Eccentricity of an ellipse

Average Accuracy: 77.1% → 100.0%

Time: 6.4s

Precision: binary64

Cost: 7232

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

Math

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

↓

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

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (/ 1.0 (sqrt (/ (/ a (- a 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 1.0 / sqrt(((a / (a - 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 = 1.0d0 / sqrt(((a / (a - 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 1.0 / Math.sqrt(((a / (a - 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 1.0 / math.sqrt(((a / (a - 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 Float64(1.0 / sqrt(Float64(Float64(a / Float64(a - b)) / Float64(Float64(a + 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 = 1.0 / sqrt(((a / (a - 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[(1.0 / N[Sqrt[N[(N[(a / N[(a - b), $MachinePrecision]), $MachinePrecision] / N[(N[(a + b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.1%

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

2. Applied egg-rr 99.9%

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

3. Simplified 99.9%

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

   [Start] 99.9	\[ \frac{1}{\sqrt{\frac{a}{a + b} \cdot \frac{a}{a - b}}} \]
   +-commutative [=>] 99.9	\[ \frac{1}{\sqrt{\frac{a}{\color{blue}{b + a}} \cdot \frac{a}{a - b}}} \]

4. Applied egg-rr 100.0%

   \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{\frac{a}{a - b}}{\frac{a + b}{a}}}}} \]

5. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	6976

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

Alternative 2
Accuracy	99.1%
Cost	704

\[1 + \frac{b}{a} \cdot \frac{b \cdot -0.5}{a} \]

Alternative 3
Accuracy	98.1%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023125 
(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)))))