Eccentricity of an ellipse

Percentage Accurate: 77.4% → 100.0%

Time: 7.6s

Precision: binary64

Cost: 13376

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

Math

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

↓

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

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

Derivation

1. Initial program 74.6%

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

2. Simplified 100.0%

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

   [Start] 74.6	\[ \sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
   div-sub [=>] 74.6	\[ \sqrt{\left|\color{blue}{\frac{a \cdot a}{a \cdot a} - \frac{b \cdot b}{a \cdot a}}\right|} \]
   *-inverses [=>] 74.6	\[ \sqrt{\left|\color{blue}{1} - \frac{b \cdot b}{a \cdot a}\right|} \]
   times-frac [=>] 100.0	\[ \sqrt{\left|1 - \color{blue}{\frac{b}{a} \cdot \frac{b}{a}}\right|} \]

3. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ \sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|} \]
   clear-num [=>] 100.0	\[ \sqrt{\left|1 - \frac{b}{a} \cdot \color{blue}{\frac{1}{\frac{a}{b}}}\right|} \]
   un-div-inv [=>] 100.0	\[ \sqrt{\left|1 - \color{blue}{\frac{\frac{b}{a}}{\frac{a}{b}}}\right|} \]

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	13376

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

Alternative 2
Accuracy	96.1%
Cost	7296

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

Alternative 3
Accuracy	18.8%
Cost	1088

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

Alternative 4
Accuracy	6.5%
Cost	192

\[\frac{b}{a} \]

Alternative 5
Accuracy	98.0%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023161 
(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)))))