Eccentricity of an ellipse

Percentage Accurate: 77.6% → 100.0%

Time: 6.1s

Alternatives: 5

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: 77.6% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. sqr-neg 81.9%

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

   2. associate-/r* 81.0%

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

   3. sqr-neg 81.0%

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

   4. associate-/r* 81.9%

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

   5. div-sub 81.9%

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

   6. fabs-sub 81.9%

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

   7. times-frac 82.0%

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

   8. *-inverses 100.0%

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

3. Simplified 100.0%

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

   1. pow1/2 100.0%

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

   2. fabs-sub 100.0%

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

   3. *-inverses 82.0%

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

   4. frac-times 81.9%

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

   5. div-sub 81.9%

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

   6. pow-to-exp 81.9%

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

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. un-div-inv 100.0%

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

7. Applied egg-rr 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return math.sqrt((((a - b) / a) * ((b + a) / a)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. difference-of-squares 81.9%

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

   2. times-frac 100.0%

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

   3. +-commutative 100.0%

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

3. Applied egg-rr 100.0%

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

   1. pow1/2 100.0%

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

   2. add-sqr-sqrt 100.0%

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

   3. fabs-sqr 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. clear-num 100.0%

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

   6. frac-times 100.0%

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

   7. *-commutative 100.0%

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

   8. *-un-lft-identity 100.0%

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

5. Applied egg-rr 100.0%

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

   1. unpow1/2 99.9%

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

   2. clear-num 99.9%

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

   3. un-div-inv 100.0%

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

7. Applied egg-rr 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

   3. *-un-lft-identity 100.0%

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

   4. associate-/r/ 100.0%

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

   5. times-frac 100.0%

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

   6. clear-num 99.9%

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

   7. +-commutative 99.9%

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

9. Applied egg-rr 99.9%

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

    1. expm1-def 99.9%

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

    2. expm1-log1p 100.0%

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

11. Simplified 100.0%

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

12. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 (sqrt (/ (+ b a) (/ a (/ (- a b) a)))))

C

double code(double a, double b) {
	return sqrt(((b + a) / (a / ((a - b) / a))));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = sqrt(((b + a) / (a / ((a - b) / a))))
end function

Java

public static double code(double a, double b) {
	return Math.sqrt(((b + a) / (a / ((a - b) / a))));
}

Python

def code(a, b):
	return math.sqrt(((b + a) / (a / ((a - b) / a))))

Julia

function code(a, b)
	return sqrt(Float64(Float64(b + a) / Float64(a / Float64(Float64(a - b) / a))))
end

MATLAB

function tmp = code(a, b)
	tmp = sqrt(((b + a) / (a / ((a - b) / a))));
end

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. difference-of-squares 81.9%

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

   2. times-frac 100.0%

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

   3. +-commutative 100.0%

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

3. Applied egg-rr 100.0%

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

   1. pow1/2 100.0%

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

   2. add-sqr-sqrt 100.0%

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

   3. fabs-sqr 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. clear-num 100.0%

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

   6. frac-times 100.0%

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

   7. *-commutative 100.0%

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

   8. *-un-lft-identity 100.0%

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

5. Applied egg-rr 100.0%

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

   1. unpow1/2 99.9%

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

   2. clear-num 99.9%

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

   3. un-div-inv 100.0%

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

7. Applied egg-rr 100.0%

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

8. Final simplification 100.0%

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

Alternative 4: 99.0% accurate, 19.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. sqr-neg 81.9%

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

   2. associate-/r* 81.0%

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

   3. sqr-neg 81.0%

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

   4. associate-/r* 81.9%

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

   5. div-sub 81.9%

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

   6. fabs-sub 81.9%

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

   7. times-frac 82.0%

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

   8. *-inverses 100.0%

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

3. Simplified 100.0%

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

   1. fabs-sub 100.0%

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

   2. *-inverses 82.0%

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

   3. frac-times 81.9%

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

   4. div-sub 81.9%

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

   5. add-sqr-sqrt 81.9%

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

   6. fabs-sqr 81.9%

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

   7. add-sqr-sqrt 81.9%

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

   8. associate-/r* 81.0%

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

   9. sqrt-div 81.0%

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

5. Applied egg-rr 81.0%

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

   1. div-sub 81.0%

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

   2. associate-/l* 99.1%

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

   3. *-inverses 99.1%

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

   4. /-rgt-identity 99.1%

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

   5. associate-*r/ 99.9%

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

7. Simplified 99.9%

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

8. Taylor expanded in a around inf 81.1%

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

   1. unpow2 81.1%

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

   2. unpow2 81.1%

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

   3. times-frac 98.7%

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

   4. unpow2 98.7%

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

10. Simplified 98.7%

    \[\leadsto \color{blue}{1 + -0.5 \cdot {\left(\frac{b}{a}\right)}^{2}} \]
11. 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. un-div-inv 100.0%

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

12. Applied egg-rr 98.7%

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

13. Final simplification 98.7%

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

Alternative 5: 97.8% 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 81.9%

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

   1. difference-of-squares 81.9%

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

   2. times-frac 100.0%

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

   3. +-commutative 100.0%

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

3. Applied egg-rr 100.0%

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

   1. expm1-log1p-u 99.9%

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

   2. expm1-udef 99.9%

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

5. Applied egg-rr 81.1%

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

   1. expm1-def 81.1%

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

   2. expm1-log1p 81.1%

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

7. Simplified 81.1%

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

8. Taylor expanded in a around inf 97.4%

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

9. Final simplification 97.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023271 
(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)))))