Eccentricity of an ellipse

Percentage Accurate: 76.9% → 100.0%

Time: 4.7s

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: 76.9% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

   1. sqr-neg 79.6%

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

   2. fabs-div 79.6%

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

   3. sqr-neg 79.6%

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

   4. fabs-sub 79.6%

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

   5. sqr-neg 79.6%

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

   6. distribute-rgt-neg-out 79.6%

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

   7. fabs-neg 79.6%

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

   8. fabs-div 79.6%

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

   9. cancel-sign-sub-inv 79.6%

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

   10. +-commutative 79.6%

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

   11. sqr-neg 79.6%

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

   12. cancel-sign-sub-inv 79.6%

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

3. Simplified 80.3%

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

   1. associate-*r/ 79.6%

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

   2. frac-times 100.0%

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

   3. clear-num 100.0%

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

   4. un-div-inv 100.0%

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

6. Applied egg-rr 100.0%

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

   1. pow1/2 100.0%

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

   2. add-sqr-sqrt 100.0%

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

   3. fabs-sqr 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. div-inv 100.0%

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

   6. clear-num 100.0%

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

   7. pow2 100.0%

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

8. Applied egg-rr 100.0%

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

   1. unpow2 100.0%

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

   2. clear-num 100.0%

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

   3. un-div-inv 100.0%

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

10. Applied egg-rr 100.0%

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

11. Final simplification 100.0%

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

Alternative 2: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5, \frac{\frac{b}{a}}{\frac{a}{b}}, 1\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

   1. sqr-neg 79.6%

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

   2. fabs-div 79.6%

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

   3. sqr-neg 79.6%

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

   4. fabs-sub 79.6%

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

   5. sqr-neg 79.6%

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

   6. distribute-rgt-neg-out 79.6%

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

   7. fabs-neg 79.6%

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

   8. fabs-div 79.6%

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

   9. cancel-sign-sub-inv 79.6%

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

   10. +-commutative 79.6%

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

   11. sqr-neg 79.6%

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

   12. cancel-sign-sub-inv 79.6%

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

3. Simplified 80.3%

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

   1. associate-*r/ 79.6%

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

   2. frac-times 100.0%

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

   3. clear-num 100.0%

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

   4. un-div-inv 100.0%

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

6. Applied egg-rr 100.0%

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

   1. pow1/2 100.0%

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

   2. add-sqr-sqrt 100.0%

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

   3. fabs-sqr 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. div-inv 100.0%

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

   6. clear-num 100.0%

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

   7. pow2 100.0%

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

8. Applied egg-rr 100.0%

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

9. Taylor expanded in b around 0 78.9%

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

    1. +-commutative 78.9%

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

    2. fma-define 78.9%

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

    3. unpow2 78.9%

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

    4. unpow2 78.9%

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

    5. times-frac 99.2%

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

    6. unpow2 99.2%

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

11. Simplified 99.2%

    \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, {\left(\frac{b}{a}\right)}^{2}, 1\right)} \]
12. Step-by-step derivation

    1. unpow2 100.0%

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

    2. clear-num 100.0%

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

    3. un-div-inv 100.0%

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

13. Applied egg-rr 99.2%

    \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\frac{\frac{b}{a}}{\frac{a}{b}}}, 1\right) \]

14. Final simplification 99.2%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{\frac{b}{a}}{\frac{a}{b}}, 1\right) \]
15. Add Preprocessing

Alternative 3: 97.9% 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 79.6%

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

   1. sqr-neg 79.6%

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

   2. fabs-div 79.6%

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

   3. sqr-neg 79.6%

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

   4. fabs-sub 79.6%

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

   5. sqr-neg 79.6%

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

   6. distribute-rgt-neg-out 79.6%

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

   7. fabs-neg 79.6%

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

   8. fabs-div 79.6%

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

   9. cancel-sign-sub-inv 79.6%

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

   10. +-commutative 79.6%

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

   11. sqr-neg 79.6%

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

   12. cancel-sign-sub-inv 79.6%

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

3. Simplified 80.3%

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

   1. associate-*r/ 79.6%

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

   2. frac-times 100.0%

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

   3. clear-num 100.0%

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

   4. un-div-inv 100.0%

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

6. Applied egg-rr 100.0%

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

   1. pow1/2 100.0%

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

   2. add-sqr-sqrt 100.0%

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

   3. fabs-sqr 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. div-inv 100.0%

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

   6. clear-num 100.0%

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

   7. pow2 100.0%

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

8. Applied egg-rr 100.0%

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

9. Taylor expanded in b around 0 98.7%

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

10. Final simplification 98.7%

    \[\leadsto 1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024080 
(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)))))