Eccentricity of an ellipse

Percentage Accurate: 77.8% → 100.0%

Time: 5.5s

Alternatives: 5

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: 77.8% 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

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

FPCore

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

C

double code(double a, double b) {
	return sqrt(fabs((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(abs((1.0d0 - (b / (a / (b / a))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.5%

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

   1. sqr-neg 80.5%

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

   2. sqr-neg 80.5%

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

   3. div-sub 80.5%

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

   4. *-inverses 80.5%

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

   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. Step-by-step derivation

   1. associate-*l/ 100.0%

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

   2. associate-/l* 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.9× speedup

Math

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

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 / Float64(a / 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 / N[(a / b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 80.5%

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

   1. sqr-neg 80.5%

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

   2. sqr-neg 80.5%

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

   3. div-sub 80.5%

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

   4. *-inverses 80.5%

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

   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. Taylor expanded in b around 0 80.5%

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

   1. fabs-sub 80.5%

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

   2. unpow2 80.5%

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

   3. unpow2 80.5%

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

   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%

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

6. Simplified 100.0%

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

   1. unpow2 100.0%

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

   2. associate-/r/ 100.0%

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

   3. associate-/r/ 100.0%

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

   4. associate-/r* 100.0%

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

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

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

Alternative 3: 99.2% accurate, 19.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.5%

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

   1. sqr-neg 80.5%

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

   2. sqr-neg 80.5%

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

   3. div-sub 80.5%

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

   4. *-inverses 80.5%

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

   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. Taylor expanded in b around 0 80.5%

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

   1. fabs-sub 80.5%

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

   2. unpow2 80.5%

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

   3. unpow2 80.5%

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

   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%

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

6. Simplified 100.0%

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

7. Taylor expanded in b around 0 79.3%

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

   1. unpow2 79.3%

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

   2. unpow2 79.3%

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

   3. times-frac 98.3%

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

   4. unpow2 98.3%

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

9. Simplified 98.3%

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

    1. unpow2 100.0%

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

    2. associate-/r/ 100.0%

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

    3. associate-/r/ 100.0%

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

    4. associate-/r* 100.0%

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

11. Applied egg-rr 98.3%

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

12. Final simplification 98.3%

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

Alternative 4: 1.6% 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 80.5%

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

   1. sqr-neg 80.5%

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

   2. sqr-neg 80.5%

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

   3. div-sub 80.5%

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

   4. *-inverses 80.5%

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

   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. Step-by-step derivation

   1. add-sqr-sqrt 100.0%

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

   2. fabs-sqr 100.0%

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

   3. add-sqr-sqrt 100.0%

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

   4. *-inverses 80.5%

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

   5. frac-times 80.5%

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

   6. div-sub 80.5%

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

   7. sqrt-div 80.5%

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

   8. sqrt-prod 80.1%

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

   9. add-sqr-sqrt 80.7%

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

   10. pow2 80.7%

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

   11. pow2 80.7%

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

5. Applied egg-rr 80.7%

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

6. Taylor expanded in a around -inf 1.6%

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

7. Final simplification 1.6%

   \[\leadsto -1 \]

Alternative 5: 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 80.5%

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

   1. sqr-neg 80.5%

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

   2. sqr-neg 80.5%

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

   3. div-sub 80.5%

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

   4. *-inverses 80.5%

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

   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. Taylor expanded in b around 0 80.5%

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

   1. fabs-sub 80.5%

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

   2. unpow2 80.5%

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

   3. unpow2 80.5%

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

   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%

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

6. Simplified 100.0%

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

   1. unpow2 100.0%

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

   2. associate-/r/ 100.0%

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

   3. associate-/r/ 100.0%

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

   4. associate-/r* 100.0%

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

8. Applied egg-rr 100.0%

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

9. Taylor expanded in b around 0 97.0%

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

10. Final simplification 97.0%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023312 
(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)))))