Eccentricity of an ellipse

Percentage Accurate: 77.3% → 100.0%

Time: 4.8s

Alternatives: 6

Speedup: 3.2×

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

\[\begin{array}{l} \\ \sqrt{\left|\frac{\mathsf{fma}\left(-b, \frac{b}{a}, a\right)}{a}\right|} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.4%

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

3. Taylor expanded in b around 0

   \[\leadsto \sqrt{\left|\color{blue}{1 + -1 \cdot \frac{{b}^{2}}{{a}^{2}}}\right|} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \sqrt{\left|1 + \color{blue}{\left(\mathsf{neg}\left(\frac{{b}^{2}}{{a}^{2}}\right)\right)}\right|} \]

   2. unsub-neg N/A

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

   3. *-inverses N/A

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

   4. div-sub N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. div-sub N/A

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

   9. unpow2 N/A

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

   10. associate-/l* N/A

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

   11. *-inverses N/A

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

   12. *-rgt-identity N/A

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

   13. sub-neg N/A

       \[\leadsto \sqrt{\left|\frac{\color{blue}{a + \left(\mathsf{neg}\left(\frac{{b}^{2}}{a}\right)\right)}}{a}\right|} \]

   14. distribute-frac-neg N/A

       \[\leadsto \sqrt{\left|\frac{a + \color{blue}{\frac{\mathsf{neg}\left({b}^{2}\right)}{a}}}{a}\right|} \]

   15. mul-1-neg N/A

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

   16. +-commutative N/A

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

   17. mul-1-neg N/A

       \[\leadsto \sqrt{\left|\frac{\frac{\color{blue}{\mathsf{neg}\left({b}^{2}\right)}}{a} + a}{a}\right|} \]

   18. unpow2 N/A

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

   19. distribute-lft-neg-in N/A

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

   20. associate-/l* N/A

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

   21. lower-fma.f64 N/A

       \[\leadsto \sqrt{\left|\frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(b\right), \frac{b}{a}, a\right)}}{a}\right|} \]

   22. lower-neg.f64 N/A

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

   23. lower-/.f64 100.0

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

5. Applied rewrites 100.0%

   \[\leadsto \sqrt{\left|\color{blue}{\frac{\mathsf{fma}\left(-b, \frac{b}{a}, a\right)}{a}}\right|} \]
6. Add Preprocessing

Alternative 2: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot a - b \cdot b}{a \cdot a} \leq 0.9999999999991839:\\ \;\;\;\;\sqrt{\left|\left(b - a\right) \cdot \frac{a + b}{a \cdot a}\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|1\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (- (* a a) (* b b)) (* a a)) 0.9999999999991839)
   (sqrt (fabs (* (- b a) (/ (+ a b) (* a a)))))
   (sqrt (fabs 1.0))))

C

double code(double a, double b) {
	double tmp;
	if ((((a * a) - (b * b)) / (a * a)) <= 0.9999999999991839) {
		tmp = sqrt(fabs(((b - a) * ((a + b) / (a * a)))));
	} else {
		tmp = sqrt(fabs(1.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((((a * a) - (b * b)) / (a * a)) <= 0.9999999999991839d0) then
        tmp = sqrt(abs(((b - a) * ((a + b) / (a * a)))))
    else
        tmp = sqrt(abs(1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((((a * a) - (b * b)) / (a * a)) <= 0.9999999999991839) {
		tmp = Math.sqrt(Math.abs(((b - a) * ((a + b) / (a * a)))));
	} else {
		tmp = Math.sqrt(Math.abs(1.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (((a * a) - (b * b)) / (a * a)) <= 0.9999999999991839:
		tmp = math.sqrt(math.fabs(((b - a) * ((a + b) / (a * a)))))
	else:
		tmp = math.sqrt(math.fabs(1.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(Float64(Float64(a * a) - Float64(b * b)) / Float64(a * a)) <= 0.9999999999991839)
		tmp = sqrt(abs(Float64(Float64(b - a) * Float64(Float64(a + b) / Float64(a * a)))));
	else
		tmp = sqrt(abs(1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((((a * a) - (b * b)) / (a * a)) <= 0.9999999999991839)
		tmp = sqrt(abs(((b - a) * ((a + b) / (a * a)))));
	else
		tmp = sqrt(abs(1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision], 0.9999999999991839], N[Sqrt[N[Abs[N[(N[(b - a), $MachinePrecision] * N[(N[(a + b), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[1.0], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 a a) (*.f64 b b)) (*.f64 a a)) < 0.999999999999183875

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.5

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

   4. Applied rewrites 99.5%

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

   if 0.999999999999183875 < (/.f64 (-.f64 (*.f64 a a) (*.f64 b b)) (*.f64 a a))

   1. Initial program 82.8%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 99.4%

      \[\leadsto \sqrt{\left|\color{blue}{1}\right|} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot a - b \cdot b}{a \cdot a} \leq 0.9999999999991839:\\ \;\;\;\;\sqrt{\left|\left(b - a\right) \cdot \frac{a + b}{a \cdot a}\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|1\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot a - b \cdot b}{a \cdot a} \leq 0.9999999999991839:\\ \;\;\;\;\frac{\sqrt{\left|\left(b - a\right) \cdot \left(a + b\right)\right|}}{a}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|1\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (- (* a a) (* b b)) (* a a)) 0.9999999999991839)
   (/ (sqrt (fabs (* (- b a) (+ a b)))) a)
   (sqrt (fabs 1.0))))

C

double code(double a, double b) {
	double tmp;
	if ((((a * a) - (b * b)) / (a * a)) <= 0.9999999999991839) {
		tmp = sqrt(fabs(((b - a) * (a + b)))) / a;
	} else {
		tmp = sqrt(fabs(1.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((((a * a) - (b * b)) / (a * a)) <= 0.9999999999991839d0) then
        tmp = sqrt(abs(((b - a) * (a + b)))) / a
    else
        tmp = sqrt(abs(1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((((a * a) - (b * b)) / (a * a)) <= 0.9999999999991839) {
		tmp = Math.sqrt(Math.abs(((b - a) * (a + b)))) / a;
	} else {
		tmp = Math.sqrt(Math.abs(1.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (((a * a) - (b * b)) / (a * a)) <= 0.9999999999991839:
		tmp = math.sqrt(math.fabs(((b - a) * (a + b)))) / a
	else:
		tmp = math.sqrt(math.fabs(1.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(Float64(Float64(a * a) - Float64(b * b)) / Float64(a * a)) <= 0.9999999999991839)
		tmp = Float64(sqrt(abs(Float64(Float64(b - a) * Float64(a + b)))) / a);
	else
		tmp = sqrt(abs(1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((((a * a) - (b * b)) / (a * a)) <= 0.9999999999991839)
		tmp = sqrt(abs(((b - a) * (a + b)))) / a;
	else
		tmp = sqrt(abs(1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision], 0.9999999999991839], N[(N[Sqrt[N[Abs[N[(N[(b - a), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision], N[Sqrt[N[Abs[1.0], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 a a) (*.f64 b b)) (*.f64 a a)) < 0.999999999999183875

   1. Initial program 99.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. fabs-div N/A

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

      5. sqrt-div N/A

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

      6. pow1/2 N/A

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

      7. lift-*.f64 N/A

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

      8. fabs-sqr N/A

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

      9. pow2 N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 99.3%

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

   if 0.999999999999183875 < (/.f64 (-.f64 (*.f64 a a) (*.f64 b b)) (*.f64 a a))

   1. Initial program 82.8%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 99.4%

      \[\leadsto \sqrt{\left|\color{blue}{1}\right|} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot a - b \cdot b}{a \cdot a} \leq 0.9999999999991839:\\ \;\;\;\;\frac{\sqrt{\left|\left(b - a\right) \cdot \left(a + b\right)\right|}}{a}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|1\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0:\\ \;\;\;\;\sqrt{\left|1\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\frac{\left(a + b\right) \cdot \left(a - b\right)}{a \cdot a}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.0)
   (sqrt (fabs 1.0))
   (sqrt (fabs (/ (* (+ a b) (- a b)) (* a a))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.0) {
		tmp = sqrt(fabs(1.0));
	} else {
		tmp = sqrt(fabs((((a + b) * (a - b)) / (a * a))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 0.0d0) then
        tmp = sqrt(abs(1.0d0))
    else
        tmp = sqrt(abs((((a + b) * (a - b)) / (a * a))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.0) {
		tmp = Math.sqrt(Math.abs(1.0));
	} else {
		tmp = Math.sqrt(Math.abs((((a + b) * (a - b)) / (a * a))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 0.0:
		tmp = math.sqrt(math.fabs(1.0))
	else:
		tmp = math.sqrt(math.fabs((((a + b) * (a - b)) / (a * a))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.0)
		tmp = sqrt(abs(1.0));
	else
		tmp = sqrt(abs(Float64(Float64(Float64(a + b) * Float64(a - b)) / Float64(a * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 0.0)
		tmp = sqrt(abs(1.0));
	else
		tmp = sqrt(abs((((a + b) * (a - b)) / (a * a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.0], N[Sqrt[N[Abs[1.0], $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(N[(N[(a + b), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.0

   1. Initial program 76.1%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 99.2%

      \[\leadsto \sqrt{\left|\color{blue}{1}\right|} \]

   if 0.0 < (*.f64 b b)

   1. Initial program 100.0%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. fabs-div N/A

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

      4. lift-*.f64 N/A

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

      5. fabs-sqr N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-fabs.f64 100.0

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. difference-of-squares N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower--.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

      \[\leadsto \sqrt{\color{blue}{\frac{\left|\left(a - b\right) \cdot \left(b + a\right)\right|}{a \cdot a}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0:\\ \;\;\;\;\sqrt{\left|1\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\frac{\left(a + b\right) \cdot \left(a - b\right)}{a \cdot a}\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.4%

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

3. Taylor expanded in b around 0

   \[\leadsto \sqrt{\color{blue}{\left|\frac{{a}^{2} - {b}^{2}}{{a}^{2}}\right|}} \]
4. Step-by-step derivation

   1. div-sub N/A

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

   2. fabs-sub N/A

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

   3. lower-fabs.f64 N/A

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

   4. *-inverses N/A

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

   5. sub-neg N/A

      \[\leadsto \sqrt{\left|\color{blue}{\frac{{b}^{2}}{{a}^{2}} + \left(\mathsf{neg}\left(1\right)\right)}\right|} \]

   6. metadata-eval N/A

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

   7. unpow2 N/A

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

   8. associate-*l/ N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 100.0

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

5. Applied rewrites 100.0%

   \[\leadsto \sqrt{\color{blue}{\left|\mathsf{fma}\left(\frac{\frac{b}{a}}{a}, b, -1\right)\right|}} \]
6. Add Preprocessing

Alternative 6: 97.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left|1\right|} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (sqrt (fabs 1.0)))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b)
	return sqrt(abs(1.0))
end

MATLAB

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

Wolfram

code[a_, b_] := N[Sqrt[N[Abs[1.0], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 83.4%

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

3. Taylor expanded in b around 0

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

5. Applied rewrites 97.7%

   \[\leadsto \sqrt{\left|\color{blue}{1}\right|} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024271 
(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)))))