Eccentricity of an ellipse

Percentage Accurate: 77.4% → 100.0%

Time: 6.3s

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: 77.4% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. sqr-neg 79.3%

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

   2. fabs-div 79.3%

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

   3. sqr-neg 79.3%

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

   4. fabs-sub 79.3%

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

   5. sqr-neg 79.3%

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

   6. distribute-rgt-neg-out 79.3%

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

   7. fabs-neg 79.3%

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

   8. fabs-div 79.3%

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

   9. cancel-sign-sub-inv 79.3%

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

   10. +-commutative 79.3%

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

   11. sqr-neg 79.3%

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

   12. cancel-sign-sub-inv 79.3%

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

3. Simplified 79.9%

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

   1. pow1/2 79.9%

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

   2. pow-to-exp 79.9%

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

   3. add-sqr-sqrt 79.3%

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

   4. fabs-sqr 79.3%

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

   5. add-sqr-sqrt 79.3%

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

   6. sub-neg 79.3%

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

   7. log1p-define 79.3%

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

   8. associate-*r/ 79.3%

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

   9. frac-times 100.0%

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

   10. pow2 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-{\left(\frac{b}{a}\right)}^{2}\right) \cdot 0.5}} \]
7. 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. div-inv 100.0%

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

   4. associate-/l/ 100.0%

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

   5. associate-/r* 100.0%

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

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 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 79.3%

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

   1. sqr-neg 79.3%

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

   2. fabs-div 79.3%

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

   3. sqr-neg 79.3%

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

   4. fabs-sub 79.3%

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

   5. sqr-neg 79.3%

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

   6. distribute-rgt-neg-out 79.3%

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

   7. fabs-neg 79.3%

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

   8. fabs-div 79.3%

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

   9. cancel-sign-sub-inv 79.3%

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

   10. +-commutative 79.3%

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

   11. sqr-neg 79.3%

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

   12. cancel-sign-sub-inv 79.3%

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

3. Simplified 79.9%

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

   1. *-inverses 79.3%

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

   2. associate-*r/ 79.3%

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

   3. div-sub 79.3%

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

   4. difference-of-squares 79.3%

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

   5. times-frac 100.0%

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

   6. +-commutative 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \sqrt{\left|\color{blue}{\frac{b + a}{a} \cdot \frac{a - b}{a}}\right|} \]
7. 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. frac-times 79.3%

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

   6. +-commutative 79.3%

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

   7. difference-of-squares 79.3%

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

   8. pow2 79.3%

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

   9. pow2 79.3%

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

   10. pow2 79.3%

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

8. Applied egg-rr 79.3%

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

   1. unpow1/2 79.3%

      \[\leadsto \color{blue}{\sqrt{\frac{{a}^{2} - {b}^{2}}{{a}^{2}}}} \]

   2. unpow2 79.3%

      \[\leadsto \sqrt{\frac{{a}^{2} - {b}^{2}}{\color{blue}{a \cdot a}}} \]

   3. associate-/l/ 79.0%

      \[\leadsto \sqrt{\color{blue}{\frac{\frac{{a}^{2} - {b}^{2}}{a}}{a}}} \]

   4. div-sub 79.0%

      \[\leadsto \sqrt{\frac{\color{blue}{\frac{{a}^{2}}{a} - \frac{{b}^{2}}{a}}}{a}} \]

   5. div-sub 79.0%

      \[\leadsto \sqrt{\color{blue}{\frac{\frac{{a}^{2}}{a}}{a} - \frac{\frac{{b}^{2}}{a}}{a}}} \]

   6. associate-/r* 79.3%

      \[\leadsto \sqrt{\color{blue}{\frac{{a}^{2}}{a \cdot a}} - \frac{\frac{{b}^{2}}{a}}{a}} \]

   7. unpow2 79.3%

      \[\leadsto \sqrt{\frac{{a}^{2}}{\color{blue}{{a}^{2}}} - \frac{\frac{{b}^{2}}{a}}{a}} \]

   8. *-inverses 98.3%

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

   9. unpow2 98.3%

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

   10. associate-*r/ 100.0%

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

   11. associate-*l/ 100.0%

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

   12. unpow2 100.0%

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

10. Simplified 100.0%

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

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

    4. associate-/l/ 100.0%

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

    5. associate-/r* 100.0%

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

12. Applied egg-rr 100.0%

    \[\leadsto \sqrt{1 - \color{blue}{\frac{\frac{b}{\frac{a}{b}}}{a}}} \]
13. Add Preprocessing

Alternative 3: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ e^{\frac{\frac{b}{\frac{a}{b}}}{a} \cdot -0.5} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. sqr-neg 79.3%

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

   2. fabs-div 79.3%

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

   3. sqr-neg 79.3%

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

   4. fabs-sub 79.3%

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

   5. sqr-neg 79.3%

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

   6. distribute-rgt-neg-out 79.3%

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

   7. fabs-neg 79.3%

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

   8. fabs-div 79.3%

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

   9. cancel-sign-sub-inv 79.3%

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

   10. +-commutative 79.3%

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

   11. sqr-neg 79.3%

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

   12. cancel-sign-sub-inv 79.3%

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

3. Simplified 79.9%

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

   1. pow1/2 79.9%

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

   2. pow-to-exp 79.9%

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

   3. add-sqr-sqrt 79.3%

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

   4. fabs-sqr 79.3%

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

   5. add-sqr-sqrt 79.3%

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

   6. sub-neg 79.3%

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

   7. log1p-define 79.3%

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

   8. associate-*r/ 79.3%

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

   9. frac-times 100.0%

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

   10. pow2 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in b around 0 78.6%

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

   1. unpow2 78.6%

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

   2. unpow2 78.6%

      \[\leadsto e^{-0.5 \cdot \frac{b \cdot b}{\color{blue}{a \cdot a}}} \]

   3. times-frac 98.2%

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

   4. unpow2 98.2%

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

9. Simplified 98.2%

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

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

    4. associate-/l/ 100.0%

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

    5. associate-/r* 100.0%

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

11. Applied egg-rr 98.2%

    \[\leadsto e^{-0.5 \cdot \color{blue}{\frac{\frac{b}{\frac{a}{b}}}{a}}} \]

12. Final simplification 98.2%

    \[\leadsto e^{\frac{\frac{b}{\frac{a}{b}}}{a} \cdot -0.5} \]
13. Add Preprocessing

Alternative 4: 97.9% accurate, 211.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 1.0)

C

double code(double a, double b) {
	return 1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0
end function

Java

public static double code(double a, double b) {
	return 1.0;
}

Python

def code(a, b):
	return 1.0

Julia

function code(a, b)
	return 1.0
end

MATLAB

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

Wolfram

code[a_, b_] := 1.0

Derivation

1. Initial program 79.3%

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

   1. sqr-neg 79.3%

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

   2. fabs-div 79.3%

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

   3. sqr-neg 79.3%

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

   4. fabs-sub 79.3%

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

   5. sqr-neg 79.3%

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

   6. distribute-rgt-neg-out 79.3%

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

   7. fabs-neg 79.3%

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

   8. fabs-div 79.3%

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

   9. cancel-sign-sub-inv 79.3%

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

   10. +-commutative 79.3%

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

   11. sqr-neg 79.3%

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

   12. cancel-sign-sub-inv 79.3%

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

3. Simplified 79.9%

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

   1. pow1/2 79.9%

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

   2. pow-to-exp 79.9%

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

   3. add-sqr-sqrt 79.3%

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

   4. fabs-sqr 79.3%

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

   5. add-sqr-sqrt 79.3%

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

   6. sub-neg 79.3%

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

   7. log1p-define 79.3%

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

   8. associate-*r/ 79.3%

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

   9. frac-times 100.0%

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

   10. pow2 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in b around 0 78.6%

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

   1. unpow2 78.6%

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

   2. unpow2 78.6%

      \[\leadsto e^{-0.5 \cdot \frac{b \cdot b}{\color{blue}{a \cdot a}}} \]

   3. times-frac 98.2%

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

   4. unpow2 98.2%

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

9. Simplified 98.2%

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

10. Taylor expanded in b around 0 96.9%

    \[\leadsto \color{blue}{1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024105 
(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)))))