Eccentricity of an ellipse

Percentage Accurate: 77.3% → 100.0%

Time: 7.8s

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: 77.3% 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

\[\begin{array}{l} \\ {\left(\frac{a}{\left|a - \frac{b}{\frac{a}{b}}\right|}\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (pow (/ a (fabs (- a (/ b (/ a b))))) -0.5))

C

double code(double a, double b) {
	return pow((a / fabs((a - (b / (a / b))))), -0.5);
}

Fortran

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

Java

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

Python

def code(a, b):
	return math.pow((a / math.fabs((a - (b / (a / b))))), -0.5)

Julia

function code(a, b)
	return Float64(a / abs(Float64(a - Float64(b / Float64(a / b))))) ^ -0.5
end

MATLAB

function tmp = code(a, b)
	tmp = (a / abs((a - (b / (a / b))))) ^ -0.5;
end

Wolfram

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

Derivation

1. Initial program 84.0%

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

   1. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|\right)\right) \]

   2. div-sub N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left|\frac{a \cdot a}{a \cdot a} - \frac{b \cdot b}{a \cdot a}\right|\right)\right) \]

   3. fabs-sub N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left|\frac{b \cdot b}{a \cdot a} - \frac{a \cdot a}{a \cdot a}\right|\right)\right) \]

   4. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\left(\frac{b \cdot b}{a \cdot a} - \frac{a \cdot a}{a \cdot a}\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\left(\frac{b \cdot b}{a \cdot a} + \left(\mathsf{neg}\left(\frac{a \cdot a}{a \cdot a}\right)\right)\right)\right)\right) \]

   6. *-inverses N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\left(\frac{b \cdot b}{a \cdot a} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\left(\frac{b \cdot b}{a \cdot a} + -1\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(\frac{b \cdot b}{a \cdot a}\right), -1\right)\right)\right) \]

   9. associate-/r* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{b \cdot b}{a}}{a}\right), -1\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{b \cdot b}{a}\right), a\right), -1\right)\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(b \cdot b\right), a\right), a\right), -1\right)\right)\right) \]

   12. *-lowering-*.f64 99.5%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), a\right), a\right), -1\right)\right)\right) \]

3. Simplified 99.5%

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

   1. neg-fabs N/A

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

   2. +-commutative N/A

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

   3. distribute-neg-in N/A

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

   4. metadata-eval N/A

      \[\leadsto \sqrt{\left|1 + \left(\mathsf{neg}\left(\frac{\frac{b \cdot b}{a}}{a}\right)\right)\right|} \]

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. *-inverses N/A

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

   9. associate-/l/ N/A

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

   10. div-sub N/A

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

   11. associate-/r* N/A

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

   12. fabs-div N/A

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

   13. sqrt-div N/A

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

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left|\frac{a \cdot a - b \cdot b}{a}\right|}\right), \color{blue}{\left(\sqrt{\left|a\right|}\right)}\right) \]

6. Applied egg-rr 100.0%

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

7. Taylor expanded in b around 0

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

9. Simplified 100.0%

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

    1. clear-num N/A

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

    2. inv-pow N/A

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

    3. sqrt-pow1 N/A

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

    4. metadata-eval N/A

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

    5. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{pow.f64}\left(\left(\frac{a}{\left|a - \frac{b}{a} \cdot b\right|}\right), \color{blue}{\frac{-1}{2}}\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(a, \left(\left|a - \frac{b}{a} \cdot b\right|\right)\right), \frac{-1}{2}\right) \]

    7. fabs-lowering-fabs.f64 N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(a, \mathsf{fabs.f64}\left(\left(a - \frac{b}{a} \cdot b\right)\right)\right), \frac{-1}{2}\right) \]

    8. --lowering--.f64 N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(a, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(a, \left(\frac{b}{a} \cdot b\right)\right)\right)\right), \frac{-1}{2}\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(a, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(a, \left(b \cdot \frac{b}{a}\right)\right)\right)\right), \frac{-1}{2}\right) \]

    10. clear-num N/A

        \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(a, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(a, \left(b \cdot \frac{1}{\frac{a}{b}}\right)\right)\right)\right), \frac{-1}{2}\right) \]

    11. un-div-inv N/A

        \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(a, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(a, \left(\frac{b}{\frac{a}{b}}\right)\right)\right)\right), \frac{-1}{2}\right) \]

    12. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(a, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(a, \mathsf{/.f64}\left(b, \left(\frac{a}{b}\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

    13. /-lowering-/.f64 100.0%

        \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(a, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(a, \mathsf{/.f64}\left(b, \mathsf{/.f64}\left(a, b\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

11. Applied egg-rr 100.0%

    \[\leadsto \color{blue}{{\left(\frac{a}{\left|a - \frac{b}{\frac{a}{b}}\right|}\right)}^{-0.5}} \]
12. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

   1. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|\right)\right) \]

   2. div-sub N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left|\frac{a \cdot a}{a \cdot a} - \frac{b \cdot b}{a \cdot a}\right|\right)\right) \]

   3. fabs-sub N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left|\frac{b \cdot b}{a \cdot a} - \frac{a \cdot a}{a \cdot a}\right|\right)\right) \]

   4. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\left(\frac{b \cdot b}{a \cdot a} - \frac{a \cdot a}{a \cdot a}\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\left(\frac{b \cdot b}{a \cdot a} + \left(\mathsf{neg}\left(\frac{a \cdot a}{a \cdot a}\right)\right)\right)\right)\right) \]

   6. *-inverses N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\left(\frac{b \cdot b}{a \cdot a} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\left(\frac{b \cdot b}{a \cdot a} + -1\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(\frac{b \cdot b}{a \cdot a}\right), -1\right)\right)\right) \]

   9. associate-/r* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{b \cdot b}{a}}{a}\right), -1\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{b \cdot b}{a}\right), a\right), -1\right)\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(b \cdot b\right), a\right), a\right), -1\right)\right)\right) \]

   12. *-lowering-*.f64 99.5%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), a\right), a\right), -1\right)\right)\right) \]

3. Simplified 99.5%

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

   1. neg-fabs N/A

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

   2. +-commutative N/A

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

   3. distribute-neg-in N/A

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

   4. metadata-eval N/A

      \[\leadsto \sqrt{\left|1 + \left(\mathsf{neg}\left(\frac{\frac{b \cdot b}{a}}{a}\right)\right)\right|} \]

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. *-inverses N/A

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

   9. associate-/l/ N/A

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

   10. div-sub N/A

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

   11. associate-/r* N/A

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

   12. fabs-div N/A

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

   13. sqrt-div N/A

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

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left|\frac{a \cdot a - b \cdot b}{a}\right|}\right), \color{blue}{\left(\sqrt{\left|a\right|}\right)}\right) \]

6. Applied egg-rr 100.0%

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

7. Taylor expanded in b around 0

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

9. Simplified 100.0%

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

10. Final simplification 100.0%

    \[\leadsto \sqrt{\frac{\left|a - b \cdot \frac{b}{a}\right|}{a}} \]
11. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	return sqrt(fabs((((b / (a / b)) / a) + -1.0)));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

   1. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|\right)\right) \]

   2. div-sub N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left|\frac{a \cdot a}{a \cdot a} - \frac{b \cdot b}{a \cdot a}\right|\right)\right) \]

   3. fabs-sub N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left|\frac{b \cdot b}{a \cdot a} - \frac{a \cdot a}{a \cdot a}\right|\right)\right) \]

   4. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\left(\frac{b \cdot b}{a \cdot a} - \frac{a \cdot a}{a \cdot a}\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\left(\frac{b \cdot b}{a \cdot a} + \left(\mathsf{neg}\left(\frac{a \cdot a}{a \cdot a}\right)\right)\right)\right)\right) \]

   6. *-inverses N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\left(\frac{b \cdot b}{a \cdot a} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\left(\frac{b \cdot b}{a \cdot a} + -1\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(\frac{b \cdot b}{a \cdot a}\right), -1\right)\right)\right) \]

   9. associate-/r* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{b \cdot b}{a}}{a}\right), -1\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{b \cdot b}{a}\right), a\right), -1\right)\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(b \cdot b\right), a\right), a\right), -1\right)\right)\right) \]

   12. *-lowering-*.f64 99.5%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), a\right), a\right), -1\right)\right)\right) \]

3. Simplified 99.5%

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

   1. associate-/l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(b \cdot \frac{b}{a}\right), a\right), -1\right)\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(b \cdot \frac{1}{\frac{a}{b}}\right), a\right), -1\right)\right)\right) \]

   3. un-div-inv N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{b}{\frac{a}{b}}\right), a\right), -1\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(b, \left(\frac{a}{b}\right)\right), a\right), -1\right)\right)\right) \]

   5. /-lowering-/.f64 100.0%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(b, \mathsf{/.f64}\left(a, b\right)\right), a\right), -1\right)\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \sqrt{\left|\frac{\color{blue}{\frac{b}{\frac{a}{b}}}}{a} + -1\right|} \]
7. 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 84.0%

   \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
2. Add Preprocessing

3. Taylor expanded in a around inf

   \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\color{blue}{1}\right)\right) \]
4. Step-by-step derivation

5. Simplified 97.5%

   \[\leadsto \sqrt{\left|\color{blue}{1}\right|} \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \sqrt{1} \]

   2. metadata-eval 97.5%

      \[\leadsto 1 \]

7. Applied egg-rr 97.5%

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

Reproduce

herbie shell --seed 2024162 
(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)))))