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

Percentage Accurate: 100.0% → 100.0%

Time: 4.9s

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: 74.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - y \leq -5 \cdot 10^{-292}:\\ \;\;\;\;\sqrt{\frac{x}{x + y} \cdot \left(-x\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- x y) -5e-292) (sqrt (* (/ x (+ x y)) (- x))) (sqrt (- x y))))

C

double code(double x, double y) {
	double tmp;
	if ((x - y) <= -5e-292) {
		tmp = sqrt(((x / (x + y)) * -x));
	} 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) <= (-5d-292)) then
        tmp = sqrt(((x / (x + y)) * -x))
    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) <= -5e-292) {
		tmp = Math.sqrt(((x / (x + y)) * -x));
	} else {
		tmp = Math.sqrt((x - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x - y) <= -5e-292:
		tmp = math.sqrt(((x / (x + y)) * -x))
	else:
		tmp = math.sqrt((x - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x - y) <= -5e-292)
		tmp = sqrt(Float64(Float64(x / Float64(x + y)) * Float64(-x)));
	else
		tmp = sqrt(Float64(x - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x - y) <= -5e-292)
		tmp = sqrt(((x / (x + y)) * -x));
	else
		tmp = sqrt((x - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x - y), $MachinePrecision], -5e-292], N[Sqrt[N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * (-x)), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(x - y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x y) < -4.99999999999999981e-292

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. fabs-sqr 0.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. flip-- 0.2%

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

      5. sqrt-div 0.1%

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

   3. Applied egg-rr 0.1%

      \[\leadsto \color{blue}{\frac{\sqrt{x \cdot x - y \cdot y}}{\sqrt{x + y}}} \]
   4. Step-by-step derivation

      1. +-commutative 0.1%

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

   5. Simplified 0.1%

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

   6. Taylor expanded in x around inf 1.5%

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

      1. add-sqr-sqrt 1.0%

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

      2. sqrt-unprod 2.1%

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

      3. sqr-neg 2.1%

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

      4. distribute-frac-neg 2.1%

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

      5. distribute-frac-neg 2.1%

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

      6. frac-times 2.0%

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

      7. sqr-neg 2.0%

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

      8. add-sqr-sqrt 2.1%

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

      9. +-commutative 2.1%

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

   8. Applied egg-rr 2.1%

      \[\leadsto \color{blue}{\sqrt{\frac{x \cdot x}{x + y}}} \]
   9. Step-by-step derivation

      1. associate-/l* 2.1%

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

      2. associate-/r/ 2.1%

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

      3. +-commutative 2.1%

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

   10. Simplified 2.1%

       \[\leadsto \color{blue}{\sqrt{\frac{x}{y + x} \cdot x}} \]
   11. Step-by-step derivation

       1. associate-*l/ 2.1%

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

       2. div-inv 2.1%

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

       3. sqr-neg 2.1%

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

       4. inv-pow 2.1%

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

       5. metadata-eval 2.1%

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

       6. pow-prod-up 2.0%

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

       7. swap-sqr 2.1%

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

       8. distribute-lft-neg-out 2.1%

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

       9. metadata-eval 2.1%

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

       10. pow-flip 2.1%

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

       11. pow1/2 2.1%

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

       12. div-inv 2.1%

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

       13. distribute-rgt-neg-out 2.1%

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

   12. Applied egg-rr 28.6%

       \[\leadsto \sqrt{\color{blue}{-\frac{x \cdot x}{x + y}}} \]
   13. Step-by-step derivation

       1. associate-*l/ 52.8%

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

       2. distribute-rgt-neg-in 52.8%

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

       3. +-commutative 52.8%

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

   14. Simplified 52.8%

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

   if -4.99999999999999981e-292 < (-.f64 x y)

   1. Initial program 100.0%

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

      1. expm1-log1p-u 93.7%

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

      2. expm1-udef 75.8%

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

      3. log1p-udef 75.8%

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

      4. add-exp-log 82.0%

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

      5. add-sqr-sqrt 82.0%

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

      6. fabs-sqr 82.0%

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

      7. add-sqr-sqrt 82.0%

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

   3. Applied egg-rr 82.0%

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

      1. +-commutative 82.0%

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

      2. associate--l+ 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - y \leq -5 \cdot 10^{-292}:\\ \;\;\;\;\sqrt{\frac{x}{x + y} \cdot \left(-x\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x - y}\\ \end{array} \]

Alternative 3: 51.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - y \leq -5 \cdot 10^{-292}:\\ \;\;\;\;\sqrt{x \cdot \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- x y) -5e-292) (sqrt (* x (/ x y))) (sqrt (- x y))))

C

double code(double x, double y) {
	double tmp;
	if ((x - y) <= -5e-292) {
		tmp = sqrt((x * (x / 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) <= (-5d-292)) then
        tmp = sqrt((x * (x / 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) <= -5e-292) {
		tmp = Math.sqrt((x * (x / y)));
	} else {
		tmp = Math.sqrt((x - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x - y) <= -5e-292:
		tmp = math.sqrt((x * (x / y)))
	else:
		tmp = math.sqrt((x - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x - y) <= -5e-292)
		tmp = sqrt(Float64(x * Float64(x / y)));
	else
		tmp = sqrt(Float64(x - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x - y) <= -5e-292)
		tmp = sqrt((x * (x / y)));
	else
		tmp = sqrt((x - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x - y), $MachinePrecision], -5e-292], N[Sqrt[N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(x - y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x y) < -4.99999999999999981e-292

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. fabs-sqr 0.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. flip-- 0.2%

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

      5. sqrt-div 0.1%

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

   3. Applied egg-rr 0.1%

      \[\leadsto \color{blue}{\frac{\sqrt{x \cdot x - y \cdot y}}{\sqrt{x + y}}} \]
   4. Step-by-step derivation

      1. +-commutative 0.1%

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

   5. Simplified 0.1%

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

   6. Taylor expanded in x around inf 1.5%

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

      1. add-sqr-sqrt 1.0%

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

      2. sqrt-unprod 2.1%

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

      3. sqr-neg 2.1%

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

      4. distribute-frac-neg 2.1%

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

      5. distribute-frac-neg 2.1%

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

      6. frac-times 2.0%

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

      7. sqr-neg 2.0%

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

      8. add-sqr-sqrt 2.1%

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

      9. +-commutative 2.1%

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

   8. Applied egg-rr 2.1%

      \[\leadsto \color{blue}{\sqrt{\frac{x \cdot x}{x + y}}} \]
   9. Step-by-step derivation

      1. associate-/l* 2.1%

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

      2. associate-/r/ 2.1%

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

      3. +-commutative 2.1%

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

   10. Simplified 2.1%

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

   11. Taylor expanded in x around 0 3.4%

       \[\leadsto \sqrt{\color{blue}{\frac{{x}^{2}}{y}}} \]
   12. Step-by-step derivation

       1. unpow2 3.4%

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

       2. associate-*r/ 3.6%

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

   13. Simplified 3.6%

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

   if -4.99999999999999981e-292 < (-.f64 x y)

   1. Initial program 100.0%

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

      1. expm1-log1p-u 93.7%

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

      2. expm1-udef 75.8%

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

      3. log1p-udef 75.8%

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

      4. add-exp-log 82.0%

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

      5. add-sqr-sqrt 82.0%

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

      6. fabs-sqr 82.0%

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

      7. add-sqr-sqrt 82.0%

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

   3. Applied egg-rr 82.0%

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

      1. +-commutative 82.0%

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

      2. associate--l+ 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - y \leq -5 \cdot 10^{-292}:\\ \;\;\;\;\sqrt{x \cdot \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x - y}\\ \end{array} \]

Alternative 4: 49.9% 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. expm1-log1p-u 93.7%

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

   2. expm1-udef 77.2%

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

   3. log1p-udef 77.2%

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

   4. add-exp-log 83.5%

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

   5. add-sqr-sqrt 44.2%

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

   6. fabs-sqr 44.2%

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

   7. add-sqr-sqrt 44.2%

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

3. Applied egg-rr 44.2%

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

   1. +-commutative 44.2%

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

   2. associate--l+ 53.9%

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

   3. metadata-eval 53.9%

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

   4. +-rgt-identity 53.9%

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

5. Simplified 53.9%

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

6. Final simplification 53.9%

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

Alternative 5: 26.3% 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. expm1-log1p-u 93.7%

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

   2. expm1-udef 77.2%

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

   3. log1p-udef 77.2%

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

   4. add-exp-log 83.5%

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

   5. add-sqr-sqrt 44.2%

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

   6. fabs-sqr 44.2%

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

   7. add-sqr-sqrt 44.2%

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

3. Applied egg-rr 44.2%

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

   1. +-commutative 44.2%

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

   2. associate--l+ 53.9%

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

   3. metadata-eval 53.9%

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

   4. +-rgt-identity 53.9%

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

5. Simplified 53.9%

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

6. Taylor expanded in y around 0 29.6%

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

7. Final simplification 29.6%

   \[\leadsto \sqrt{x} \]

Reproduce

herbie shell --seed 2023230 
(FPCore (x y)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, C"
  :precision binary64
  (sqrt (fabs (- x y))))