Eccentricity of an ellipse

Percentage Accurate: 77.5% → 100.0%

Time: 9.3s

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: 77.5% 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

\[\begin{array}{l} \\ {\left({\left(1 - {\left(\frac{b}{a}\right)}^{2}\right)}^{1.5}\right)}^{0.3333333333333333} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (pow (pow (- 1.0 (pow (/ b a) 2.0)) 1.5) 0.3333333333333333))

C

double code(double a, double b) {
	return pow(pow((1.0 - pow((b / a), 2.0)), 1.5), 0.3333333333333333);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((1.0d0 - ((b / a) ** 2.0d0)) ** 1.5d0) ** 0.3333333333333333d0
end function

Java

public static double code(double a, double b) {
	return Math.pow(Math.pow((1.0 - Math.pow((b / a), 2.0)), 1.5), 0.3333333333333333);
}

Python

def code(a, b):
	return math.pow(math.pow((1.0 - math.pow((b / a), 2.0)), 1.5), 0.3333333333333333)

Julia

function code(a, b)
	return (Float64(1.0 - (Float64(b / a) ^ 2.0)) ^ 1.5) ^ 0.3333333333333333
end

MATLAB

function tmp = code(a, b)
	tmp = ((1.0 - ((b / a) ^ 2.0)) ^ 1.5) ^ 0.3333333333333333;
end

Wolfram

code[a_, b_] := N[Power[N[Power[N[(1.0 - N[Power[N[(b / a), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 0.3333333333333333], $MachinePrecision]

Derivation

1. Initial program 76.9%

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

   1. fabs-div 76.9%

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

   2. sqr-neg 76.9%

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

   3. fabs-div 76.9%

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

   4. div-sub 76.9%

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

   5. *-inverses 76.9%

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

   6. sqr-neg 76.9%

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

   7. 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-cbrt-cube 100.0%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|} \cdot \sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\right) \cdot \sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}}} \]

   2. pow1/3 100.0%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|} \cdot \sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\right) \cdot \sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\right)}^{0.3333333333333333}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{\left({\left(1 - {\left(\frac{b}{a}\right)}^{2}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

6. Final simplification 100.0%

   \[\leadsto {\left({\left(1 - {\left(\frac{b}{a}\right)}^{2}\right)}^{1.5}\right)}^{0.3333333333333333} \]

Alternative 2: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	return sqrt((1.0 + (1.0 + (-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((1.0d0 + (1.0d0 + ((-1.0d0) - ((b / (a / b)) / a)))))
end function

Java

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

Python

def code(a, b):
	return math.sqrt((1.0 + (1.0 + (-1.0 - ((b / (a / b)) / a)))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

   1. fabs-div 76.9%

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

   2. sqr-neg 76.9%

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

   3. fabs-div 76.9%

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

   4. div-sub 76.9%

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

   5. *-inverses 76.9%

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

   6. sqr-neg 76.9%

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

   7. 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. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

   3. add-sqr-sqrt 100.0%

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

   4. fabs-sqr 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. pow2 100.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

7. Simplified 100.0%

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

   1. +-rgt-identity 100.0%

      \[\leadsto \sqrt{1 - \color{blue}{\left({\left(\frac{b}{a}\right)}^{2} + 0\right)}} \]

   2. metadata-eval 100.0%

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

   3. associate--l+ 100.0%

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

9. Applied egg-rr 100.0%

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

    1. unpow2 98.9%

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

    2. clear-num 98.9%

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

    3. frac-times 98.9%

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

    4. *-un-lft-identity 98.9%

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

    5. associate-/r* 98.9%

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

11. Applied egg-rr 100.0%

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

12. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	return sqrt((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((1.0d0 - ((b / (a / b)) / a)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

   1. fabs-div 76.9%

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

   2. sqr-neg 76.9%

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

   3. fabs-div 76.9%

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

   4. div-sub 76.9%

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

   5. *-inverses 76.9%

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

   6. sqr-neg 76.9%

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

   7. 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. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

   3. add-sqr-sqrt 100.0%

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

   4. fabs-sqr 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. pow2 100.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

7. Simplified 100.0%

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

   1. unpow2 98.9%

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

   2. clear-num 98.9%

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

   3. frac-times 98.9%

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

   4. *-un-lft-identity 98.9%

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

   5. associate-/r* 98.9%

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

9. Applied egg-rr 100.0%

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

10. Final simplification 100.0%

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

Alternative 4: 99.0% accurate, 19.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

   1. fabs-div 76.9%

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

   2. sqr-neg 76.9%

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

   3. fabs-div 76.9%

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

   4. div-sub 76.9%

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

   5. *-inverses 76.9%

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

   6. sqr-neg 76.9%

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

   7. 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-cbrt-cube 100.0%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|} \cdot \sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\right) \cdot \sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}}} \]

   2. pow1/3 100.0%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|} \cdot \sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\right) \cdot \sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\right)}^{0.3333333333333333}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{\left({\left(1 - {\left(\frac{b}{a}\right)}^{2}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
6. Step-by-step derivation

   1. unpow1/3 100.0%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(1 - {\left(\frac{b}{a}\right)}^{2}\right)}^{1.5}}} \]

7. Simplified 100.0%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(1 - {\left(\frac{b}{a}\right)}^{2}\right)}^{1.5}}} \]

8. Taylor expanded in b around 0 76.1%

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

   1. unpow2 76.1%

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

   2. associate-/l/ 98.5%

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

   3. unpow2 98.5%

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

   4. associate-*r/ 98.9%

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

   5. associate-*l/ 98.9%

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

   6. unpow2 98.9%

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

10. Simplified 98.9%

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

    1. unpow2 98.9%

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

    2. clear-num 98.9%

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

    3. frac-times 98.9%

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

    4. *-un-lft-identity 98.9%

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

    5. associate-/r* 98.9%

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

12. Applied egg-rr 98.9%

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

13. Final simplification 98.9%

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

Alternative 5: 98.0% 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 76.9%

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

   1. fabs-div 76.9%

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

   2. sqr-neg 76.9%

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

   3. fabs-div 76.9%

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

   4. div-sub 76.9%

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

   5. *-inverses 76.9%

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

   6. sqr-neg 76.9%

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

   7. 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-cbrt-cube 100.0%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|} \cdot \sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\right) \cdot \sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}}} \]

   2. pow1/3 100.0%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|} \cdot \sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\right) \cdot \sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\right)}^{0.3333333333333333}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{\left({\left(1 - {\left(\frac{b}{a}\right)}^{2}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
6. Step-by-step derivation

   1. unpow1/3 100.0%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(1 - {\left(\frac{b}{a}\right)}^{2}\right)}^{1.5}}} \]

7. Simplified 100.0%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(1 - {\left(\frac{b}{a}\right)}^{2}\right)}^{1.5}}} \]

8. Taylor expanded in b around 0 98.3%

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

9. Final simplification 98.3%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023301 
(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)))))