Eccentricity of an ellipse

Average Error: 14.6 → 0.7

Time: 3.6s

Precision: binary64

Cost: 13764

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 0:\\ \;\;\;\;\sqrt{\left|\left(\frac{b}{a} + 1\right) - \frac{b}{a}\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.0)
   (sqrt (fabs (- (+ (/ b a) 1.0) (/ b a))))
   (sqrt (fabs (/ (- (* a a) (* b b)) (* a a))))))

C

double code(double a, double b) {
	return sqrt(fabs((((a * a) - (b * b)) / (a * a))));
}

↓

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.0) {
		tmp = sqrt(fabs((((b / a) + 1.0) - (b / a))));
	} else {
		tmp = sqrt(fabs((((a * a) - (b * b)) / (a * a))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((b * b) <= 0.0d0) then
        tmp = sqrt(abs((((b / a) + 1.0d0) - (b / a))))
    else
        tmp = sqrt(abs((((a * a) - (b * b)) / (a * a))))
    end if
    code = tmp
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) {
	double tmp;
	if ((b * b) <= 0.0) {
		tmp = Math.sqrt(Math.abs((((b / a) + 1.0) - (b / a))));
	} else {
		tmp = Math.sqrt(Math.abs((((a * a) - (b * b)) / (a * a))));
	}
	return tmp;
}

Python

def code(a, b):
	return math.sqrt(math.fabs((((a * a) - (b * b)) / (a * a))))

↓

def code(a, b):
	tmp = 0
	if (b * b) <= 0.0:
		tmp = math.sqrt(math.fabs((((b / a) + 1.0) - (b / a))))
	else:
		tmp = math.sqrt(math.fabs((((a * a) - (b * b)) / (a * a))))
	return tmp

Julia

function code(a, b)
	return sqrt(abs(Float64(Float64(Float64(a * a) - Float64(b * b)) / Float64(a * a))))
end

↓

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.0)
		tmp = sqrt(abs(Float64(Float64(Float64(b / a) + 1.0) - Float64(b / a))));
	else
		tmp = sqrt(abs(Float64(Float64(Float64(a * a) - Float64(b * b)) / Float64(a * a))));
	end
	return tmp
end

MATLAB

function tmp = code(a, b)
	tmp = sqrt(abs((((a * a) - (b * b)) / (a * a))));
end

↓

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 0.0)
		tmp = sqrt(abs((((b / a) + 1.0) - (b / a))));
	else
		tmp = sqrt(abs((((a * a) - (b * b)) / (a * a))));
	end
	tmp_2 = tmp;
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_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.0], N[Sqrt[N[Abs[N[(N[(N[(b / a), $MachinePrecision] + 1.0), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.0

   1. Initial program 20.2

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

   2. Simplified 20.2

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

      [Start] 20.2	\[ \sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
      rational_best_oopsla_all_46_json_45_simplify-42 [=>] 20.2	\[ \sqrt{\left|\frac{\color{blue}{\left(a - b\right) \cdot \left(a + b\right)}}{a \cdot a}\right|} \]

   3. Taylor expanded in a around 0 20.2

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

   4. Simplified 20.2

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

      [Start] 20.2	\[ \sqrt{\left|\frac{b}{a} + \left(1 + \left(-1 \cdot \frac{{b}^{2}}{{a}^{2}} + -1 \cdot \frac{b}{a}\right)\right)\right|} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 20.2	\[ \sqrt{\left|\frac{b}{a} + \color{blue}{\left(\left(-1 \cdot \frac{{b}^{2}}{{a}^{2}} + -1 \cdot \frac{b}{a}\right) + 1\right)}\right|} \]
      rational_best_oopsla_all_46_json_45_simplify-82 [=>] 20.2	\[ \sqrt{\left|\color{blue}{\left(-1 \cdot \frac{{b}^{2}}{{a}^{2}} + -1 \cdot \frac{b}{a}\right) + \left(\frac{b}{a} + 1\right)}\right|} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 20.2	\[ \sqrt{\left|\left(\color{blue}{\frac{{b}^{2}}{{a}^{2}} \cdot -1} + -1 \cdot \frac{b}{a}\right) + \left(\frac{b}{a} + 1\right)\right|} \]
      rational_best_oopsla_all_46_json_45_simplify-23 [=>] 20.2	\[ \sqrt{\left|\color{blue}{-1 \cdot \left(\frac{{b}^{2}}{{a}^{2}} + \frac{b}{a}\right)} + \left(\frac{b}{a} + 1\right)\right|} \]

   5. Applied egg-rr 20.2

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

   6. Taylor expanded in b around 0 0.9

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

   if 0.0 < (*.f64 b b)

   1. Initial program 0.1

      \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0:\\ \;\;\;\;\sqrt{\left|\left(\frac{b}{a} + 1\right) - \frac{b}{a}\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.4
Cost	13636

\[\begin{array}{l} \mathbf{if}\;b \leq 1.02 \cdot 10^{-196}:\\ \;\;\;\;\sqrt{\left|\left(\frac{b}{a} + 1\right) - \frac{b}{a}\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\frac{\left(a - b\right) \cdot \left(a + b\right)}{a \cdot a}\right|}\\ \end{array} \]

Alternative 2
Error	1.4
Cost	13376

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

Alternative 3
Error	1.4
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023090 
(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)))))