Eccentricity of an ellipse

Percentage Accurate: 77.5% → 100.0%

Time: 4.6s

Alternatives: 5

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: 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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.1%

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

   1. div-sub 71.1%

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

   2. *-inverses 71.1%

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

   3. 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. pow1/2 100.0%

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

   2. pow-to-exp 100.0%

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

   3. add-sqr-sqrt 100.0%

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

   4. fabs-sqr 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. sub-neg 100.0%

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

   7. log1p-def 100.0%

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

   8. pow2 100.0%

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

5. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-{\left(\frac{b}{a}\right)}^{2}\right) \cdot 0.5}} \]
6. Step-by-step derivation

   1. unpow2 100.0%

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

   2. clear-num 100.0%

      \[\leadsto e^{\mathsf{log1p}\left(-\frac{b}{a} \cdot \color{blue}{\frac{1}{\frac{a}{b}}}\right) \cdot 0.5} \]

   3. div-inv 100.0%

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

   4. associate-/l/ 100.0%

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

   5. associate-/r* 100.0%

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

7. Applied egg-rr 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 71.1%

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

   1. div-sub 71.1%

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

   2. *-inverses 71.1%

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

   3. 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. Final simplification 100.0%

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

Alternative 3: 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 71.1%

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

   1. div-sub 71.1%

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

   2. *-inverses 71.1%

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

   3. 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. clear-num 100.0%

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

   2. un-div-inv 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 4: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.1%

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

   1. div-sub 71.1%

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

   2. *-inverses 71.1%

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

   3. 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. pow1/2 100.0%

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

   2. pow-to-exp 100.0%

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

   3. add-sqr-sqrt 100.0%

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

   4. fabs-sqr 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. sub-neg 100.0%

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

   7. log1p-def 100.0%

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

   8. pow2 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in b around 0 70.8%

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

   1. unpow2 70.8%

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

   2. unpow2 70.8%

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

   3. times-frac 98.4%

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

   4. unpow2 98.4%

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

8. Simplified 98.4%

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

   1. unpow2 100.0%

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

   2. clear-num 100.0%

      \[\leadsto e^{\mathsf{log1p}\left(-\frac{b}{a} \cdot \color{blue}{\frac{1}{\frac{a}{b}}}\right) \cdot 0.5} \]

   3. div-inv 100.0%

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

   4. associate-/l/ 100.0%

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

   5. associate-/r* 100.0%

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

10. Applied egg-rr 98.4%

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

11. Final simplification 98.4%

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

Alternative 5: 98.0% 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 71.1%

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

   1. div-sub 71.1%

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

   2. *-inverses 71.1%

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

   3. 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%

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

   3. add-sqr-sqrt 100.0%

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

   4. fabs-sub 100.0%

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

   5. fma-neg 100.0%

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

   6. metadata-eval 100.0%

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

   7. add-sqr-sqrt 0.0%

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

   8. fabs-sqr 0.0%

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

   9. add-sqr-sqrt 100.0%

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

   10. fma-udef 100.0%

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

   11. difference-of-sqr--1 99.9%

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

5. Applied egg-rr 99.9%

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

7. Applied egg-rr 97.2%

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

   1. expm1-def 97.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, \frac{b}{a}\right)\right)\right)} \]

   2. expm1-log1p 97.2%

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

9. Simplified 97.2%

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

10. Final simplification 97.2%

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

Reproduce

herbie shell --seed 2023193 
(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)))))