Eccentricity of an ellipse

Percentage Accurate: 78.1% → 100.0%

Time: 5.9s

Alternatives: 4

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: 78.1% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.6%

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

   1. lift-fabs.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. fabs-div N/A

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

   4. lift-*.f64 N/A

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

   5. fabs-sqr N/A

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

   6. lift-*.f64 N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. clear-num N/A

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

   10. lift-*.f64 N/A

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

   11. fabs-sqr N/A

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

   12. lift-*.f64 N/A

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

   13. fabs-div N/A

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

   14. lift-/.f64 N/A

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

   15. lift-fabs.f64 N/A

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

   16. inv-pow N/A

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

   17. lower-pow.f64 72.6

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.6%

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

3. Taylor expanded in b around 0

   \[\leadsto \sqrt{\left|\color{blue}{1 + -1 \cdot \frac{{b}^{2}}{{a}^{2}}}\right|} \]
4. Step-by-step derivation

   1. *-inverses N/A

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

   2. mul-1-neg N/A

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

   3. sub-neg N/A

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

   4. div-sub N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. div-sub N/A

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

   9. unpow2 N/A

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

   10. associate-/l* N/A

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

   11. *-inverses N/A

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

   12. *-rgt-identity N/A

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

   13. sub-neg N/A

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

   14. distribute-frac-neg N/A

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

   15. mul-1-neg N/A

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

   16. +-commutative N/A

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

   17. mul-1-neg N/A

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

   18. unpow2 N/A

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

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

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

   20. associate-/l* N/A

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

   21. lower-fma.f64 N/A

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

   22. lower-neg.f64 N/A

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

   23. lower-/.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 3: 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 72.6%

   \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
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 4: 97.8% 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 72.6%

   \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
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 2024251 
(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)))))