Eccentricity of an ellipse

Percentage Accurate: 76.6% → 100.0%

Time: 5.6s

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: 76.6% 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, 0.7× speedup

Math

\[\begin{array}{l} \\ e^{\mathsf{log1p}\left(-{\left(\frac{b}{a}\right)}^{2}\right) \cdot 0.5} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (exp (* (log1p (- (pow (/ b a) 2.0))) 0.5)))

C

double code(double a, double b) {
	return exp((log1p(-pow((b / a), 2.0)) * 0.5));
}

Java

public static double code(double a, double b) {
	return Math.exp((Math.log1p(-Math.pow((b / a), 2.0)) * 0.5));
}

Python

def code(a, b):
	return math.exp((math.log1p(-math.pow((b / a), 2.0)) * 0.5))

Julia

function code(a, b)
	return exp(Float64(log1p(Float64(-(Float64(b / a) ^ 2.0))) * 0.5))
end

Wolfram

code[a_, b_] := N[Exp[N[(N[Log[1 + (-N[Power[N[(b / a), $MachinePrecision], 2.0], $MachinePrecision])], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 78.5%

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

   1. sqr-neg 78.5%

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

   2. fabs-div 78.5%

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

   3. sqr-neg 78.5%

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

   4. fabs-sub 78.5%

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

   5. sqr-neg 78.5%

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

   6. distribute-rgt-neg-out 78.5%

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

   7. fabs-neg 78.5%

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

   8. fabs-div 78.5%

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

   9. cancel-sign-sub-inv 78.5%

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

   10. +-commutative 78.5%

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

   11. sqr-neg 78.5%

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

   12. cancel-sign-sub-inv 78.5%

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

3. Simplified 79.2%

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

   1. pow1/2 79.2%

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

   2. pow-to-exp 79.2%

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

   3. add-sqr-sqrt 78.5%

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

   4. fabs-sqr 78.5%

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

   5. add-sqr-sqrt 78.5%

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

   6. sub-neg 78.5%

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

   7. log1p-define 78.5%

      \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(-b \cdot \frac{b}{a \cdot a}\right)} \cdot 0.5} \]

   8. associate-*r/ 78.5%

      \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{b \cdot b}{a \cdot a}}\right) \cdot 0.5} \]

   9. frac-times 100.0%

      \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{b}{a} \cdot \frac{b}{a}}\right) \cdot 0.5} \]

   10. pow2 100.0%

       \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{{\left(\frac{b}{a}\right)}^{2}}\right) \cdot 0.5} \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-{\left(\frac{b}{a}\right)}^{2}\right) \cdot 0.5}} \]
7. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 78.5%

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

   1. sqr-neg 78.5%

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

   2. fabs-div 78.5%

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

   3. sqr-neg 78.5%

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

   4. fabs-sub 78.5%

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

   5. sqr-neg 78.5%

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

   6. distribute-rgt-neg-out 78.5%

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

   7. fabs-neg 78.5%

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

   8. fabs-div 78.5%

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

   9. cancel-sign-sub-inv 78.5%

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

   10. +-commutative 78.5%

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

   11. sqr-neg 78.5%

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

   12. cancel-sign-sub-inv 78.5%

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

3. Simplified 79.2%

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

   1. associate-*r/ 78.5%

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

   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. Add Preprocessing

Alternative 3: 98.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ e^{-0.5 \cdot \frac{\frac{b}{\frac{a}{b}}}{a}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return math.exp((-0.5 * ((b / (a / b)) / a)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

   1. sqr-neg 78.5%

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

   2. fabs-div 78.5%

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

   3. sqr-neg 78.5%

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

   4. fabs-sub 78.5%

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

   5. sqr-neg 78.5%

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

   6. distribute-rgt-neg-out 78.5%

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

   7. fabs-neg 78.5%

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

   8. fabs-div 78.5%

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

   9. cancel-sign-sub-inv 78.5%

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

   10. +-commutative 78.5%

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

   11. sqr-neg 78.5%

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

   12. cancel-sign-sub-inv 78.5%

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

3. Simplified 79.2%

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

   1. pow1/2 79.2%

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

   2. pow-to-exp 79.2%

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

   3. add-sqr-sqrt 78.5%

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

   4. fabs-sqr 78.5%

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

   5. add-sqr-sqrt 78.5%

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

   6. sub-neg 78.5%

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

   7. log1p-define 78.5%

      \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(-b \cdot \frac{b}{a \cdot a}\right)} \cdot 0.5} \]

   8. associate-*r/ 78.5%

      \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{b \cdot b}{a \cdot a}}\right) \cdot 0.5} \]

   9. frac-times 100.0%

      \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{b}{a} \cdot \frac{b}{a}}\right) \cdot 0.5} \]

   10. pow2 100.0%

       \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{{\left(\frac{b}{a}\right)}^{2}}\right) \cdot 0.5} \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-{\left(\frac{b}{a}\right)}^{2}\right) \cdot 0.5}} \]

7. Taylor expanded in b around 0 77.3%

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

   1. unpow2 77.3%

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

   2. unpow2 77.3%

      \[\leadsto e^{-0.5 \cdot \frac{\color{blue}{b \cdot b}}{a \cdot a}} \]

   3. times-frac 98.1%

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

   4. unpow2 98.1%

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

9. Simplified 98.1%

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

    1. unpow2 98.1%

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

    2. clear-num 98.1%

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

    3. div-inv 98.1%

       \[\leadsto e^{-0.5 \cdot \color{blue}{\frac{\frac{b}{a}}{\frac{a}{b}}}} \]

    4. associate-/l/ 98.1%

       \[\leadsto e^{-0.5 \cdot \color{blue}{\frac{b}{\frac{a}{b} \cdot a}}} \]

    5. associate-/r* 98.1%

       \[\leadsto e^{-0.5 \cdot \color{blue}{\frac{\frac{b}{\frac{a}{b}}}{a}}} \]

11. Applied egg-rr 98.1%

    \[\leadsto e^{-0.5 \cdot \color{blue}{\frac{\frac{b}{\frac{a}{b}}}{a}}} \]
12. Add Preprocessing

Alternative 4: 97.6% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 (hypot 1.0 (/ b a)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[Sqrt[1.0 ^ 2 + N[(b / a), $MachinePrecision] ^ 2], $MachinePrecision]

Derivation

1. Initial program 78.5%

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

   1. sqr-neg 78.5%

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

   2. fabs-div 78.5%

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

   3. sqr-neg 78.5%

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

   4. fabs-sub 78.5%

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

   5. sqr-neg 78.5%

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

   6. distribute-rgt-neg-out 78.5%

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

   7. fabs-neg 78.5%

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

   8. fabs-div 78.5%

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

   9. cancel-sign-sub-inv 78.5%

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

   10. +-commutative 78.5%

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

   11. sqr-neg 78.5%

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

   12. cancel-sign-sub-inv 78.5%

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

3. Simplified 79.2%

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

   1. associate-*r/ 78.5%

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

   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. expm1-log1p-u 99.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left|1 - \frac{\frac{b}{a}}{\frac{a}{b}}\right|}\right)\right)} \]

   2. expm1-undefine 99.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\left|1 - \frac{\frac{b}{a}}{\frac{a}{b}}\right|}\right)} - 1} \]

8. Applied egg-rr 97.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \frac{b}{a}\right)\right)} - 1} \]
9. Step-by-step derivation

   1. log1p-undefine 97.1%

      \[\leadsto e^{\color{blue}{\log \left(1 + \mathsf{hypot}\left(1, \frac{b}{a}\right)\right)}} - 1 \]

   2. rem-exp-log 97.1%

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

   3. associate-+r- 97.1%

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

   4. +-commutative 97.1%

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

   5. associate-+l- 97.1%

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

   6. metadata-eval 97.1%

      \[\leadsto \mathsf{hypot}\left(1, \frac{b}{a}\right) - \color{blue}{0} \]

   7. --rgt-identity 97.1%

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

10. Simplified 97.1%

    \[\leadsto \color{blue}{\mathsf{hypot}\left(1, \frac{b}{a}\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024088 
(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)))))