Eccentricity of an ellipse

Percentage Accurate: 77.4% → 100.0%

Time: 2.4s

Alternatives: 3

Speedup: 211.0×

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: 77.4% 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, 1.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{1 - \frac{b}{a} \cdot \frac{b}{a}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

2. Taylor expanded in a around 0 75.8%

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

   1. unpow2 75.8%

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

   2. unpow2 75.8%

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

   3. unpow2 75.8%

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

   4. difference-of-squares 75.8%

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

   5. times-frac 99.9%

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

   6. associate-*r/ 100.0%

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

   7. associate-/l* 100.0%

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

   8. fabs-div 100.0%

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

   9. *-lft-identity 100.0%

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

   10. associate-/l* 100.0%

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

   11. fabs-div 100.0%

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

   12. associate-/r* 100.0%

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

   13. unpow-1 100.0%

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

   14. sqr-pow 100.0%

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

4. Simplified 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \sqrt{1 - \frac{b}{a} \cdot \frac{b}{a}} \]

Alternative 2: 99.1% accurate, 19.2× speedup

Math

\[\begin{array}{l} \\ 1 + -0.5 \cdot \frac{\frac{b}{a}}{\frac{a}{b}} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (+ 1.0 (* -0.5 (/ (/ b a) (/ a b)))))

C

double code(double a, double b) {
	return 1.0 + (-0.5 * ((b / a) / (a / b)));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 + ((-0.5d0) * ((b / a) / (a / b)))
end function

Java

public static double code(double a, double b) {
	return 1.0 + (-0.5 * ((b / a) / (a / b)));
}

Python

def code(a, b):
	return 1.0 + (-0.5 * ((b / a) / (a / b)))

Julia

function code(a, b)
	return Float64(1.0 + Float64(-0.5 * Float64(Float64(b / a) / Float64(a / b))))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 + (-0.5 * ((b / a) / (a / b)));
end

Wolfram

code[a_, b_] := N[(1.0 + N[(-0.5 * N[(N[(b / a), $MachinePrecision] / N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.8%

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

2. Taylor expanded in a around 0 75.8%

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

   1. unpow2 75.8%

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

   2. unpow2 75.8%

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

   3. unpow2 75.8%

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

   4. difference-of-squares 75.8%

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

   5. times-frac 99.9%

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

   6. associate-*r/ 100.0%

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

   7. associate-/l* 100.0%

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

   8. fabs-div 100.0%

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

   9. *-lft-identity 100.0%

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

   10. associate-/l* 100.0%

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

   11. fabs-div 100.0%

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

   12. associate-/r* 100.0%

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

   13. unpow-1 100.0%

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

   14. sqr-pow 100.0%

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

4. Simplified 100.0%

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

5. Taylor expanded in b around 0 75.7%

   \[\leadsto \color{blue}{1 + -0.5 \cdot \frac{{b}^{2}}{{a}^{2}}} \]
6. Step-by-step derivation

   1. unpow2 75.7%

      \[\leadsto 1 + -0.5 \cdot \frac{\color{blue}{b \cdot b}}{{a}^{2}} \]

   2. unpow2 75.7%

      \[\leadsto 1 + -0.5 \cdot \frac{b \cdot b}{\color{blue}{a \cdot a}} \]

   3. times-frac 99.2%

      \[\leadsto 1 + -0.5 \cdot \color{blue}{\left(\frac{b}{a} \cdot \frac{b}{a}\right)} \]

   4. unpow2 99.2%

      \[\leadsto 1 + -0.5 \cdot \color{blue}{{\left(\frac{b}{a}\right)}^{2}} \]

7. Simplified 99.2%

   \[\leadsto \color{blue}{1 + -0.5 \cdot {\left(\frac{b}{a}\right)}^{2}} \]
8. Step-by-step derivation

   1. unpow2 99.2%

      \[\leadsto 1 + -0.5 \cdot \color{blue}{\left(\frac{b}{a} \cdot \frac{b}{a}\right)} \]

   2. clear-num 99.2%

      \[\leadsto 1 + -0.5 \cdot \left(\frac{b}{a} \cdot \color{blue}{\frac{1}{\frac{a}{b}}}\right) \]

   3. un-div-inv 99.2%

      \[\leadsto 1 + -0.5 \cdot \color{blue}{\frac{\frac{b}{a}}{\frac{a}{b}}} \]

9. Applied egg-rr 99.2%

   \[\leadsto 1 + -0.5 \cdot \color{blue}{\frac{\frac{b}{a}}{\frac{a}{b}}} \]

10. Final simplification 99.2%

    \[\leadsto 1 + -0.5 \cdot \frac{\frac{b}{a}}{\frac{a}{b}} \]

Alternative 3: 98.1% accurate, 211.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 1.0

Julia

function code(a, b)
	return 1.0
end

MATLAB

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

Wolfram

code[a_, b_] := 1.0

Derivation

1. Initial program 75.8%

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

2. Taylor expanded in a around 0 75.8%

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

   1. unpow2 75.8%

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

   2. unpow2 75.8%

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

   3. unpow2 75.8%

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

   4. difference-of-squares 75.8%

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

   5. times-frac 99.9%

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

   6. associate-*r/ 100.0%

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

   7. associate-/l* 100.0%

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

   8. fabs-div 100.0%

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

   9. *-lft-identity 100.0%

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

   10. associate-/l* 100.0%

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

   11. fabs-div 100.0%

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

   12. associate-/r* 100.0%

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

   13. unpow-1 100.0%

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

   14. sqr-pow 100.0%

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

4. Simplified 100.0%

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

5. Taylor expanded in b around 0 98.1%

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

6. Final simplification 98.1%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023187 
(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)))))