Eccentricity of an ellipse

Percentage Accurate: 77.4% → 100.0%

Time: 4.2s

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: 77.4% 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

\[\begin{array}{l} \\ {\left({\left(\frac{\frac{b}{a}}{\frac{a}{b}} + -1\right)}^{2}\right)}^{0.25} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (pow (pow (+ (/ (/ b a) (/ a b)) -1.0) 2.0) 0.25))

C

double code(double a, double b) {
	return pow(pow((((b / a) / (a / b)) + -1.0), 2.0), 0.25);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((b / a) / (a / b)) + (-1.0d0)) ** 2.0d0) ** 0.25d0
end function

Java

public static double code(double a, double b) {
	return Math.pow(Math.pow((((b / a) / (a / b)) + -1.0), 2.0), 0.25);
}

Python

def code(a, b):
	return math.pow(math.pow((((b / a) / (a / b)) + -1.0), 2.0), 0.25)

Julia

function code(a, b)
	return (Float64(Float64(Float64(b / a) / Float64(a / b)) + -1.0) ^ 2.0) ^ 0.25
end

MATLAB

function tmp = code(a, b)
	tmp = ((((b / a) / (a / b)) + -1.0) ^ 2.0) ^ 0.25;
end

Wolfram

code[a_, b_] := N[Power[N[Power[N[(N[(N[(b / a), $MachinePrecision] / N[(a / b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], 2.0], $MachinePrecision], 0.25], $MachinePrecision]

Derivation

1. Initial program 75.0%

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

   1. sqr-neg 75.0%

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

   2. associate-/r* 75.5%

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

   3. sqr-neg 75.5%

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

   4. associate-/r* 75.0%

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

   5. div-sub 75.0%

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

   6. fabs-sub 75.0%

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

   7. times-frac 75.0%

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

   8. *-inverses 100.0%

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

3. Simplified 100.0%

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

   1. pow1/2 100.0%

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

   2. add-sqr-sqrt 100.0%

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

   3. sqrt-unprod 100.0%

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

   4. pow1/2 100.0%

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

   5. pow-pow 100.0%

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

5. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{\left({\left({\left(\frac{a}{b}\right)}^{-2} + -1\right)}^{2}\right)}^{0.25}} \]
6. Step-by-step derivation

   1. metadata-eval 100.0%

      \[\leadsto {\left({\left({\left(\frac{a}{b}\right)}^{\color{blue}{\left(2 \cdot -1\right)}} + -1\right)}^{2}\right)}^{0.25} \]

   2. pow-sqr 100.0%

      \[\leadsto {\left({\left(\color{blue}{{\left(\frac{a}{b}\right)}^{-1} \cdot {\left(\frac{a}{b}\right)}^{-1}} + -1\right)}^{2}\right)}^{0.25} \]

   3. inv-pow 100.0%

      \[\leadsto {\left({\left(\color{blue}{\frac{1}{\frac{a}{b}}} \cdot {\left(\frac{a}{b}\right)}^{-1} + -1\right)}^{2}\right)}^{0.25} \]

   4. clear-num 100.0%

      \[\leadsto {\left({\left(\color{blue}{\frac{b}{a}} \cdot {\left(\frac{a}{b}\right)}^{-1} + -1\right)}^{2}\right)}^{0.25} \]

   5. inv-pow 100.0%

      \[\leadsto {\left({\left(\frac{b}{a} \cdot \color{blue}{\frac{1}{\frac{a}{b}}} + -1\right)}^{2}\right)}^{0.25} \]

   6. un-div-inv 100.0%

      \[\leadsto {\left({\left(\color{blue}{\frac{\frac{b}{a}}{\frac{a}{b}}} + -1\right)}^{2}\right)}^{0.25} \]

7. Applied egg-rr 100.0%

   \[\leadsto {\left({\left(\color{blue}{\frac{\frac{b}{a}}{\frac{a}{b}}} + -1\right)}^{2}\right)}^{0.25} \]

8. Final simplification 100.0%

   \[\leadsto {\left({\left(\frac{\frac{b}{a}}{\frac{a}{b}} + -1\right)}^{2}\right)}^{0.25} \]

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

   1. sqr-neg 75.0%

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

   2. associate-/r* 75.5%

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

   3. sqr-neg 75.5%

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

   4. associate-/r* 75.0%

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

   5. div-sub 75.0%

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

   6. fabs-sub 75.0%

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

   7. times-frac 75.0%

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

   8. *-inverses 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5, {\left(\frac{b}{a}\right)}^{2}, 1\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (fma -0.5 (pow (/ b a) 2.0) 1.0))

C

double code(double a, double b) {
	return fma(-0.5, pow((b / a), 2.0), 1.0);
}

Julia

function code(a, b)
	return fma(-0.5, (Float64(b / a) ^ 2.0), 1.0)
end

Wolfram

code[a_, b_] := N[(-0.5 * N[Power[N[(b / a), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 75.0%

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

   1. sqr-neg 75.0%

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

   2. associate-/r* 75.5%

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

   3. sqr-neg 75.5%

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

   4. associate-/r* 75.0%

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

   5. div-sub 75.0%

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

   6. fabs-sub 75.0%

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

   7. times-frac 75.0%

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

   8. *-inverses 100.0%

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

3. Simplified 100.0%

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

   1. pow1/2 100.0%

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

   2. add-sqr-sqrt 100.0%

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

   3. sqrt-unprod 100.0%

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

   4. pow1/2 100.0%

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

   5. pow-pow 100.0%

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

5. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{\left({\left({\left(\frac{a}{b}\right)}^{-2} + -1\right)}^{2}\right)}^{0.25}} \]
6. Step-by-step derivation

   1. metadata-eval 100.0%

      \[\leadsto {\left({\left({\left(\frac{a}{b}\right)}^{\color{blue}{\left(2 \cdot -1\right)}} + -1\right)}^{2}\right)}^{0.25} \]

   2. pow-sqr 100.0%

      \[\leadsto {\left({\left(\color{blue}{{\left(\frac{a}{b}\right)}^{-1} \cdot {\left(\frac{a}{b}\right)}^{-1}} + -1\right)}^{2}\right)}^{0.25} \]

   3. inv-pow 100.0%

      \[\leadsto {\left({\left(\color{blue}{\frac{1}{\frac{a}{b}}} \cdot {\left(\frac{a}{b}\right)}^{-1} + -1\right)}^{2}\right)}^{0.25} \]

   4. clear-num 100.0%

      \[\leadsto {\left({\left(\color{blue}{\frac{b}{a}} \cdot {\left(\frac{a}{b}\right)}^{-1} + -1\right)}^{2}\right)}^{0.25} \]

   5. inv-pow 100.0%

      \[\leadsto {\left({\left(\frac{b}{a} \cdot \color{blue}{\frac{1}{\frac{a}{b}}} + -1\right)}^{2}\right)}^{0.25} \]

   6. un-div-inv 100.0%

      \[\leadsto {\left({\left(\color{blue}{\frac{\frac{b}{a}}{\frac{a}{b}}} + -1\right)}^{2}\right)}^{0.25} \]

7. Applied egg-rr 100.0%

   \[\leadsto {\left({\left(\color{blue}{\frac{\frac{b}{a}}{\frac{a}{b}}} + -1\right)}^{2}\right)}^{0.25} \]

8. Taylor expanded in b around 0 74.6%

   \[\leadsto {\color{blue}{\left(1 + -2 \cdot \frac{{b}^{2}}{{a}^{2}}\right)}}^{0.25} \]
9. Step-by-step derivation

   1. *-commutative 74.6%

      \[\leadsto {\left(1 + \color{blue}{\frac{{b}^{2}}{{a}^{2}} \cdot -2}\right)}^{0.25} \]

   2. unpow2 74.6%

      \[\leadsto {\left(1 + \frac{\color{blue}{b \cdot b}}{{a}^{2}} \cdot -2\right)}^{0.25} \]

   3. unpow2 74.6%

      \[\leadsto {\left(1 + \frac{b \cdot b}{\color{blue}{a \cdot a}} \cdot -2\right)}^{0.25} \]

10. Simplified 74.6%

    \[\leadsto {\color{blue}{\left(1 + \frac{b \cdot b}{a \cdot a} \cdot -2\right)}}^{0.25} \]

11. Taylor expanded in b around 0 74.7%

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

    1. +-commutative 74.7%

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

    2. fma-def 74.7%

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

    3. unpow2 74.7%

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

    4. unpow2 74.7%

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

    5. times-frac 99.1%

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

    6. unpow2 99.1%

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

13. Simplified 99.1%

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

14. Final simplification 99.1%

    \[\leadsto \mathsf{fma}\left(-0.5, {\left(\frac{b}{a}\right)}^{2}, 1\right) \]

Alternative 4: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ {\left(1 + \frac{b \cdot -2}{\frac{a}{\frac{b}{a}}}\right)}^{0.25} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (pow (+ 1.0 (/ (* b -2.0) (/ a (/ b a)))) 0.25))

C

double code(double a, double b) {
	return pow((1.0 + ((b * -2.0) / (a / (b / a)))), 0.25);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (1.0d0 + ((b * (-2.0d0)) / (a / (b / a)))) ** 0.25d0
end function

Java

public static double code(double a, double b) {
	return Math.pow((1.0 + ((b * -2.0) / (a / (b / a)))), 0.25);
}

Python

def code(a, b):
	return math.pow((1.0 + ((b * -2.0) / (a / (b / a)))), 0.25)

Julia

function code(a, b)
	return Float64(1.0 + Float64(Float64(b * -2.0) / Float64(a / Float64(b / a)))) ^ 0.25
end

MATLAB

function tmp = code(a, b)
	tmp = (1.0 + ((b * -2.0) / (a / (b / a)))) ^ 0.25;
end

Wolfram

code[a_, b_] := N[Power[N[(1.0 + N[(N[(b * -2.0), $MachinePrecision] / N[(a / N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]

Derivation

1. Initial program 75.0%

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

   1. sqr-neg 75.0%

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

   2. associate-/r* 75.5%

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

   3. sqr-neg 75.5%

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

   4. associate-/r* 75.0%

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

   5. div-sub 75.0%

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

   6. fabs-sub 75.0%

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

   7. times-frac 75.0%

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

   8. *-inverses 100.0%

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

3. Simplified 100.0%

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

   1. pow1/2 100.0%

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

   2. add-sqr-sqrt 100.0%

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

   3. sqrt-unprod 100.0%

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

   4. pow1/2 100.0%

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

   5. pow-pow 100.0%

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

5. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{\left({\left({\left(\frac{a}{b}\right)}^{-2} + -1\right)}^{2}\right)}^{0.25}} \]
6. Step-by-step derivation

   1. metadata-eval 100.0%

      \[\leadsto {\left({\left({\left(\frac{a}{b}\right)}^{\color{blue}{\left(2 \cdot -1\right)}} + -1\right)}^{2}\right)}^{0.25} \]

   2. pow-sqr 100.0%

      \[\leadsto {\left({\left(\color{blue}{{\left(\frac{a}{b}\right)}^{-1} \cdot {\left(\frac{a}{b}\right)}^{-1}} + -1\right)}^{2}\right)}^{0.25} \]

   3. inv-pow 100.0%

      \[\leadsto {\left({\left(\color{blue}{\frac{1}{\frac{a}{b}}} \cdot {\left(\frac{a}{b}\right)}^{-1} + -1\right)}^{2}\right)}^{0.25} \]

   4. clear-num 100.0%

      \[\leadsto {\left({\left(\color{blue}{\frac{b}{a}} \cdot {\left(\frac{a}{b}\right)}^{-1} + -1\right)}^{2}\right)}^{0.25} \]

   5. inv-pow 100.0%

      \[\leadsto {\left({\left(\frac{b}{a} \cdot \color{blue}{\frac{1}{\frac{a}{b}}} + -1\right)}^{2}\right)}^{0.25} \]

   6. un-div-inv 100.0%

      \[\leadsto {\left({\left(\color{blue}{\frac{\frac{b}{a}}{\frac{a}{b}}} + -1\right)}^{2}\right)}^{0.25} \]

7. Applied egg-rr 100.0%

   \[\leadsto {\left({\left(\color{blue}{\frac{\frac{b}{a}}{\frac{a}{b}}} + -1\right)}^{2}\right)}^{0.25} \]

8. Taylor expanded in b around 0 74.6%

   \[\leadsto {\color{blue}{\left(1 + -2 \cdot \frac{{b}^{2}}{{a}^{2}}\right)}}^{0.25} \]
9. Step-by-step derivation

   1. *-commutative 74.6%

      \[\leadsto {\left(1 + \color{blue}{\frac{{b}^{2}}{{a}^{2}} \cdot -2}\right)}^{0.25} \]

   2. unpow2 74.6%

      \[\leadsto {\left(1 + \frac{\color{blue}{b \cdot b}}{{a}^{2}} \cdot -2\right)}^{0.25} \]

   3. unpow2 74.6%

      \[\leadsto {\left(1 + \frac{b \cdot b}{\color{blue}{a \cdot a}} \cdot -2\right)}^{0.25} \]

10. Simplified 74.6%

    \[\leadsto {\color{blue}{\left(1 + \frac{b \cdot b}{a \cdot a} \cdot -2\right)}}^{0.25} \]
11. Step-by-step derivation

    1. associate-/l* 75.4%

       \[\leadsto {\left(1 + \color{blue}{\frac{b}{\frac{a \cdot a}{b}}} \cdot -2\right)}^{0.25} \]

    2. associate-*l/ 75.4%

       \[\leadsto {\left(1 + \color{blue}{\frac{b \cdot -2}{\frac{a \cdot a}{b}}}\right)}^{0.25} \]

    3. associate-/l* 99.1%

       \[\leadsto {\left(1 + \frac{b \cdot -2}{\color{blue}{\frac{a}{\frac{b}{a}}}}\right)}^{0.25} \]

12. Applied egg-rr 99.1%

    \[\leadsto {\left(1 + \color{blue}{\frac{b \cdot -2}{\frac{a}{\frac{b}{a}}}}\right)}^{0.25} \]

13. Final simplification 99.1%

    \[\leadsto {\left(1 + \frac{b \cdot -2}{\frac{a}{\frac{b}{a}}}\right)}^{0.25} \]

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 75.0%

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

   1. sqr-neg 75.0%

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

   2. associate-/r* 75.5%

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

   3. sqr-neg 75.5%

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

   4. associate-/r* 75.0%

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

   5. div-sub 75.0%

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

   6. fabs-sub 75.0%

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

   7. times-frac 75.0%

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

   8. *-inverses 100.0%

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

3. Simplified 100.0%

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

   1. pow1/2 100.0%

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

   2. add-sqr-sqrt 100.0%

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

   3. sqrt-unprod 100.0%

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

   4. pow1/2 100.0%

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

   5. pow-pow 100.0%

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

5. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{\left({\left({\left(\frac{a}{b}\right)}^{-2} + -1\right)}^{2}\right)}^{0.25}} \]
6. Step-by-step derivation

   1. metadata-eval 100.0%

      \[\leadsto {\left({\left({\left(\frac{a}{b}\right)}^{\color{blue}{\left(2 \cdot -1\right)}} + -1\right)}^{2}\right)}^{0.25} \]

   2. pow-sqr 100.0%

      \[\leadsto {\left({\left(\color{blue}{{\left(\frac{a}{b}\right)}^{-1} \cdot {\left(\frac{a}{b}\right)}^{-1}} + -1\right)}^{2}\right)}^{0.25} \]

   3. inv-pow 100.0%

      \[\leadsto {\left({\left(\color{blue}{\frac{1}{\frac{a}{b}}} \cdot {\left(\frac{a}{b}\right)}^{-1} + -1\right)}^{2}\right)}^{0.25} \]

   4. clear-num 100.0%

      \[\leadsto {\left({\left(\color{blue}{\frac{b}{a}} \cdot {\left(\frac{a}{b}\right)}^{-1} + -1\right)}^{2}\right)}^{0.25} \]

   5. inv-pow 100.0%

      \[\leadsto {\left({\left(\frac{b}{a} \cdot \color{blue}{\frac{1}{\frac{a}{b}}} + -1\right)}^{2}\right)}^{0.25} \]

   6. un-div-inv 100.0%

      \[\leadsto {\left({\left(\color{blue}{\frac{\frac{b}{a}}{\frac{a}{b}}} + -1\right)}^{2}\right)}^{0.25} \]

7. Applied egg-rr 100.0%

   \[\leadsto {\left({\left(\color{blue}{\frac{\frac{b}{a}}{\frac{a}{b}}} + -1\right)}^{2}\right)}^{0.25} \]

8. Taylor expanded in b around 0 74.6%

   \[\leadsto {\color{blue}{\left(1 + -2 \cdot \frac{{b}^{2}}{{a}^{2}}\right)}}^{0.25} \]
9. Step-by-step derivation

   1. *-commutative 74.6%

      \[\leadsto {\left(1 + \color{blue}{\frac{{b}^{2}}{{a}^{2}} \cdot -2}\right)}^{0.25} \]

   2. unpow2 74.6%

      \[\leadsto {\left(1 + \frac{\color{blue}{b \cdot b}}{{a}^{2}} \cdot -2\right)}^{0.25} \]

   3. unpow2 74.6%

      \[\leadsto {\left(1 + \frac{b \cdot b}{\color{blue}{a \cdot a}} \cdot -2\right)}^{0.25} \]

10. Simplified 74.6%

    \[\leadsto {\color{blue}{\left(1 + \frac{b \cdot b}{a \cdot a} \cdot -2\right)}}^{0.25} \]

11. Taylor expanded in b around 0 98.2%

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

12. Final simplification 98.2%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023292 
(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)))))