Eccentricity of an ellipse

Percentage Accurate: 77.5% → 100.0%

Time: 8.7s

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: 77.5% 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

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

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 74.6%

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

   1. sqr-neg 74.6%

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

   2. fabs-div 74.6%

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

   3. sqr-neg 74.6%

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

   4. fabs-sub 74.6%

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

   5. sqr-neg 74.6%

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

   6. distribute-rgt-neg-out 74.6%

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

   7. fabs-neg 74.6%

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

   8. fabs-div 74.6%

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

   9. cancel-sign-sub-inv 74.6%

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

   10. +-commutative 74.6%

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

   11. sqr-neg 74.6%

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

   12. cancel-sign-sub-inv 74.6%

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

3. Simplified 75.4%

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

   1. *-inverses 74.6%

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

   2. associate-*r/ 74.6%

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

   3. div-sub 74.6%

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

   4. difference-of-squares 74.6%

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

   5. times-frac 100.0%

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

   6. +-commutative 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in b around 0 74.6%

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

   1. fabs-div 74.6%

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

   2. difference-of-squares 74.6%

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

   3. unpow2 74.6%

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

   4. unpow2 74.6%

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

   5. *-rgt-identity 74.6%

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

   6. fabs-div 74.6%

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

   7. *-rgt-identity 74.6%

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

   8. div-sub 74.6%

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

   9. fabs-sub 74.6%

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

   10. unpow2 74.6%

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

   11. unpow2 74.6%

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

   12. times-frac 74.6%

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

   13. unpow2 74.6%

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

   14. *-inverses 100.0%

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

   15. fabs-sub 100.0%

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

   16. rem-square-sqrt 100.0%

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

   17. 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. Simplified 100.0%

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

    1. unpow2 100.0%

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

    2. clear-num 100.0%

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

    3. un-div-inv 100.0%

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

11. Applied egg-rr 100.0%

    \[\leadsto \sqrt{1 - \color{blue}{\frac{\frac{b}{a}}{\frac{a}{b}}}} \]
12. Add Preprocessing

Alternative 2: 98.2% 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 74.6%

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

   1. sqr-neg 74.6%

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

   2. fabs-div 74.6%

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

   3. sqr-neg 74.6%

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

   4. fabs-sub 74.6%

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

   5. sqr-neg 74.6%

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

   6. distribute-rgt-neg-out 74.6%

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

   7. fabs-neg 74.6%

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

   8. fabs-div 74.6%

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

   9. cancel-sign-sub-inv 74.6%

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

   10. +-commutative 74.6%

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

   11. sqr-neg 74.6%

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

   12. cancel-sign-sub-inv 74.6%

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

3. Simplified 75.4%

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

   1. *-inverses 74.6%

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

   2. associate-*r/ 74.6%

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

   3. div-sub 74.6%

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

   4. difference-of-squares 74.6%

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

   5. times-frac 100.0%

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

   6. +-commutative 100.0%

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

6. Applied egg-rr 100.0%

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

   1. add-sqr-sqrt 100.0%

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

   2. fabs-sqr 100.0%

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

   3. add-sqr-sqrt 100.0%

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

   4. associate-*l/ 100.0%

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

   5. associate-/l* 99.7%

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

   6. div-sub 99.7%

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

   7. *-inverses 99.7%

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

8. Applied egg-rr 99.7%

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

9. Taylor expanded in a around inf 96.5%

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

    1. associate-*r/ 96.5%

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

    2. distribute-rgt1-in 96.5%

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

    3. metadata-eval 96.5%

       \[\leadsto 1 + \frac{0.5 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

    4. mul0-lft 96.5%

       \[\leadsto 1 + \frac{0.5 \cdot \color{blue}{0}}{a} \]

    5. metadata-eval 96.5%

       \[\leadsto 1 + \frac{\color{blue}{0}}{a} \]

11. Simplified 96.5%

    \[\leadsto \color{blue}{1 + \frac{0}{a}} \]

12. Taylor expanded in a around 0 96.5%

    \[\leadsto 1 + \color{blue}{0} \]

13. Final simplification 96.5%

    \[\leadsto 1 \]
14. Add Preprocessing

Alternative 3: 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 74.6%

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

   1. sqr-neg 74.6%

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

   2. fabs-div 74.6%

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

   3. sqr-neg 74.6%

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

   4. fabs-sub 74.6%

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

   5. sqr-neg 74.6%

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

   6. distribute-rgt-neg-out 74.6%

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

   7. fabs-neg 74.6%

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

   8. fabs-div 74.6%

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

   9. cancel-sign-sub-inv 74.6%

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

   10. +-commutative 74.6%

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

   11. sqr-neg 74.6%

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

   12. cancel-sign-sub-inv 74.6%

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

3. Simplified 75.4%

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

   1. *-inverses 74.6%

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

   2. associate-*r/ 74.6%

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

   3. div-sub 74.6%

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

   4. difference-of-squares 74.6%

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

   5. times-frac 100.0%

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

   6. +-commutative 100.0%

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

6. Applied egg-rr 100.0%

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

   1. add-sqr-sqrt 100.0%

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

   2. fabs-sqr 100.0%

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

   3. add-sqr-sqrt 100.0%

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

   4. associate-*l/ 100.0%

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

   5. associate-/l* 99.7%

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

   6. div-sub 99.7%

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

   7. *-inverses 99.7%

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

8. Applied egg-rr 99.7%

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

9. Taylor expanded in a around -inf 0.0%

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

    1. unpow2 0.0%

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

    2. rem-square-sqrt 1.6%

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

11. Simplified 1.6%

    \[\leadsto \color{blue}{-1} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024112 
(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)))))