Eccentricity of an ellipse

Average Accuracy: 77.5% → 100.0%

Time: 6.6s

Precision: binary64

Cost: 13504

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

Math

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

↓

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

FPCore

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

↓

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

Derivation

1. Initial program 77.5%

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

2. Simplified 77.5%

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

   [Start] 77.5	\[ \sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
   difference-of-squares [=>] 77.5	\[ \sqrt{\left|\frac{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}}{a \cdot a}\right|} \]

3. Applied egg-rr 100.0%

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

4. Simplified 100.0%

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

   [Start] 100.0	\[ \frac{\sqrt{a + b}}{\sqrt{\frac{a}{\frac{a - b}{a}}}} \]
   +-commutative [=>] 100.0	\[ \frac{\sqrt{\color{blue}{b + a}}}{\sqrt{\frac{a}{\frac{a - b}{a}}}} \]
   div-sub [=>] 100.0	\[ \frac{\sqrt{b + a}}{\sqrt{\frac{a}{\color{blue}{\frac{a}{a} - \frac{b}{a}}}}} \]
   *-inverses [=>] 100.0	\[ \frac{\sqrt{b + a}}{\sqrt{\frac{a}{\color{blue}{1} - \frac{b}{a}}}} \]

5. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	6976

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

Alternative 2
Accuracy	99.0%
Cost	704

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

Alternative 3
Accuracy	97.8%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023129 
(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)))))