Eccentricity of an ellipse

Percentage Accurate: 76.9% → 100.0%

Time: 6.5s

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: 76.9% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left|\frac{1 - {\left(\frac{b}{a}\right)}^{4}}{{\left(\frac{b}{a}\right)}^{2} + 1}\right|} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (sqrt (fabs (/ (- 1.0 (pow (/ b a) 4.0)) (+ (pow (/ b a) 2.0) 1.0)))))

C

double code(double a, double b) {
	return sqrt(fabs(((1.0 - pow((b / a), 4.0)) / (pow((b / a), 2.0) + 1.0))));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = sqrt(abs(((1.0d0 - ((b / a) ** 4.0d0)) / (((b / a) ** 2.0d0) + 1.0d0))))
end function

Java

public static double code(double a, double b) {
	return Math.sqrt(Math.abs(((1.0 - Math.pow((b / a), 4.0)) / (Math.pow((b / a), 2.0) + 1.0))));
}

Python

def code(a, b):
	return math.sqrt(math.fabs(((1.0 - math.pow((b / a), 4.0)) / (math.pow((b / a), 2.0) + 1.0))))

Julia

function code(a, b)
	return sqrt(abs(Float64(Float64(1.0 - (Float64(b / a) ^ 4.0)) / Float64((Float64(b / a) ^ 2.0) + 1.0))))
end

MATLAB

function tmp = code(a, b)
	tmp = sqrt(abs(((1.0 - ((b / a) ^ 4.0)) / (((b / a) ^ 2.0) + 1.0))));
end

Wolfram

code[a_, b_] := N[Sqrt[N[Abs[N[(N[(1.0 - N[Power[N[(b / a), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(b / a), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 74.6%

   \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. *-inverses N/A

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

   5. flip-- N/A

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

   6. lower-/.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower--.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. times-frac N/A

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

   13. pow-prod-down N/A

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

   14. pow-prod-up N/A

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

   15. lower-pow.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

       \[\leadsto \sqrt{\left|\frac{1 - {\left(\frac{b}{a}\right)}^{\color{blue}{4}}}{1 + \frac{b \cdot b}{a \cdot a}}\right|} \]

   18. lower-+.f64 N/A

       \[\leadsto \sqrt{\left|\frac{1 - {\left(\frac{b}{a}\right)}^{4}}{\color{blue}{1 + \frac{b \cdot b}{a \cdot a}}}\right|} \]

4. Applied rewrites 100.0%

   \[\leadsto \sqrt{\left|\color{blue}{\frac{1 - {\left(\frac{b}{a}\right)}^{4}}{1 + {\left(\frac{b}{a}\right)}^{2}}}\right|} \]

5. Final simplification 100.0%

   \[\leadsto \sqrt{\left|\frac{1 - {\left(\frac{b}{a}\right)}^{4}}{{\left(\frac{b}{a}\right)}^{2} + 1}\right|} \]
6. Add Preprocessing

Alternative 2: 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 74.6%

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

3. Taylor expanded in b 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 3: 98.0% 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 74.6%

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

3. Taylor expanded in b around 0

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

5. Applied rewrites 98.0%

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

Reproduce

herbie shell --seed 2024243 
(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)))))