Optimisation.CirclePacking:place from circle-packing-0.1.0.4, C

Percentage Accurate: 100.0% → 100.0%

Time: 3.1s

Alternatives: 5

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \sqrt{\left|x - y\right|} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (sqrt (fabs (- x y))))

C

double code(double x, double y) {
	return sqrt(fabs((x - y)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(abs((x - y)))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt(Math.abs((x - y)));
}

Python

def code(x, y):
	return math.sqrt(math.fabs((x - y)))

Julia

function code(x, y)
	return sqrt(abs(Float64(x - y)))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt(abs((x - y)));
end

Wolfram

code[x_, y_] := N[Sqrt[N[Abs[N[(x - y), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left|x - y\right|} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (sqrt (fabs (- x y))))

C

double code(double x, double y) {
	return sqrt(fabs((x - y)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(abs((x - y)))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt(Math.abs((x - y)));
}

Python

def code(x, y):
	return math.sqrt(math.fabs((x - y)))

Julia

function code(x, y)
	return sqrt(abs(Float64(x - y)))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt(abs((x - y)));
end

Wolfram

code[x_, y_] := N[Sqrt[N[Abs[N[(x - y), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left|x - y\right|} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (sqrt (fabs (- x y))))

C

double code(double x, double y) {
	return sqrt(fabs((x - y)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(abs((x - y)))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt(Math.abs((x - y)));
}

Python

def code(x, y):
	return math.sqrt(math.fabs((x - y)))

Julia

function code(x, y)
	return sqrt(abs(Float64(x - y)))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt(abs((x - y)));
end

Wolfram

code[x_, y_] := N[Sqrt[N[Abs[N[(x - y), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\sqrt{\left|x - y\right|} \]

2. Final simplification 100.0%

   \[\leadsto \sqrt{\left|x - y\right|} \]

Alternative 2: 75.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - y \leq -2 \cdot 10^{-271}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- x y) -2e-271) (sqrt y) (sqrt (- x y))))

C

double code(double x, double y) {
	double tmp;
	if ((x - y) <= -2e-271) {
		tmp = sqrt(y);
	} else {
		tmp = sqrt((x - y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x - y) <= (-2d-271)) then
        tmp = sqrt(y)
    else
        tmp = sqrt((x - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x - y) <= -2e-271) {
		tmp = Math.sqrt(y);
	} else {
		tmp = Math.sqrt((x - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x - y) <= -2e-271:
		tmp = math.sqrt(y)
	else:
		tmp = math.sqrt((x - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x - y) <= -2e-271)
		tmp = sqrt(y);
	else
		tmp = sqrt(Float64(x - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x - y) <= -2e-271)
		tmp = sqrt(y);
	else
		tmp = sqrt((x - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x - y), $MachinePrecision], -2e-271], N[Sqrt[y], $MachinePrecision], N[Sqrt[N[(x - y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x y) < -1.99999999999999993e-271

   1. Initial program 100.0%

      \[\sqrt{\left|x - y\right|} \]
   2. Step-by-step derivation

      1. add-cbrt-cube 76.3%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\left|x - y\right|} \cdot \sqrt{\left|x - y\right|}\right) \cdot \sqrt{\left|x - y\right|}}} \]

      2. pow1/3 71.2%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\left|x - y\right|} \cdot \sqrt{\left|x - y\right|}\right) \cdot \sqrt{\left|x - y\right|}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 71.2%

         \[\leadsto {\left(\color{blue}{\left|x - y\right|} \cdot \sqrt{\left|x - y\right|}\right)}^{0.3333333333333333} \]

      4. pow1 71.2%

         \[\leadsto {\left(\color{blue}{{\left(\left|x - y\right|\right)}^{1}} \cdot \sqrt{\left|x - y\right|}\right)}^{0.3333333333333333} \]

      5. pow1/2 71.2%

         \[\leadsto {\left({\left(\left|x - y\right|\right)}^{1} \cdot \color{blue}{{\left(\left|x - y\right|\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 71.2%

         \[\leadsto {\color{blue}{\left({\left(\left|x - y\right|\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. add-sqr-sqrt 0.0%

         \[\leadsto {\left({\left(\left|\color{blue}{\sqrt{x - y} \cdot \sqrt{x - y}}\right|\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. fabs-sqr 0.0%

         \[\leadsto {\left({\color{blue}{\left(\sqrt{x - y} \cdot \sqrt{x - y}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      9. add-sqr-sqrt 0.0%

         \[\leadsto {\left({\color{blue}{\left(x - y\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      10. metadata-eval 0.0%

          \[\leadsto {\left({\left(x - y\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   3. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{{\left({\left(x - y\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   4. Step-by-step derivation

      1. unpow1/3 0.0%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(x - y\right)}^{1.5}}} \]

   5. Simplified 0.0%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(x - y\right)}^{1.5}}} \]
   6. Step-by-step derivation

      1. pow1/3 0.0%

         \[\leadsto \color{blue}{{\left({\left(x - y\right)}^{1.5}\right)}^{0.3333333333333333}} \]

      2. pow-pow 0.0%

         \[\leadsto \color{blue}{{\left(x - y\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      3. metadata-eval 0.0%

         \[\leadsto {\left(x - y\right)}^{\color{blue}{0.5}} \]

      4. pow1/2 0.0%

         \[\leadsto \color{blue}{\sqrt{x - y}} \]

      5. flip3-- 0.1%

         \[\leadsto \sqrt{\color{blue}{\frac{{x}^{3} - {y}^{3}}{x \cdot x + \left(y \cdot y + x \cdot y\right)}}} \]

      6. sqrt-div 0.1%

         \[\leadsto \color{blue}{\frac{\sqrt{{x}^{3} - {y}^{3}}}{\sqrt{x \cdot x + \left(y \cdot y + x \cdot y\right)}}} \]

      7. distribute-rgt-in 0.1%

         \[\leadsto \frac{\sqrt{{x}^{3} - {y}^{3}}}{\sqrt{x \cdot x + \color{blue}{y \cdot \left(y + x\right)}}} \]

      8. +-commutative 0.1%

         \[\leadsto \frac{\sqrt{{x}^{3} - {y}^{3}}}{\sqrt{\color{blue}{y \cdot \left(y + x\right) + x \cdot x}}} \]

      9. fma-udef 0.1%

         \[\leadsto \frac{\sqrt{{x}^{3} - {y}^{3}}}{\sqrt{\color{blue}{\mathsf{fma}\left(y, y + x, x \cdot x\right)}}} \]

      10. clear-num 0.1%

          \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(y, y + x, x \cdot x\right)}}{\sqrt{{x}^{3} - {y}^{3}}}}} \]

      11. clear-num 0.1%

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\sqrt{{x}^{3} - {y}^{3}}}{\sqrt{\mathsf{fma}\left(y, y + x, x \cdot x\right)}}}}} \]

      12. fma-udef 0.1%

          \[\leadsto \frac{1}{\frac{1}{\frac{\sqrt{{x}^{3} - {y}^{3}}}{\sqrt{\color{blue}{y \cdot \left(y + x\right) + x \cdot x}}}}} \]

      13. +-commutative 0.1%

          \[\leadsto \frac{1}{\frac{1}{\frac{\sqrt{{x}^{3} - {y}^{3}}}{\sqrt{\color{blue}{x \cdot x + y \cdot \left(y + x\right)}}}}} \]

      14. distribute-rgt-in 0.1%

          \[\leadsto \frac{1}{\frac{1}{\frac{\sqrt{{x}^{3} - {y}^{3}}}{\sqrt{x \cdot x + \color{blue}{\left(y \cdot y + x \cdot y\right)}}}}} \]

   7. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\sqrt{x - y}}}} \]
   8. Step-by-step derivation

      1. add-log-exp 0.0%

         \[\leadsto \color{blue}{\log \left(e^{\frac{1}{\frac{1}{\sqrt{x - y}}}}\right)} \]

      2. remove-double-div 0.0%

         \[\leadsto \log \left(e^{\color{blue}{\sqrt{x - y}}}\right) \]

      3. *-un-lft-identity 0.0%

         \[\leadsto \log \color{blue}{\left(1 \cdot e^{\sqrt{x - y}}\right)} \]

      4. log-prod 0.0%

         \[\leadsto \color{blue}{\log 1 + \log \left(e^{\sqrt{x - y}}\right)} \]

      5. metadata-eval 0.0%

         \[\leadsto \color{blue}{0} + \log \left(e^{\sqrt{x - y}}\right) \]

      6. add-log-exp 0.0%

         \[\leadsto 0 + \color{blue}{\sqrt{x - y}} \]

      7. sub-neg 0.0%

         \[\leadsto 0 + \sqrt{\color{blue}{x + \left(-y\right)}} \]

      8. add-sqr-sqrt 0.0%

         \[\leadsto 0 + \sqrt{x + \color{blue}{\sqrt{-y} \cdot \sqrt{-y}}} \]

      9. sqrt-unprod 25.4%

         \[\leadsto 0 + \sqrt{x + \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}} \]

      10. sqr-neg 25.4%

          \[\leadsto 0 + \sqrt{x + \sqrt{\color{blue}{y \cdot y}}} \]

      11. sqrt-unprod 46.9%

          \[\leadsto 0 + \sqrt{x + \color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

      12. add-sqr-sqrt 46.9%

          \[\leadsto 0 + \sqrt{x + \color{blue}{y}} \]

   9. Applied egg-rr 46.9%

      \[\leadsto \color{blue}{0 + \sqrt{x + y}} \]
   10. Step-by-step derivation

       1. +-lft-identity 46.9%

          \[\leadsto \color{blue}{\sqrt{x + y}} \]

       2. +-commutative 46.9%

          \[\leadsto \sqrt{\color{blue}{y + x}} \]

   11. Simplified 46.9%

       \[\leadsto \color{blue}{\sqrt{y + x}} \]

   12. Taylor expanded in x around 0 48.4%

       \[\leadsto \color{blue}{\sqrt{y}} \]

   if -1.99999999999999993e-271 < (-.f64 x y)

   1. Initial program 100.0%

      \[\sqrt{\left|x - y\right|} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 94.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left|x - y\right|}\right)\right)} \]

      2. expm1-udef 66.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\left|x - y\right|}\right)} - 1} \]

      3. log1p-udef 66.3%

         \[\leadsto e^{\color{blue}{\log \left(1 + \sqrt{\left|x - y\right|}\right)}} - 1 \]

      4. add-exp-log 71.4%

         \[\leadsto \color{blue}{\left(1 + \sqrt{\left|x - y\right|}\right)} - 1 \]

      5. add-sqr-sqrt 71.4%

         \[\leadsto \left(1 + \sqrt{\left|\color{blue}{\sqrt{x - y} \cdot \sqrt{x - y}}\right|}\right) - 1 \]

      6. fabs-sqr 71.4%

         \[\leadsto \left(1 + \sqrt{\color{blue}{\sqrt{x - y} \cdot \sqrt{x - y}}}\right) - 1 \]

      7. add-sqr-sqrt 71.4%

         \[\leadsto \left(1 + \sqrt{\color{blue}{x - y}}\right) - 1 \]

   3. Applied egg-rr 71.4%

      \[\leadsto \color{blue}{\left(1 + \sqrt{x - y}\right) - 1} \]
   4. Step-by-step derivation

      1. +-commutative 71.4%

         \[\leadsto \color{blue}{\left(\sqrt{x - y} + 1\right)} - 1 \]

      2. associate--l+ 99.2%

         \[\leadsto \color{blue}{\sqrt{x - y} + \left(1 - 1\right)} \]

      3. metadata-eval 99.2%

         \[\leadsto \sqrt{x - y} + \color{blue}{0} \]

      4. +-rgt-identity 99.2%

         \[\leadsto \color{blue}{\sqrt{x - y}} \]

   5. Simplified 99.2%

      \[\leadsto \color{blue}{\sqrt{x - y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - y \leq -2 \cdot 10^{-271}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x - y}\\ \end{array} \]

Alternative 3: 41.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.2e-167) (sqrt x) (sqrt y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.2e-167) {
		tmp = sqrt(x);
	} else {
		tmp = sqrt(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.2d-167) then
        tmp = sqrt(x)
    else
        tmp = sqrt(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.2e-167) {
		tmp = Math.sqrt(x);
	} else {
		tmp = Math.sqrt(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.2e-167:
		tmp = math.sqrt(x)
	else:
		tmp = math.sqrt(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.2e-167)
		tmp = sqrt(x);
	else
		tmp = sqrt(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.2e-167)
		tmp = sqrt(x);
	else
		tmp = sqrt(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.2e-167], N[Sqrt[x], $MachinePrecision], N[Sqrt[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.2e-167

   1. Initial program 100.0%

      \[\sqrt{\left|x - y\right|} \]
   2. Step-by-step derivation

      1. add-cbrt-cube 72.2%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\left|x - y\right|} \cdot \sqrt{\left|x - y\right|}\right) \cdot \sqrt{\left|x - y\right|}}} \]

      2. pow1/3 67.5%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\left|x - y\right|} \cdot \sqrt{\left|x - y\right|}\right) \cdot \sqrt{\left|x - y\right|}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 67.5%

         \[\leadsto {\left(\color{blue}{\left|x - y\right|} \cdot \sqrt{\left|x - y\right|}\right)}^{0.3333333333333333} \]

      4. pow1 67.5%

         \[\leadsto {\left(\color{blue}{{\left(\left|x - y\right|\right)}^{1}} \cdot \sqrt{\left|x - y\right|}\right)}^{0.3333333333333333} \]

      5. pow1/2 67.5%

         \[\leadsto {\left({\left(\left|x - y\right|\right)}^{1} \cdot \color{blue}{{\left(\left|x - y\right|\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 67.5%

         \[\leadsto {\color{blue}{\left({\left(\left|x - y\right|\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. add-sqr-sqrt 45.2%

         \[\leadsto {\left({\left(\left|\color{blue}{\sqrt{x - y} \cdot \sqrt{x - y}}\right|\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. fabs-sqr 45.2%

         \[\leadsto {\left({\color{blue}{\left(\sqrt{x - y} \cdot \sqrt{x - y}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      9. add-sqr-sqrt 45.2%

         \[\leadsto {\left({\color{blue}{\left(x - y\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      10. metadata-eval 45.2%

          \[\leadsto {\left({\left(x - y\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   3. Applied egg-rr 45.2%

      \[\leadsto \color{blue}{{\left({\left(x - y\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   4. Step-by-step derivation

      1. unpow1/3 48.3%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(x - y\right)}^{1.5}}} \]

   5. Simplified 48.3%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(x - y\right)}^{1.5}}} \]

   6. Taylor expanded in y around 0 30.5%

      \[\leadsto \color{blue}{\sqrt{x}} \]

   if 2.2e-167 < y

   1. Initial program 100.0%

      \[\sqrt{\left|x - y\right|} \]
   2. Step-by-step derivation

      1. add-cbrt-cube 70.0%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\left|x - y\right|} \cdot \sqrt{\left|x - y\right|}\right) \cdot \sqrt{\left|x - y\right|}}} \]

      2. pow1/3 65.1%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\left|x - y\right|} \cdot \sqrt{\left|x - y\right|}\right) \cdot \sqrt{\left|x - y\right|}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 65.1%

         \[\leadsto {\left(\color{blue}{\left|x - y\right|} \cdot \sqrt{\left|x - y\right|}\right)}^{0.3333333333333333} \]

      4. pow1 65.1%

         \[\leadsto {\left(\color{blue}{{\left(\left|x - y\right|\right)}^{1}} \cdot \sqrt{\left|x - y\right|}\right)}^{0.3333333333333333} \]

      5. pow1/2 65.1%

         \[\leadsto {\left({\left(\left|x - y\right|\right)}^{1} \cdot \color{blue}{{\left(\left|x - y\right|\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 65.1%

         \[\leadsto {\color{blue}{\left({\left(\left|x - y\right|\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. add-sqr-sqrt 7.3%

         \[\leadsto {\left({\left(\left|\color{blue}{\sqrt{x - y} \cdot \sqrt{x - y}}\right|\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. fabs-sqr 7.3%

         \[\leadsto {\left({\color{blue}{\left(\sqrt{x - y} \cdot \sqrt{x - y}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      9. add-sqr-sqrt 7.3%

         \[\leadsto {\left({\color{blue}{\left(x - y\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      10. metadata-eval 7.3%

          \[\leadsto {\left({\left(x - y\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   3. Applied egg-rr 7.3%

      \[\leadsto \color{blue}{{\left({\left(x - y\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   4. Step-by-step derivation

      1. unpow1/3 7.9%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(x - y\right)}^{1.5}}} \]

   5. Simplified 7.9%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(x - y\right)}^{1.5}}} \]
   6. Step-by-step derivation

      1. pow1/3 7.3%

         \[\leadsto \color{blue}{{\left({\left(x - y\right)}^{1.5}\right)}^{0.3333333333333333}} \]

      2. pow-pow 14.1%

         \[\leadsto \color{blue}{{\left(x - y\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      3. metadata-eval 14.1%

         \[\leadsto {\left(x - y\right)}^{\color{blue}{0.5}} \]

      4. pow1/2 14.1%

         \[\leadsto \color{blue}{\sqrt{x - y}} \]

      5. flip3-- 4.1%

         \[\leadsto \sqrt{\color{blue}{\frac{{x}^{3} - {y}^{3}}{x \cdot x + \left(y \cdot y + x \cdot y\right)}}} \]

      6. sqrt-div 4.0%

         \[\leadsto \color{blue}{\frac{\sqrt{{x}^{3} - {y}^{3}}}{\sqrt{x \cdot x + \left(y \cdot y + x \cdot y\right)}}} \]

      7. distribute-rgt-in 4.0%

         \[\leadsto \frac{\sqrt{{x}^{3} - {y}^{3}}}{\sqrt{x \cdot x + \color{blue}{y \cdot \left(y + x\right)}}} \]

      8. +-commutative 4.0%

         \[\leadsto \frac{\sqrt{{x}^{3} - {y}^{3}}}{\sqrt{\color{blue}{y \cdot \left(y + x\right) + x \cdot x}}} \]

      9. fma-udef 4.0%

         \[\leadsto \frac{\sqrt{{x}^{3} - {y}^{3}}}{\sqrt{\color{blue}{\mathsf{fma}\left(y, y + x, x \cdot x\right)}}} \]

      10. clear-num 4.1%

          \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(y, y + x, x \cdot x\right)}}{\sqrt{{x}^{3} - {y}^{3}}}}} \]

      11. clear-num 4.0%

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\sqrt{{x}^{3} - {y}^{3}}}{\sqrt{\mathsf{fma}\left(y, y + x, x \cdot x\right)}}}}} \]

      12. fma-udef 4.0%

          \[\leadsto \frac{1}{\frac{1}{\frac{\sqrt{{x}^{3} - {y}^{3}}}{\sqrt{\color{blue}{y \cdot \left(y + x\right) + x \cdot x}}}}} \]

      13. +-commutative 4.0%

          \[\leadsto \frac{1}{\frac{1}{\frac{\sqrt{{x}^{3} - {y}^{3}}}{\sqrt{\color{blue}{x \cdot x + y \cdot \left(y + x\right)}}}}} \]

      14. distribute-rgt-in 4.0%

          \[\leadsto \frac{1}{\frac{1}{\frac{\sqrt{{x}^{3} - {y}^{3}}}{\sqrt{x \cdot x + \color{blue}{\left(y \cdot y + x \cdot y\right)}}}}} \]

   7. Applied egg-rr 14.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\sqrt{x - y}}}} \]
   8. Step-by-step derivation

      1. add-log-exp 0.6%

         \[\leadsto \color{blue}{\log \left(e^{\frac{1}{\frac{1}{\sqrt{x - y}}}}\right)} \]

      2. remove-double-div 0.6%

         \[\leadsto \log \left(e^{\color{blue}{\sqrt{x - y}}}\right) \]

      3. *-un-lft-identity 0.6%

         \[\leadsto \log \color{blue}{\left(1 \cdot e^{\sqrt{x - y}}\right)} \]

      4. log-prod 0.6%

         \[\leadsto \color{blue}{\log 1 + \log \left(e^{\sqrt{x - y}}\right)} \]

      5. metadata-eval 0.6%

         \[\leadsto \color{blue}{0} + \log \left(e^{\sqrt{x - y}}\right) \]

      6. add-log-exp 14.1%

         \[\leadsto 0 + \color{blue}{\sqrt{x - y}} \]

      7. sub-neg 14.1%

         \[\leadsto 0 + \sqrt{\color{blue}{x + \left(-y\right)}} \]

      8. add-sqr-sqrt 0.0%

         \[\leadsto 0 + \sqrt{x + \color{blue}{\sqrt{-y} \cdot \sqrt{-y}}} \]

      9. sqrt-unprod 51.1%

         \[\leadsto 0 + \sqrt{x + \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}} \]

      10. sqr-neg 51.1%

          \[\leadsto 0 + \sqrt{x + \sqrt{\color{blue}{y \cdot y}}} \]

      11. sqrt-unprod 81.9%

          \[\leadsto 0 + \sqrt{x + \color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

      12. add-sqr-sqrt 81.9%

          \[\leadsto 0 + \sqrt{x + \color{blue}{y}} \]

   9. Applied egg-rr 81.9%

      \[\leadsto \color{blue}{0 + \sqrt{x + y}} \]
   10. Step-by-step derivation

       1. +-lft-identity 81.9%

          \[\leadsto \color{blue}{\sqrt{x + y}} \]

       2. +-commutative 81.9%

          \[\leadsto \sqrt{\color{blue}{y + x}} \]

   11. Simplified 81.9%

       \[\leadsto \color{blue}{\sqrt{y + x}} \]

   12. Taylor expanded in x around 0 70.0%

       \[\leadsto \color{blue}{\sqrt{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \]

Alternative 4: 48.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{x + y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (sqrt (+ x y)))

C

double code(double x, double y) {
	return sqrt((x + y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt((x + y))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt((x + y));
}

Python

def code(x, y):
	return math.sqrt((x + y))

Julia

function code(x, y)
	return sqrt(Float64(x + y))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt((x + y));
end

Wolfram

code[x_, y_] := N[Sqrt[N[(x + y), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\sqrt{\left|x - y\right|} \]
2. Step-by-step derivation

   1. add-cbrt-cube 71.5%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\left|x - y\right|} \cdot \sqrt{\left|x - y\right|}\right) \cdot \sqrt{\left|x - y\right|}}} \]

   2. pow1/3 66.8%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{\left|x - y\right|} \cdot \sqrt{\left|x - y\right|}\right) \cdot \sqrt{\left|x - y\right|}\right)}^{0.3333333333333333}} \]

   3. add-sqr-sqrt 66.8%

      \[\leadsto {\left(\color{blue}{\left|x - y\right|} \cdot \sqrt{\left|x - y\right|}\right)}^{0.3333333333333333} \]

   4. pow1 66.8%

      \[\leadsto {\left(\color{blue}{{\left(\left|x - y\right|\right)}^{1}} \cdot \sqrt{\left|x - y\right|}\right)}^{0.3333333333333333} \]

   5. pow1/2 66.8%

      \[\leadsto {\left({\left(\left|x - y\right|\right)}^{1} \cdot \color{blue}{{\left(\left|x - y\right|\right)}^{0.5}}\right)}^{0.3333333333333333} \]

   6. pow-prod-up 66.8%

      \[\leadsto {\color{blue}{\left({\left(\left|x - y\right|\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

   7. add-sqr-sqrt 33.7%

      \[\leadsto {\left({\left(\left|\color{blue}{\sqrt{x - y} \cdot \sqrt{x - y}}\right|\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

   8. fabs-sqr 33.7%

      \[\leadsto {\left({\color{blue}{\left(\sqrt{x - y} \cdot \sqrt{x - y}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

   9. add-sqr-sqrt 33.7%

      \[\leadsto {\left({\color{blue}{\left(x - y\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

   10. metadata-eval 33.7%

       \[\leadsto {\left({\left(x - y\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

3. Applied egg-rr 33.7%

   \[\leadsto \color{blue}{{\left({\left(x - y\right)}^{1.5}\right)}^{0.3333333333333333}} \]
4. Step-by-step derivation

   1. unpow1/3 36.0%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(x - y\right)}^{1.5}}} \]

5. Simplified 36.0%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(x - y\right)}^{1.5}}} \]
6. Step-by-step derivation

   1. pow1/3 33.7%

      \[\leadsto \color{blue}{{\left({\left(x - y\right)}^{1.5}\right)}^{0.3333333333333333}} \]

   2. pow-pow 53.1%

      \[\leadsto \color{blue}{{\left(x - y\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

   3. metadata-eval 53.1%

      \[\leadsto {\left(x - y\right)}^{\color{blue}{0.5}} \]

   4. pow1/2 53.1%

      \[\leadsto \color{blue}{\sqrt{x - y}} \]

   5. flip3-- 21.4%

      \[\leadsto \sqrt{\color{blue}{\frac{{x}^{3} - {y}^{3}}{x \cdot x + \left(y \cdot y + x \cdot y\right)}}} \]

   6. sqrt-div 21.3%

      \[\leadsto \color{blue}{\frac{\sqrt{{x}^{3} - {y}^{3}}}{\sqrt{x \cdot x + \left(y \cdot y + x \cdot y\right)}}} \]

   7. distribute-rgt-in 21.3%

      \[\leadsto \frac{\sqrt{{x}^{3} - {y}^{3}}}{\sqrt{x \cdot x + \color{blue}{y \cdot \left(y + x\right)}}} \]

   8. +-commutative 21.3%

      \[\leadsto \frac{\sqrt{{x}^{3} - {y}^{3}}}{\sqrt{\color{blue}{y \cdot \left(y + x\right) + x \cdot x}}} \]

   9. fma-udef 21.3%

      \[\leadsto \frac{\sqrt{{x}^{3} - {y}^{3}}}{\sqrt{\color{blue}{\mathsf{fma}\left(y, y + x, x \cdot x\right)}}} \]

   10. clear-num 21.3%

       \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(y, y + x, x \cdot x\right)}}{\sqrt{{x}^{3} - {y}^{3}}}}} \]

   11. clear-num 21.3%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\sqrt{{x}^{3} - {y}^{3}}}{\sqrt{\mathsf{fma}\left(y, y + x, x \cdot x\right)}}}}} \]

   12. fma-udef 21.3%

       \[\leadsto \frac{1}{\frac{1}{\frac{\sqrt{{x}^{3} - {y}^{3}}}{\sqrt{\color{blue}{y \cdot \left(y + x\right) + x \cdot x}}}}} \]

   13. +-commutative 21.3%

       \[\leadsto \frac{1}{\frac{1}{\frac{\sqrt{{x}^{3} - {y}^{3}}}{\sqrt{\color{blue}{x \cdot x + y \cdot \left(y + x\right)}}}}} \]

   14. distribute-rgt-in 21.3%

       \[\leadsto \frac{1}{\frac{1}{\frac{\sqrt{{x}^{3} - {y}^{3}}}{\sqrt{x \cdot x + \color{blue}{\left(y \cdot y + x \cdot y\right)}}}}} \]

7. Applied egg-rr 53.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{\sqrt{x - y}}}} \]
8. Step-by-step derivation

   1. add-log-exp 4.8%

      \[\leadsto \color{blue}{\log \left(e^{\frac{1}{\frac{1}{\sqrt{x - y}}}}\right)} \]

   2. remove-double-div 4.8%

      \[\leadsto \log \left(e^{\color{blue}{\sqrt{x - y}}}\right) \]

   3. *-un-lft-identity 4.8%

      \[\leadsto \log \color{blue}{\left(1 \cdot e^{\sqrt{x - y}}\right)} \]

   4. log-prod 4.8%

      \[\leadsto \color{blue}{\log 1 + \log \left(e^{\sqrt{x - y}}\right)} \]

   5. metadata-eval 4.8%

      \[\leadsto \color{blue}{0} + \log \left(e^{\sqrt{x - y}}\right) \]

   6. add-log-exp 53.1%

      \[\leadsto 0 + \color{blue}{\sqrt{x - y}} \]

   7. sub-neg 53.1%

      \[\leadsto 0 + \sqrt{\color{blue}{x + \left(-y\right)}} \]

   8. add-sqr-sqrt 42.6%

      \[\leadsto 0 + \sqrt{x + \color{blue}{\sqrt{-y} \cdot \sqrt{-y}}} \]

   9. sqrt-unprod 53.9%

      \[\leadsto 0 + \sqrt{x + \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}} \]

   10. sqr-neg 53.9%

       \[\leadsto 0 + \sqrt{x + \sqrt{\color{blue}{y \cdot y}}} \]

   11. sqrt-unprod 32.2%

       \[\leadsto 0 + \sqrt{x + \color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

   12. add-sqr-sqrt 46.2%

       \[\leadsto 0 + \sqrt{x + \color{blue}{y}} \]

9. Applied egg-rr 46.2%

   \[\leadsto \color{blue}{0 + \sqrt{x + y}} \]
10. Step-by-step derivation

    1. +-lft-identity 46.2%

       \[\leadsto \color{blue}{\sqrt{x + y}} \]

    2. +-commutative 46.2%

       \[\leadsto \sqrt{\color{blue}{y + x}} \]

11. Simplified 46.2%

    \[\leadsto \color{blue}{\sqrt{y + x}} \]

12. Final simplification 46.2%

    \[\leadsto \sqrt{x + y} \]

Alternative 5: 26.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (sqrt x))

C

double code(double x, double y) {
	return sqrt(x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(x)
end function

Java

public static double code(double x, double y) {
	return Math.sqrt(x);
}

Python

def code(x, y):
	return math.sqrt(x)

Julia

function code(x, y)
	return sqrt(x)
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt(x);
end

Wolfram

code[x_, y_] := N[Sqrt[x], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\sqrt{\left|x - y\right|} \]
2. Step-by-step derivation

   1. add-cbrt-cube 71.5%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\left|x - y\right|} \cdot \sqrt{\left|x - y\right|}\right) \cdot \sqrt{\left|x - y\right|}}} \]

   2. pow1/3 66.8%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{\left|x - y\right|} \cdot \sqrt{\left|x - y\right|}\right) \cdot \sqrt{\left|x - y\right|}\right)}^{0.3333333333333333}} \]

   3. add-sqr-sqrt 66.8%

      \[\leadsto {\left(\color{blue}{\left|x - y\right|} \cdot \sqrt{\left|x - y\right|}\right)}^{0.3333333333333333} \]

   4. pow1 66.8%

      \[\leadsto {\left(\color{blue}{{\left(\left|x - y\right|\right)}^{1}} \cdot \sqrt{\left|x - y\right|}\right)}^{0.3333333333333333} \]

   5. pow1/2 66.8%

      \[\leadsto {\left({\left(\left|x - y\right|\right)}^{1} \cdot \color{blue}{{\left(\left|x - y\right|\right)}^{0.5}}\right)}^{0.3333333333333333} \]

   6. pow-prod-up 66.8%

      \[\leadsto {\color{blue}{\left({\left(\left|x - y\right|\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

   7. add-sqr-sqrt 33.7%

      \[\leadsto {\left({\left(\left|\color{blue}{\sqrt{x - y} \cdot \sqrt{x - y}}\right|\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

   8. fabs-sqr 33.7%

      \[\leadsto {\left({\color{blue}{\left(\sqrt{x - y} \cdot \sqrt{x - y}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

   9. add-sqr-sqrt 33.7%

      \[\leadsto {\left({\color{blue}{\left(x - y\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

   10. metadata-eval 33.7%

       \[\leadsto {\left({\left(x - y\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

3. Applied egg-rr 33.7%

   \[\leadsto \color{blue}{{\left({\left(x - y\right)}^{1.5}\right)}^{0.3333333333333333}} \]
4. Step-by-step derivation

   1. unpow1/3 36.0%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(x - y\right)}^{1.5}}} \]

5. Simplified 36.0%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(x - y\right)}^{1.5}}} \]

6. Taylor expanded in y around 0 26.2%

   \[\leadsto \color{blue}{\sqrt{x}} \]

7. Final simplification 26.2%

   \[\leadsto \sqrt{x} \]

Reproduce

herbie shell --seed 2023199 
(FPCore (x y)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, C"
  :precision binary64
  (sqrt (fabs (- x y))))