Eccentricity of an ellipse

Percentage Accurate: 78.0% → 100.0%

Time: 8.6s

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: 78.0% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.7%

   \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
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. associate-/l* N/A

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

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

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

   13. /-lowering-/.f64 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 98.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.7%

   \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
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. associate-/l* N/A

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

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

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

   13. /-lowering-/.f64 100.0%

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

3. Simplified 100.0%

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

6. Applied egg-rr 72.9%

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

7. Taylor expanded in b around 0

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

9. Simplified 73.8%

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

    1. pow1/2 N/A

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

    2. sqr-pow N/A

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

    3. pow-prod-down N/A

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

11. Applied egg-rr 73.8%

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

    1. frac-times N/A

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

    2. metadata-eval N/A

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

    3. inv-pow N/A

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

    4. pow-pow N/A

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

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

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

13. Applied egg-rr 98.2%

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

Alternative 3: 98.5% accurate, 19.2× speedup

Math

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

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(Float64(-0.5 * 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[(N[(-0.5 * N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.7%

   \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
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. associate-/l* N/A

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

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

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

   13. /-lowering-/.f64 100.0%

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

3. Simplified 100.0%

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

6. Applied egg-rr 72.9%

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

7. Taylor expanded in b around 0

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

9. Simplified 73.8%

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

    1. pow1/2 N/A

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

    2. sqr-pow N/A

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

    3. pow-prod-down N/A

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

11. Applied egg-rr 73.8%

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

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. associate-*r/ N/A

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

    3. unpow2 N/A

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

    4. associate-/r* N/A

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

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

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

    6. associate-/l* N/A

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

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

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

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

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

    9. unpow2 N/A

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

    10. *-lowering-*.f64 97.9%

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

14. Simplified 97.9%

    \[\leadsto \color{blue}{1 + \frac{-0.5 \cdot \frac{b \cdot b}{a}}{a}} \]
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 73.7%

   \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
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. associate-/l* N/A

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

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

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

   13. /-lowering-/.f64 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\sqrt{\left|\frac{b \cdot \frac{b}{a}}{a} + -1\right|}} \]
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 97.3%

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

   1. metadata-eval N/A

      \[\leadsto \sqrt{1} \]

   2. metadata-eval 97.3%

      \[\leadsto 1 \]

9. Applied egg-rr 97.3%

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

Reproduce

herbie shell --seed 2024155 
(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)))))