Eccentricity of an ellipse

Percentage Accurate: 77.3% → 100.0%

Time: 6.1s

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: 77.3% 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, 1.9× 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 79.3%

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

   1. sqr-neg 79.3%

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

   2. fabs-div 79.3%

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

   3. sqr-neg 79.3%

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

   4. fabs-sub 79.3%

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

   5. sqr-neg 79.3%

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

   6. distribute-rgt-neg-out 79.3%

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

   7. fabs-neg 79.3%

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

   8. fabs-div 79.3%

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

   9. cancel-sign-sub-inv 79.3%

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

   10. +-commutative 79.3%

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

   11. sqr-neg 79.3%

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

   12. cancel-sign-sub-inv 79.3%

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

   13. div-sub 79.3%

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

3. Simplified 79.9%

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

   1. associate-*r/ 79.3%

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

   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. Taylor expanded in b around 0 79.3%

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

   1. fabs-sub 79.3%

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

   2. unpow2 79.3%

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

   3. unpow2 79.3%

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

   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%

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

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. Final simplification 100.0%

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

Alternative 2: 99.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. sqr-neg 79.3%

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

   2. fabs-div 79.3%

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

   3. sqr-neg 79.3%

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

   4. fabs-sub 79.3%

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

   5. sqr-neg 79.3%

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

   6. distribute-rgt-neg-out 79.3%

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

   7. fabs-neg 79.3%

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

   8. fabs-div 79.3%

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

   9. cancel-sign-sub-inv 79.3%

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

   10. +-commutative 79.3%

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

   11. sqr-neg 79.3%

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

   12. cancel-sign-sub-inv 79.3%

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

   13. div-sub 79.3%

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

3. Simplified 79.9%

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

   1. associate-*r/ 79.3%

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

   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. Taylor expanded in b around 0 79.3%

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

   1. fabs-sub 79.3%

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

   2. unpow2 79.3%

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

   3. unpow2 79.3%

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

   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%

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

9. Simplified 100.0%

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

10. Taylor expanded in b around 0 78.5%

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

    1. +-commutative 78.5%

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

    2. fma-define 78.5%

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

    3. unpow2 78.5%

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

    4. unpow2 78.5%

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

    5. times-frac 98.9%

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

    6. unpow2 98.9%

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

12. Simplified 98.9%

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

    1. fma-undefine 98.9%

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

    2. unpow2 98.9%

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

    3. clear-num 98.9%

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

    4. associate-/r/ 98.9%

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

    5. un-div-inv 98.9%

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

    6. div-inv 98.9%

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

    7. clear-num 98.9%

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

    8. pow2 98.9%

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

14. Applied egg-rr 98.9%

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

15. Final simplification 98.9%

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

Alternative 3: 97.7% 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 79.3%

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

   1. sqr-neg 79.3%

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

   2. fabs-div 79.3%

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

   3. sqr-neg 79.3%

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

   4. fabs-sub 79.3%

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

   5. sqr-neg 79.3%

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

   6. distribute-rgt-neg-out 79.3%

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

   7. fabs-neg 79.3%

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

   8. fabs-div 79.3%

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

   9. cancel-sign-sub-inv 79.3%

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

   10. +-commutative 79.3%

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

   11. sqr-neg 79.3%

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

   12. cancel-sign-sub-inv 79.3%

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

   13. div-sub 79.3%

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

3. Simplified 79.9%

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

   1. associate-*r/ 79.3%

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

   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. Taylor expanded in b around 0 79.3%

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

   1. fabs-sub 79.3%

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

   2. unpow2 79.3%

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

   3. unpow2 79.3%

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

   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%

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

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. Taylor expanded in b around 0 98.0%

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

13. Final simplification 98.0%

    \[\leadsto 1 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024130 
(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)))))