Eccentricity of an ellipse

Percentage Accurate: 77.8% → 100.0%

Time: 9.0s

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: 77.8% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.4%

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

   1. sqr-neg 75.4%

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

   2. associate-/r* 75.2%

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

   3. sqr-neg 75.2%

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

   4. associate-/r* 75.4%

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

   5. div-sub 75.4%

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

   6. fabs-sub 75.4%

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

   7. times-frac 75.4%

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

   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 75.4%

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

   3. frac-times 75.4%

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

   4. div-sub 75.4%

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

   5. add-sqr-sqrt 75.4%

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

   6. fabs-sqr 75.4%

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

   7. add-sqr-sqrt 75.4%

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

   8. associate-/r* 75.2%

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

   9. sqrt-div 75.2%

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

5. Applied egg-rr 75.2%

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

   1. div-sub 75.2%

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

   2. associate-/l* 99.0%

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

   3. *-inverses 99.0%

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

   4. /-rgt-identity 99.0%

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

   5. associate-*r/ 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

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 75.4%

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

   1. sqr-neg 75.4%

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

   2. associate-/r* 75.2%

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

   3. sqr-neg 75.2%

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

   4. associate-/r* 75.4%

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

   5. div-sub 75.4%

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

   6. fabs-sub 75.4%

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

   7. times-frac 75.4%

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

   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 75.4%

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

   3. frac-times 75.4%

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

   4. div-sub 75.4%

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

   5. add-sqr-sqrt 75.4%

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

   6. fabs-sqr 75.4%

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

   7. add-sqr-sqrt 75.4%

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

   8. associate-/r* 75.2%

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

   9. sqrt-div 75.2%

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

5. Applied egg-rr 75.2%

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

   1. div-sub 75.2%

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

   2. associate-/l* 99.0%

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

   3. *-inverses 99.0%

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

   4. /-rgt-identity 99.0%

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

   5. associate-*r/ 100.0%

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

7. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

      \[\leadsto 0 + \color{blue}{\frac{\sqrt{a - b \cdot \frac{b}{a}}}{\sqrt{a}}} \]

   6. sqrt-undiv 100.0%

      \[\leadsto 0 + \color{blue}{\sqrt{\frac{a - b \cdot \frac{b}{a}}{a}}} \]

   7. div-sub 100.0%

      \[\leadsto 0 + \sqrt{\color{blue}{\frac{a}{a} - \frac{b \cdot \frac{b}{a}}{a}}} \]

   8. associate-*l/ 100.0%

      \[\leadsto 0 + \sqrt{\frac{a}{a} - \color{blue}{\frac{b}{a} \cdot \frac{b}{a}}} \]

   9. pow2 100.0%

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

9. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{0 + \sqrt{\frac{a}{a} - {\left(\frac{b}{a}\right)}^{2}}} \]
10. Step-by-step derivation

    1. +-lft-identity 100.0%

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

    2. *-inverses 100.0%

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

11. Simplified 100.0%

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

    1. unpow2 99.3%

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

    2. clear-num 99.3%

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

    3. un-div-inv 99.3%

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

13. Applied egg-rr 100.0%

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

14. Final simplification 100.0%

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

Alternative 3: 99.1% 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 75.4%

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

   1. sqr-neg 75.4%

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

   2. associate-/r* 75.2%

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

   3. sqr-neg 75.2%

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

   4. associate-/r* 75.4%

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

   5. div-sub 75.4%

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

   6. fabs-sub 75.4%

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

   7. times-frac 75.4%

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

   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 75.4%

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

   3. frac-times 75.4%

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

   4. div-sub 75.4%

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

   5. add-sqr-sqrt 75.4%

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

   6. fabs-sqr 75.4%

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

   7. add-sqr-sqrt 75.4%

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

   8. associate-/r* 75.2%

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

   9. sqrt-div 75.2%

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

5. Applied egg-rr 75.2%

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

   1. div-sub 75.2%

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

   2. associate-/l* 99.0%

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

   3. *-inverses 99.0%

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

   4. /-rgt-identity 99.0%

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

   5. associate-*r/ 100.0%

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

7. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

      \[\leadsto 0 + \color{blue}{\frac{\sqrt{a - b \cdot \frac{b}{a}}}{\sqrt{a}}} \]

   6. sqrt-undiv 100.0%

      \[\leadsto 0 + \color{blue}{\sqrt{\frac{a - b \cdot \frac{b}{a}}{a}}} \]

   7. div-sub 100.0%

      \[\leadsto 0 + \sqrt{\color{blue}{\frac{a}{a} - \frac{b \cdot \frac{b}{a}}{a}}} \]

   8. associate-*l/ 100.0%

      \[\leadsto 0 + \sqrt{\frac{a}{a} - \color{blue}{\frac{b}{a} \cdot \frac{b}{a}}} \]

   9. pow2 100.0%

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

9. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{0 + \sqrt{\frac{a}{a} - {\left(\frac{b}{a}\right)}^{2}}} \]
10. Step-by-step derivation

    1. +-lft-identity 100.0%

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

    2. *-inverses 100.0%

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

11. Simplified 100.0%

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

12. Taylor expanded in b around 0 75.0%

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

    1. unpow2 75.0%

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

    2. unpow2 75.0%

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

    3. times-frac 99.3%

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

    4. unpow2 99.3%

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

14. Simplified 99.3%

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

    1. unpow2 99.3%

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

    2. clear-num 99.3%

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

    3. un-div-inv 99.3%

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

16. Applied egg-rr 99.3%

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

17. Final simplification 99.3%

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

Alternative 4: 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 75.4%

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

   1. sqr-neg 75.4%

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

   2. associate-/r* 75.2%

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

   3. sqr-neg 75.2%

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

   4. associate-/r* 75.4%

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

   5. div-sub 75.4%

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

   6. fabs-sub 75.4%

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

   7. times-frac 75.4%

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

   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 75.4%

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

   3. frac-times 75.4%

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

   4. div-sub 75.4%

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

   5. add-sqr-sqrt 75.4%

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

   6. fabs-sqr 75.4%

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

   7. add-sqr-sqrt 75.4%

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

   8. associate-/r* 75.2%

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

   9. sqrt-div 75.2%

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

5. Applied egg-rr 75.2%

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

   1. div-sub 75.2%

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

   2. associate-/l* 99.0%

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

   3. *-inverses 99.0%

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

   4. /-rgt-identity 99.0%

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

   5. associate-*r/ 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in a around inf 98.2%

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

9. Final simplification 98.2%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023283 
(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)))))