Eccentricity of an ellipse

Percentage Accurate: 75.7% → 100.0%

Time: 8.4s

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: 75.7% 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

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

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

Derivation

1. Initial program 72.6%

   \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
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. /-lowering-/.f64 N/A

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

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

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

   11. *-lowering-*.f64 72.6%

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

3. Simplified 72.6%

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. sub-neg N/A

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

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

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

   5. associate-/r* N/A

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

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

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

   7. associate-/l* N/A

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

   8. clear-num N/A

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

   9. un-div-inv N/A

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

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

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

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

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

   12. metadata-eval 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%

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

7. Final simplification 100.0%

   \[\leadsto \sqrt{\left|1 - \frac{\frac{b}{\frac{a}{b}}}{a}\right|} \]
8. Add Preprocessing

Alternative 2: 99.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 - \frac{b}{a \cdot \frac{a}{b}}\\ {\left(t\_0 \cdot t\_0\right)}^{-0.25} \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (- -1.0 (/ b (* a (/ a b)))))) (pow (* t_0 t_0) -0.25)))

C

double code(double a, double b) {
	double t_0 = -1.0 - (b / (a * (a / b)));
	return pow((t_0 * t_0), -0.25);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    t_0 = (-1.0d0) - (b / (a * (a / b)))
    code = (t_0 * t_0) ** (-0.25d0)
end function

Java

public static double code(double a, double b) {
	double t_0 = -1.0 - (b / (a * (a / b)));
	return Math.pow((t_0 * t_0), -0.25);
}

Python

def code(a, b):
	t_0 = -1.0 - (b / (a * (a / b)))
	return math.pow((t_0 * t_0), -0.25)

Julia

function code(a, b)
	t_0 = Float64(-1.0 - Float64(b / Float64(a * Float64(a / b))))
	return Float64(t_0 * t_0) ^ -0.25
end

MATLAB

function tmp = code(a, b)
	t_0 = -1.0 - (b / (a * (a / b)));
	tmp = (t_0 * t_0) ^ -0.25;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(-1.0 - N[(b / N[(a * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], -0.25], $MachinePrecision]]

Derivation

1. Initial program 72.6%

   \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
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. /-lowering-/.f64 N/A

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

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

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

   11. *-lowering-*.f64 72.6%

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

3. Simplified 72.6%

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

   1. +-commutative N/A

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

   2. flip-+ N/A

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

   3. *-rgt-identity N/A

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

   4. metadata-eval N/A

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

   5. swap-sqr N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

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

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

6. Applied egg-rr 88.7%

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

7. Taylor expanded in b around 0

   \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(b, \mathsf{/.f64}\left(a, b\right)\right), a\right)\right)\right)\right)\right) \]
8. Step-by-step derivation

9. Simplified 99.0%

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

    1. pow1/2 N/A

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

    2. sqr-pow N/A

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

    3. pow-prod-down N/A

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

11. Applied egg-rr 99.0%

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

    1. frac-times N/A

       \[\leadsto {\left(\frac{1 \cdot 1}{\left(-1 - \frac{b}{\frac{a}{\frac{b}{a}}}\right) \cdot \left(-1 - \frac{b}{\frac{a}{\frac{b}{a}}}\right)}\right)}^{\frac{1}{4}} \]

    2. metadata-eval N/A

       \[\leadsto {\left(\frac{1}{\left(-1 - \frac{b}{\frac{a}{\frac{b}{a}}}\right) \cdot \left(-1 - \frac{b}{\frac{a}{\frac{b}{a}}}\right)}\right)}^{\frac{1}{4}} \]

    3. inv-pow N/A

       \[\leadsto {\left({\left(\left(-1 - \frac{b}{\frac{a}{\frac{b}{a}}}\right) \cdot \left(-1 - \frac{b}{\frac{a}{\frac{b}{a}}}\right)\right)}^{-1}\right)}^{\frac{1}{4}} \]

    4. pow-pow N/A

       \[\leadsto {\left(\left(-1 - \frac{b}{\frac{a}{\frac{b}{a}}}\right) \cdot \left(-1 - \frac{b}{\frac{a}{\frac{b}{a}}}\right)\right)}^{\color{blue}{\left(-1 \cdot \frac{1}{4}\right)}} \]

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

       \[\leadsto \mathsf{pow.f64}\left(\left(\left(-1 - \frac{b}{\frac{a}{\frac{b}{a}}}\right) \cdot \left(-1 - \frac{b}{\frac{a}{\frac{b}{a}}}\right)\right), \color{blue}{\left(-1 \cdot \frac{1}{4}\right)}\right) \]

13. Applied egg-rr 99.0%

    \[\leadsto \color{blue}{{\left(\left(-1 - \frac{b}{a \cdot \frac{a}{b}}\right) \cdot \left(-1 - \frac{b}{a \cdot \frac{a}{b}}\right)\right)}^{-0.25}} \]
14. Add Preprocessing

Alternative 3: 98.5% accurate, 19.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.6%

   \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
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. /-lowering-/.f64 N/A

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

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

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

   11. *-lowering-*.f64 72.6%

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

3. Simplified 72.6%

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

   1. +-commutative N/A

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

   2. flip-+ N/A

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

   3. *-rgt-identity N/A

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

   4. metadata-eval N/A

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

   5. swap-sqr N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

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

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

6. Applied egg-rr 88.7%

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

7. Taylor expanded in b around 0

   \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(b, \mathsf{/.f64}\left(a, b\right)\right), a\right)\right)\right)\right)\right) \]
8. Step-by-step derivation

9. Simplified 99.0%

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

    1. pow1/2 N/A

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

    2. sqr-pow N/A

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

    3. pow-prod-down N/A

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

11. Applied egg-rr 99.0%

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

12. Taylor expanded in b around 0

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

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

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

    2. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{{b}^{2}}{{a}^{2}}\right)}\right)\right) \]

    3. unpow2 N/A

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

    4. associate-/r* N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{\frac{{b}^{2}}{a}}{\color{blue}{a}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{{b}^{2}}{a}\right), \color{blue}{a}\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({b}^{2}\right), a\right), a\right)\right)\right) \]

    7. unpow2 N/A

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

    8. *-lowering-*.f64 98.4%

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

14. Simplified 98.4%

    \[\leadsto \color{blue}{1 + -0.5 \cdot \frac{\frac{b \cdot b}{a}}{a}} \]
15. 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 72.6%

   \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
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. /-lowering-/.f64 N/A

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

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

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

   11. *-lowering-*.f64 72.6%

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

3. Simplified 72.6%

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

5. Taylor expanded in b around 0

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

7. Simplified 97.8%

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

   1. metadata-eval N/A

      \[\leadsto \sqrt{1} \]

   2. metadata-eval 97.8%

      \[\leadsto 1 \]

9. Applied egg-rr 97.8%

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

Reproduce

herbie shell --seed 2024139 
(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)))))