Eccentricity of an ellipse

Percentage Accurate: 77.5% → 100.0%

Time: 4.9s

Alternatives: 4

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, 0.7× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (sqrt (fabs (fma (* b (/ (- b) a)) (/ 1.0 a) 1.0))))

C

double code(double a, double b) {
	return sqrt(fabs(fma((b * (-b / a)), (1.0 / a), 1.0)));
}

Julia

function code(a, b)
	return sqrt(abs(fma(Float64(b * Float64(Float64(-b) / a)), Float64(1.0 / a), 1.0)))
end

Wolfram

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

Derivation

1. Initial program 78.1%

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

   1. div-sub 78.1%

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

   2. *-inverses 78.1%

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

   3. frac-times 100.0%

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

   4. sub-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. distribute-lft-neg-in 100.0%

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

   7. div-inv 100.0%

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

   8. associate-*r* 100.0%

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

   9. fma-def 100.0%

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

   10. distribute-neg-frac 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 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 78.1%

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

   1. div-sub 78.1%

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

   2. *-inverses 78.1%

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

   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. Step-by-step derivation

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 99.9%

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

   3. add-sqr-sqrt 99.9%

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

   4. fabs-sqr 99.9%

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

   5. add-sqr-sqrt 99.9%

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

   6. div-inv 99.9%

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

   7. clear-num 99.9%

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

   8. pow2 99.9%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{1 - \color{blue}{{\left(\frac{b}{a}\right)}^{2}}}\right)} - 1 \]

7. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{1 - {\left(\frac{b}{a}\right)}^{2}}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 99.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - {\left(\frac{b}{a}\right)}^{2}}\right)\right)} \]

   2. expm1-log1p 100.0%

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

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. Final simplification 100.0%

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

Alternative 3: 99.0% accurate, 19.2× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{b}{\frac{a}{b}} \cdot \frac{-0.5}{a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double a, double b) {
	return 1.0 + ((b / (a / b)) * (-0.5 / a));
}

Python

def code(a, b):
	return 1.0 + ((b / (a / b)) * (-0.5 / a))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

   1. div-sub 78.1%

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

   2. *-inverses 78.1%

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

   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%

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

   3. add-sqr-sqrt 100.0%

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

   4. *-inverses 78.1%

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

   5. frac-times 78.1%

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

   6. div-sub 78.1%

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

   7. associate-/r* 77.7%

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

   8. sqrt-div 77.7%

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

5. Applied egg-rr 77.7%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{a \cdot a - b \cdot b}{a}}}{\sqrt{a}}} \]
6. Step-by-step derivation

   1. difference-of-squares 77.7%

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

   2. +-commutative 77.7%

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

   3. associate-*r/ 100.0%

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

   4. div-sub 99.9%

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

   5. *-inverses 99.9%

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

7. Simplified 99.9%

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

8. Taylor expanded in b around 0 77.6%

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

   1. unpow2 77.6%

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

   2. associate-*r/ 77.6%

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

   3. unpow2 77.6%

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

10. Simplified 77.6%

    \[\leadsto \color{blue}{1 + \frac{-0.5 \cdot \left(b \cdot b\right)}{a \cdot a}} \]
11. Step-by-step derivation

    1. *-commutative 77.6%

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

    2. times-frac 98.1%

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

    3. associate-/l* 99.0%

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

12. Applied egg-rr 99.0%

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

13. Final simplification 99.0%

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

Alternative 4: 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 78.1%

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

   1. div-sub 78.1%

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

   2. *-inverses 78.1%

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

   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. Step-by-step derivation

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 99.9%

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

   3. add-sqr-sqrt 99.9%

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

   4. fabs-sqr 99.9%

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

   5. add-sqr-sqrt 99.9%

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

   6. div-inv 99.9%

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

   7. clear-num 99.9%

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

   8. pow2 99.9%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{1 - \color{blue}{{\left(\frac{b}{a}\right)}^{2}}}\right)} - 1 \]

7. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{1 - {\left(\frac{b}{a}\right)}^{2}}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 99.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - {\left(\frac{b}{a}\right)}^{2}}\right)\right)} \]

   2. expm1-log1p 100.0%

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

9. Simplified 100.0%

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

10. Taylor expanded in b around 0 97.6%

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

11. Final simplification 97.6%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023173 
(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)))))