Eccentricity of an ellipse

Percentage Accurate: 77.3% → 100.0%

Time: 5.3s

Alternatives: 3

Speedup: 3.8×

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: 77.3% 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 81.2%

   \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
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. fabs-neg N/A

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

   2. lower-fabs.f64 N/A

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

   3. distribute-neg-frac2 N/A

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

   4. unpow2 N/A

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

   5. fp-cancel-sub-sign-inv N/A

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

   6. distribute-lft-neg-in N/A

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

   7. unpow2 N/A

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

   8. mul-1-neg N/A

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

   9. +-commutative N/A

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

   10. unpow2 N/A

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

   11. distribute-lft-neg-in N/A

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

   12. div-add N/A

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

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: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.7 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\mathsf{fma}\left(\frac{b}{a \cdot a}, b, -1\right)\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.7e-179) (sqrt 1.0) (sqrt (fabs (fma (/ b (* a a)) b -1.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.7e-179) {
		tmp = sqrt(1.0);
	} else {
		tmp = sqrt(fabs(fma((b / (a * a)), b, -1.0)));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.7e-179)
		tmp = sqrt(1.0);
	else
		tmp = sqrt(abs(fma(Float64(b / Float64(a * a)), b, -1.0)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.7e-179], N[Sqrt[1.0], $MachinePrecision], N[Sqrt[N[Abs[N[(N[(b / N[(a * a), $MachinePrecision]), $MachinePrecision] * b + -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.6999999999999999e-179

   1. Initial program 71.4%

      \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
   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 99.1%

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

      1. lift-fabs.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left|1\right|}} \]

      2. rem-sqrt-square-rev N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{1 \cdot 1}}} \]

      3. sqrt-prod N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{1} \cdot \sqrt{1}}} \]

      4. rem-square-sqrt 99.1

         \[\leadsto \sqrt{\color{blue}{1}} \]

   7. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\sqrt{1}} \]

   if 1.6999999999999999e-179 < b

   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. fabs-neg N/A

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

      2. lower-fabs.f64 N/A

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

      3. distribute-neg-frac2 N/A

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

      4. unpow2 N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. distribute-lft-neg-in N/A

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

      7. unpow2 N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. distribute-lft-neg-in N/A

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

      12. div-add N/A

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

   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%

      \[\leadsto \sqrt{\left|\mathsf{fma}\left(\frac{b}{a \cdot a}, b, -1\right)\right|} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

   \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
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 97.8%

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

   1. lift-fabs.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left|1\right|}} \]

   2. rem-sqrt-square-rev N/A

      \[\leadsto \sqrt{\color{blue}{\sqrt{1 \cdot 1}}} \]

   3. sqrt-prod N/A

      \[\leadsto \sqrt{\color{blue}{\sqrt{1} \cdot \sqrt{1}}} \]

   4. rem-square-sqrt 97.8

      \[\leadsto \sqrt{\color{blue}{1}} \]

7. Applied rewrites 97.8%

   \[\leadsto \color{blue}{\sqrt{1}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024343 
(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)))))