Eccentricity of an ellipse

Percentage Accurate: 77.3% → 100.0%

Time: 5.0s

Alternatives: 4

Speedup: 2.0×

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: 77.3% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.0%

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

   1. sqr-neg 76.0%

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

   2. sqr-neg 76.0%

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

   3. div-sub 76.0%

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

   4. *-inverses 76.0%

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

   5. times-frac 100.0%

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

3. Simplified 100.0%

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

   1. clear-num 100.0%

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

   2. frac-times 100.0%

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

   3. *-un-lft-identity 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

   \[\leadsto \sqrt{\left|1 - \frac{b}{a \cdot \frac{a}{b}}\right|} \]
8. Add Preprocessing

Alternative 2: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.0%

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

   1. sqr-neg 76.0%

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

   2. sqr-neg 76.0%

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

   3. div-sub 76.0%

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

   4. *-inverses 76.0%

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

   5. times-frac 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in b around 0 76.0%

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

   1. fabs-sub 76.0%

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

   2. unpow2 76.0%

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

   3. unpow2 76.0%

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

   4. times-frac 100.0%

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

   5. unpow2 100.0%

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

   6. fabs-sub 100.0%

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

   7. rem-square-sqrt 100.0%

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

   8. fabs-sqr 100.0%

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

   9. rem-square-sqrt 100.0%

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

7. Simplified 100.0%

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

   1. unpow2 100.0%

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

   2. clear-num 100.0%

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

   3. un-div-inv 100.0%

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

9. Applied egg-rr 100.0%

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

10. Final simplification 100.0%

    \[\leadsto \sqrt{1 - \frac{\frac{b}{a}}{\frac{a}{b}}} \]
11. Add Preprocessing

Alternative 3: 99.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 1 + -0.5 \cdot {\left(\frac{b}{a}\right)}^{2} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (+ 1.0 (* -0.5 (pow (/ b a) 2.0))))

C

double code(double a, double b) {
	return 1.0 + (-0.5 * pow((b / a), 2.0));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 + ((-0.5d0) * ((b / a) ** 2.0d0))
end function

Java

public static double code(double a, double b) {
	return 1.0 + (-0.5 * Math.pow((b / a), 2.0));
}

Python

def code(a, b):
	return 1.0 + (-0.5 * math.pow((b / a), 2.0))

Julia

function code(a, b)
	return Float64(1.0 + Float64(-0.5 * (Float64(b / a) ^ 2.0)))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 + (-0.5 * ((b / a) ^ 2.0));
end

Wolfram

code[a_, b_] := N[(1.0 + N[(-0.5 * N[Power[N[(b / a), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.0%

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

   1. sqr-neg 76.0%

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

   2. sqr-neg 76.0%

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

   3. div-sub 76.0%

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

   4. *-inverses 76.0%

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

   5. times-frac 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in b around 0 76.0%

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

   1. fabs-sub 76.0%

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

   2. unpow2 76.0%

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

   3. unpow2 76.0%

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

   4. times-frac 100.0%

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

   5. unpow2 100.0%

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

   6. fabs-sub 100.0%

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

   7. rem-square-sqrt 100.0%

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

   8. fabs-sqr 100.0%

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

   9. rem-square-sqrt 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in b around 0 75.3%

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

   1. unpow2 75.3%

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

   2. unpow2 75.3%

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

   3. times-frac 98.8%

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

   4. unpow2 98.8%

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

10. Simplified 98.8%

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

11. Final simplification 98.8%

    \[\leadsto 1 + -0.5 \cdot {\left(\frac{b}{a}\right)}^{2} \]
12. Add Preprocessing

Alternative 4: 0.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.0%

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

   1. sqr-neg 76.0%

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

   2. sqr-neg 76.0%

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

   3. div-sub 76.0%

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

   4. *-inverses 76.0%

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

   5. times-frac 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in b around 0 76.0%

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

   1. fabs-sub 76.0%

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

   2. unpow2 76.0%

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

   3. unpow2 76.0%

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

   4. times-frac 100.0%

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

   5. unpow2 100.0%

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

   6. fabs-sub 100.0%

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

   7. rem-square-sqrt 100.0%

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

   8. fabs-sqr 100.0%

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

   9. rem-square-sqrt 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in b around inf 0.0%

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

9. Final simplification 0.0%

   \[\leadsto \frac{b \cdot \sqrt{-1}}{a} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(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)))))