left(\frac{b}{a}\right)}^{2}\right|}} \]
   rem-square-sqrt [<=] 100.0%	\[ \sqrt{\left|\color{blue}{\sqrt{1 - {\left(\frac{b}{a}\right)}^{2}} \cdot \sqrt{1 - {\left(\frac{b}{a}\right)}^{2}}}\right|} \]
   fabs-sqr [=>] 100.0%	\[ \sqrt{\color{blue}{\sqrt{1 - {\left(\frac{b}{a}\right)}^{2}} \cdot \sqrt{1 - {\left(\frac{b}{a}\right)}^{2}}}} \]
   rem-square-sqrt [=>] 100.0%	\[ \sqrt{\color{blue}{1 - {\left(\frac{b}{a}\right)}^{2}}} \]

5. Applied egg-rr 100.0%

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

   [Start] 100.0%	\[ \sqrt{1 - {\left(\frac{b}{a}\right)}^{2}} \]
   expm1-log1p-u [=>] 100.0%	\[ \sqrt{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{b}{a}\right)}^{2}\right)\right)}} \]
   expm1-udef [=>] 100.0%	\[ \sqrt{1 - \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{b}{a}\right)}^{2}\right)} - 1\right)}} \]
   log1p-udef [=>] 100.0%	\[ \sqrt{1 - \left(e^{\color{blue}{\log \left(1 + {\left(\frac{b}{a}\right)}^{2}\right)}} - 1\right)} \]
   add-exp-log [<=] 100.0%	\[ \sqrt{1 - \left(\color{blue}{\left(1 + {\left(\frac{b}{a}\right)}^{2}\right)} - 1\right)} \]
   +-commutative [=>] 100.0%	\[ \sqrt{1 - \left(\color{blue}{\left({\left(\frac{b}{a}\right)}^{2} + 1\right)} - 1\right)} \]

6. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	13440

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

Alternative 2
Accuracy	100.0%
Cost	6976

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

Alternative 3
Accuracy	99.1%
Cost	6912

\[1 + {\left(\frac{b}{a}\right)}^{2} \cdot -0.5 \]

Alternative 4
Accuracy	0.0%
Cost	6720

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

Reproduce

herbie shell --seed 2023167 
(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)))))