Eccentricity of an ellipse

Percentage Accurate: 76.2% → 100.0%

Time: 5.3s

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: 76.2% 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

\[\begin{array}{l} \\ e^{\mathsf{log1p}\left(-{\left(\frac{b}{a}\right)}^{2}\right) \cdot 0.5} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (exp (* (log1p (- (pow (/ b a) 2.0))) 0.5)))

C

double code(double a, double b) {
	return exp((log1p(-pow((b / a), 2.0)) * 0.5));
}

Java

public static double code(double a, double b) {
	return Math.exp((Math.log1p(-Math.pow((b / a), 2.0)) * 0.5));
}

Python

def code(a, b):
	return math.exp((math.log1p(-math.pow((b / a), 2.0)) * 0.5))

Julia

function code(a, b)
	return exp(Float64(log1p(Float64(-(Float64(b / a) ^ 2.0))) * 0.5))
end

Wolfram

code[a_, b_] := N[Exp[N[(N[Log[1 + (-N[Power[N[(b / a), $MachinePrecision], 2.0], $MachinePrecision])], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 78.5%

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

   1. sqr-neg 78.5%

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

   2. associate-/r* 78.4%

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

   3. sqr-neg 78.4%

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

   4. associate-/r* 78.5%

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

   5. div-sub 78.5%

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

   6. fabs-sub 78.5%

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

   7. times-frac 78.5%

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

   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 78.5%

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

   4. frac-times 78.5%

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

   5. div-sub 78.5%

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

   6. pow-to-exp 78.5%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.9× speedup

Math

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

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

Derivation

1. Initial program 78.5%

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

   1. sqr-neg 78.5%

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

   2. associate-/r* 78.4%

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

   3. sqr-neg 78.4%

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

   4. associate-/r* 78.5%

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

   5. div-sub 78.5%

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

   6. fabs-sub 78.5%

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

   7. times-frac 78.5%

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

   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 78.5%

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

   3. frac-times 78.5%

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

   4. div-sub 78.5%

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

   5. add-sqr-sqrt 78.5%

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

   6. fabs-sqr 78.5%

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

   7. add-sqr-sqrt 78.5%

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

   8. associate-/r* 78.4%

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

   9. sqrt-div 78.4%

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

5. Applied egg-rr 78.4%

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

   1. div-sub 78.4%

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

   2. associate-/l* 98.7%

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

   3. *-inverses 98.7%

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

   4. /-rgt-identity 98.7%

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

   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. Step-by-step derivation

   1. add-log-exp 99.9%

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

   2. *-un-lft-identity 99.9%

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

   3. log-prod 99.9%

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

   4. metadata-eval 99.9%

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

   5. add-log-exp 99.9%

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

   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. unpow2 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 100.0%

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

    2. clear-num 100.0%

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

    3. frac-times 100.0%

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

    4. *-un-lft-identity 100.0%

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

13. Applied egg-rr 100.0%

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

14. Final simplification 100.0%

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

Alternative 3: 99.0% accurate, 16.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

   1. sqr-neg 78.5%

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

   2. associate-/r* 78.4%

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

   3. sqr-neg 78.4%

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

   4. associate-/r* 78.5%

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

   5. div-sub 78.5%

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

   6. fabs-sub 78.5%

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

   7. times-frac 78.5%

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

   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 78.5%

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

   3. frac-times 78.5%

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

   4. div-sub 78.5%

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

   5. add-sqr-sqrt 78.5%

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

   6. fabs-sqr 78.5%

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

   7. add-sqr-sqrt 78.5%

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

   8. associate-/r* 78.4%

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

   9. sqrt-div 78.4%

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

5. Applied egg-rr 78.4%

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

   1. div-sub 78.4%

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

   2. associate-/l* 98.7%

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

   3. *-inverses 98.7%

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

   4. /-rgt-identity 98.7%

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

   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 78.0%

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

   1. unpow2 78.0%

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

   2. unpow2 78.0%

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

   3. times-frac 99.0%

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

   4. unpow2 99.0%

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

10. Simplified 99.0%

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

    1. unpow2 99.0%

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

    2. clear-num 99.0%

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

    3. clear-num 99.0%

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

    4. frac-times 99.0%

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

    5. metadata-eval 99.0%

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

12. Applied egg-rr 99.0%

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

13. Final simplification 99.0%

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

Alternative 4: 99.0% accurate, 19.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

   1. sqr-neg 78.5%

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

   2. associate-/r* 78.4%

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

   3. sqr-neg 78.4%

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

   4. associate-/r* 78.5%

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

   5. div-sub 78.5%

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

   6. fabs-sub 78.5%

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

   7. times-frac 78.5%

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

   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 78.5%

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

   3. frac-times 78.5%

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

   4. div-sub 78.5%

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

   5. add-sqr-sqrt 78.5%

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

   6. fabs-sqr 78.5%

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

   7. add-sqr-sqrt 78.5%

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

   8. associate-/r* 78.4%

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

   9. sqrt-div 78.4%

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

5. Applied egg-rr 78.4%

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

   1. div-sub 78.4%

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

   2. associate-/l* 98.7%

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

   3. *-inverses 98.7%

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

   4. /-rgt-identity 98.7%

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

   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 78.0%

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

   1. unpow2 78.0%

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

   2. unpow2 78.0%

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

   3. times-frac 99.0%

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

   4. unpow2 99.0%

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

10. Simplified 99.0%

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

    1. unpow2 99.0%

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

    2. associate-*l/ 99.0%

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

    3. clear-num 99.0%

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

    4. un-div-inv 99.0%

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

12. Applied egg-rr 99.0%

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

13. Final simplification 99.0%

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

Alternative 5: 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 78.5%

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

   1. sqr-neg 78.5%

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

   2. associate-/r* 78.4%

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

   3. sqr-neg 78.4%

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

   4. associate-/r* 78.5%

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

   5. div-sub 78.5%

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

   6. fabs-sub 78.5%

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

   7. times-frac 78.5%

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

   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 78.5%

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

   3. frac-times 78.5%

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

   4. div-sub 78.5%

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

   5. add-sqr-sqrt 78.5%

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

   6. fabs-sqr 78.5%

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

   7. add-sqr-sqrt 78.5%

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

   8. associate-/r* 78.4%

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

   9. sqrt-div 78.4%

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

5. Applied egg-rr 78.4%

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

   1. div-sub 78.4%

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

   2. associate-/l* 98.7%

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

   3. *-inverses 98.7%

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

   4. /-rgt-identity 98.7%

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

   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 97.4%

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

9. Final simplification 97.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023264 
(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)))))