Eccentricity of an ellipse

Percentage Accurate: 76.8% → 100.0%

Time: 4.7s

Alternatives: 3

Speedup: 3.2×

Specification

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

Math

\[\begin{array}{l} \\ \sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \end{array} \]

FPCore

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

C

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

Java

public static double code(double a, double b) {
	return Math.sqrt(Math.abs((((a * a) - (b * b)) / (a * a))));
}

Python

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

Julia

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

MATLAB

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

Initial Program: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \end{array} \]

FPCore

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

C

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

Java

public static double code(double a, double b) {
	return Math.sqrt(Math.abs((((a * a) - (b * b)) / (a * a))));
}

Python

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

Julia

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

MATLAB

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left|\mathsf{fma}\left(\frac{\frac{b}{a}}{a}, b, -1\right)\right|} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (sqrt (fabs (fma (/ (/ b a) a) b -1.0))))

C

double code(double a, double b) {
	return sqrt(fabs(fma(((b / a) / a), b, -1.0)));
}

Julia

function code(a, b)
	return sqrt(abs(fma(Float64(Float64(b / a) / a), b, -1.0)))
end

Wolfram

code[a_, b_] := N[Sqrt[N[Abs[N[(N[(N[(b / a), $MachinePrecision] / a), $MachinePrecision] * b + -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 75.4%

   \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
2. Add Preprocessing

3. Taylor expanded in a around 0

   \[\leadsto \sqrt{\color{blue}{\left|\frac{{a}^{2} - {b}^{2}}{{a}^{2}}\right|}} \]
4. Step-by-step derivation

   1. div-sub N/A

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

   2. fabs-sub N/A

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

   3. lower-fabs.f64 N/A

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

   4. *-inverses N/A

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

   5. sub-neg N/A

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

   6. unpow2 N/A

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

   7. associate-*l/ N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

       \[\leadsto \sqrt{\left|\mathsf{fma}\left(\frac{b}{\color{blue}{a \cdot a}}, b, -1\right)\right|} \]

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 100.0

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

5. Applied rewrites 100.0%

   \[\leadsto \sqrt{\color{blue}{\left|\mathsf{fma}\left(\frac{\frac{b}{a}}{a}, b, -1\right)\right|}} \]
6. Add Preprocessing

Alternative 2: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot a - b \cdot b}{a \cdot a} \leq 0.9985529502280344:\\ \;\;\;\;\sqrt{\left|\mathsf{fma}\left(\frac{b}{a \cdot a}, b, -1\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|1\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (- (* a a) (* b b)) (* a a)) 0.9985529502280344)
   (sqrt (fabs (fma (/ b (* a a)) b -1.0)))
   (sqrt (fabs 1.0))))

C

double code(double a, double b) {
	double tmp;
	if ((((a * a) - (b * b)) / (a * a)) <= 0.9985529502280344) {
		tmp = sqrt(fabs(fma((b / (a * a)), b, -1.0)));
	} else {
		tmp = sqrt(fabs(1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(Float64(Float64(a * a) - Float64(b * b)) / Float64(a * a)) <= 0.9985529502280344)
		tmp = sqrt(abs(fma(Float64(b / Float64(a * a)), b, -1.0)));
	else
		tmp = sqrt(abs(1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision], 0.9985529502280344], N[Sqrt[N[Abs[N[(N[(b / N[(a * a), $MachinePrecision]), $MachinePrecision] * b + -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[1.0], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 a a) (*.f64 b b)) (*.f64 a a)) < 0.99855295022803436

   1. Initial program 100.0%

      \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \sqrt{\color{blue}{\left|\frac{{a}^{2} - {b}^{2}}{{a}^{2}}\right|}} \]
   4. Step-by-step derivation

      1. div-sub N/A

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

      2. fabs-sub N/A

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

      3. lower-fabs.f64 N/A

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

      4. *-inverses N/A

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

      5. sub-neg N/A

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

      6. unpow2 N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

          \[\leadsto \sqrt{\left|\mathsf{fma}\left(\frac{b}{\color{blue}{a \cdot a}}, b, -1\right)\right|} \]

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \sqrt{\color{blue}{\left|\mathsf{fma}\left(\frac{\frac{b}{a}}{a}, b, -1\right)\right|}} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

      \[\leadsto \sqrt{\left|\mathsf{fma}\left(\frac{b}{a \cdot a}, b, -1\right)\right|} \]

   if 0.99855295022803436 < (/.f64 (-.f64 (*.f64 a a) (*.f64 b b)) (*.f64 a a))

   1. Initial program 74.8%

      \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \sqrt{\left|\color{blue}{1}\right|} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.3%

      \[\leadsto \sqrt{\left|\color{blue}{1}\right|} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left|1\right|} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (sqrt (fabs 1.0)))

C

double code(double a, double b) {
	return sqrt(fabs(1.0));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = sqrt(abs(1.0d0))
end function

Java

public static double code(double a, double b) {
	return Math.sqrt(Math.abs(1.0));
}

Python

def code(a, b):
	return math.sqrt(math.fabs(1.0))

Julia

function code(a, b)
	return sqrt(abs(1.0))
end

MATLAB

function tmp = code(a, b)
	tmp = sqrt(abs(1.0));
end

Wolfram

code[a_, b_] := N[Sqrt[N[Abs[1.0], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 75.4%

   \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
2. Add Preprocessing

3. Taylor expanded in a around inf

   \[\leadsto \sqrt{\left|\color{blue}{1}\right|} \]
4. Step-by-step derivation

5. Applied rewrites 97.6%

   \[\leadsto \sqrt{\left|\color{blue}{1}\right|} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024331 
(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)))))