Eccentricity of an ellipse

Percentage Accurate: 78.0% → 100.0%

Time: 7.5s

Alternatives: 4

Speedup: 42.0×

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: 78.0% 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, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = sqrt(abs((((a + b) / a) / (a / (a - b)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. clear-num N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. difference-of-squares N/A

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

   11. times-frac N/A

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

   12. associate-/r* N/A

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

   13. clear-num N/A

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

   14. lower-/.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-+.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower--.f64 100.0

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

4. Applied egg-rr 100.0%

   \[\leadsto \sqrt{\left|\color{blue}{\frac{\frac{a + b}{a}}{\frac{a}{a - b}}}\right|} \]
5. Add Preprocessing

Alternative 2: 100.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. clear-num N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. difference-of-squares N/A

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

   11. times-frac N/A

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

   12. associate-/r* N/A

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

   13. clear-num N/A

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

   14. lower-/.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-+.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower--.f64 100.0

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

4. Applied egg-rr 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. div-inv N/A

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

   6. associate-*r/ N/A

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

   7. associate-*r/ N/A

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

   8. div-inv N/A

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

   9. lift-/.f64 N/A

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

   10. frac-2neg N/A

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

   11. associate-/l/ N/A

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

   12. fabs-div N/A

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

   13. neg-fabs N/A

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

   14. lower-/.f64 N/A

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

   15. lower-fabs.f64 N/A

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

   16. lower-fabs.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. lower-neg.f64 100.0

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Reproduce

herbie shell --seed 2024218 
(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)))))