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

Percentage Accurate: 100.0% → 100.0%

Time: 5.6s

Alternatives: 9

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: 74.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x \cdot \left(x + y \cdot -2\right)\right)}^{0.25}\\ \mathbf{if}\;x - y \leq -6 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{elif}\;x - y \leq -500000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x - y \leq -5 \cdot 10^{-53}:\\ \;\;\;\;{\left(y \cdot y\right)}^{0.25}\\ \mathbf{elif}\;x - y \leq -5 \cdot 10^{-130}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x - y \leq -2 \cdot 10^{-183}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (* x (+ x (* y -2.0))) 0.25)))
   (if (<= (- x y) -6e+147)
     (sqrt y)
     (if (<= (- x y) -500000.0)
       t_0
       (if (<= (- x y) -5e-53)
         (pow (* y y) 0.25)
         (if (<= (- x y) -5e-130)
           t_0
           (if (<= (- x y) -2e-183) (sqrt y) (sqrt (- x y)))))))))

C

double code(double x, double y) {
	double t_0 = pow((x * (x + (y * -2.0))), 0.25);
	double tmp;
	if ((x - y) <= -6e+147) {
		tmp = sqrt(y);
	} else if ((x - y) <= -500000.0) {
		tmp = t_0;
	} else if ((x - y) <= -5e-53) {
		tmp = pow((y * y), 0.25);
	} else if ((x - y) <= -5e-130) {
		tmp = t_0;
	} else if ((x - y) <= -2e-183) {
		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) :: t_0
    real(8) :: tmp
    t_0 = (x * (x + (y * (-2.0d0)))) ** 0.25d0
    if ((x - y) <= (-6d+147)) then
        tmp = sqrt(y)
    else if ((x - y) <= (-500000.0d0)) then
        tmp = t_0
    else if ((x - y) <= (-5d-53)) then
        tmp = (y * y) ** 0.25d0
    else if ((x - y) <= (-5d-130)) then
        tmp = t_0
    else if ((x - y) <= (-2d-183)) 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 t_0 = Math.pow((x * (x + (y * -2.0))), 0.25);
	double tmp;
	if ((x - y) <= -6e+147) {
		tmp = Math.sqrt(y);
	} else if ((x - y) <= -500000.0) {
		tmp = t_0;
	} else if ((x - y) <= -5e-53) {
		tmp = Math.pow((y * y), 0.25);
	} else if ((x - y) <= -5e-130) {
		tmp = t_0;
	} else if ((x - y) <= -2e-183) {
		tmp = Math.sqrt(y);
	} else {
		tmp = Math.sqrt((x - y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.pow((x * (x + (y * -2.0))), 0.25)
	tmp = 0
	if (x - y) <= -6e+147:
		tmp = math.sqrt(y)
	elif (x - y) <= -500000.0:
		tmp = t_0
	elif (x - y) <= -5e-53:
		tmp = math.pow((y * y), 0.25)
	elif (x - y) <= -5e-130:
		tmp = t_0
	elif (x - y) <= -2e-183:
		tmp = math.sqrt(y)
	else:
		tmp = math.sqrt((x - y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x + Float64(y * -2.0))) ^ 0.25
	tmp = 0.0
	if (Float64(x - y) <= -6e+147)
		tmp = sqrt(y);
	elseif (Float64(x - y) <= -500000.0)
		tmp = t_0;
	elseif (Float64(x - y) <= -5e-53)
		tmp = Float64(y * y) ^ 0.25;
	elseif (Float64(x - y) <= -5e-130)
		tmp = t_0;
	elseif (Float64(x - y) <= -2e-183)
		tmp = sqrt(y);
	else
		tmp = sqrt(Float64(x - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * (x + (y * -2.0))) ^ 0.25;
	tmp = 0.0;
	if ((x - y) <= -6e+147)
		tmp = sqrt(y);
	elseif ((x - y) <= -500000.0)
		tmp = t_0;
	elseif ((x - y) <= -5e-53)
		tmp = (y * y) ^ 0.25;
	elseif ((x - y) <= -5e-130)
		tmp = t_0;
	elseif ((x - y) <= -2e-183)
		tmp = sqrt(y);
	else
		tmp = sqrt((x - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[(x * N[(x + N[(y * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]}, If[LessEqual[N[(x - y), $MachinePrecision], -6e+147], N[Sqrt[y], $MachinePrecision], If[LessEqual[N[(x - y), $MachinePrecision], -500000.0], t$95$0, If[LessEqual[N[(x - y), $MachinePrecision], -5e-53], N[Power[N[(y * y), $MachinePrecision], 0.25], $MachinePrecision], If[LessEqual[N[(x - y), $MachinePrecision], -5e-130], t$95$0, If[LessEqual[N[(x - y), $MachinePrecision], -2e-183], N[Sqrt[y], $MachinePrecision], N[Sqrt[N[(x - y), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 x y) < -5.99999999999999987e147 or -4.9999999999999996e-130 < (-.f64 x y) < -2.00000000000000001e-183

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

   3. Simplified 100.0%

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

      1. pow1/2 100.0%

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

      2. sqr-pow 99.2%

         \[\leadsto \color{blue}{{\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      3. pow-prod-down 11.2%

         \[\leadsto \color{blue}{{\left(\left|y - x\right| \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      4. neg-fabs 11.2%

         \[\leadsto {\left(\color{blue}{\left|-\left(y - x\right)\right|} \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \]

      5. neg-fabs 11.2%

         \[\leadsto {\left(\left|-\left(y - x\right)\right| \cdot \color{blue}{\left|-\left(y - x\right)\right|}\right)}^{\left(\frac{0.5}{2}\right)} \]

      6. sqr-abs 11.2%

         \[\leadsto {\color{blue}{\left(\left(-\left(y - x\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)} \]

      7. sub-neg 11.2%

         \[\leadsto {\left(\left(-\color{blue}{\left(y + \left(-x\right)\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      8. +-commutative 11.2%

         \[\leadsto {\left(\left(-\color{blue}{\left(\left(-x\right) + y\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      9. distribute-neg-in 11.2%

         \[\leadsto {\left(\color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      10. remove-double-neg 11.2%

          \[\leadsto {\left(\left(\color{blue}{x} + \left(-y\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      11. sub-neg 11.2%

          \[\leadsto {\left(\color{blue}{\left(x - y\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      12. sub-neg 11.2%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(y + \left(-x\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      13. +-commutative 11.2%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(\left(-x\right) + y\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      14. distribute-neg-in 11.2%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      15. remove-double-neg 11.2%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(\color{blue}{x} + \left(-y\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      16. sub-neg 11.2%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(x - y\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      17. metadata-eval 11.2%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{\color{blue}{0.25}} \]

   5. Applied egg-rr 11.2%

      \[\leadsto \color{blue}{{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}} \]

   6. Taylor expanded in x around 0 59.1%

      \[\leadsto \color{blue}{{1}^{0.25} \cdot \sqrt{y}} \]
   7. Step-by-step derivation

      1. pow-base-1 59.1%

         \[\leadsto \color{blue}{1} \cdot \sqrt{y} \]

      2. *-lft-identity 59.1%

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

   8. Simplified 59.1%

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

   if -5.99999999999999987e147 < (-.f64 x y) < -5e5 or -5e-53 < (-.f64 x y) < -4.9999999999999996e-130

   1. Initial program 99.9%

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

      1. fabs-sub 99.9%

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

   3. Simplified 99.9%

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

      1. pow1/2 99.9%

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

      2. sqr-pow 99.1%

         \[\leadsto \color{blue}{{\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      3. pow-prod-down 99.9%

         \[\leadsto \color{blue}{{\left(\left|y - x\right| \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      4. neg-fabs 99.9%

         \[\leadsto {\left(\color{blue}{\left|-\left(y - x\right)\right|} \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \]

      5. neg-fabs 99.9%

         \[\leadsto {\left(\left|-\left(y - x\right)\right| \cdot \color{blue}{\left|-\left(y - x\right)\right|}\right)}^{\left(\frac{0.5}{2}\right)} \]

      6. sqr-abs 99.9%

         \[\leadsto {\color{blue}{\left(\left(-\left(y - x\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)} \]

      7. sub-neg 99.9%

         \[\leadsto {\left(\left(-\color{blue}{\left(y + \left(-x\right)\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      8. +-commutative 99.9%

         \[\leadsto {\left(\left(-\color{blue}{\left(\left(-x\right) + y\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      9. distribute-neg-in 99.9%

         \[\leadsto {\left(\color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      10. remove-double-neg 99.9%

          \[\leadsto {\left(\left(\color{blue}{x} + \left(-y\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      11. sub-neg 99.9%

          \[\leadsto {\left(\color{blue}{\left(x - y\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      12. sub-neg 99.9%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(y + \left(-x\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      13. +-commutative 99.9%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(\left(-x\right) + y\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      14. distribute-neg-in 99.9%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      15. remove-double-neg 99.9%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(\color{blue}{x} + \left(-y\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      16. sub-neg 99.9%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(x - y\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      17. metadata-eval 99.9%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{\color{blue}{0.25}} \]

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}} \]
   6. Step-by-step derivation

      1. sub-neg 99.9%

         \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(x + \left(-y\right)\right)}\right)}^{0.25} \]

      2. +-commutative 99.9%

         \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(\left(-y\right) + x\right)}\right)}^{0.25} \]

      3. distribute-rgt-in 100.0%

         \[\leadsto {\color{blue}{\left(\left(-y\right) \cdot \left(x - y\right) + x \cdot \left(x - y\right)\right)}}^{0.25} \]

      4. sub-neg 100.0%

         \[\leadsto {\left(\left(-y\right) \cdot \color{blue}{\left(x + \left(-y\right)\right)} + x \cdot \left(x - y\right)\right)}^{0.25} \]

      5. +-commutative 100.0%

         \[\leadsto {\left(\left(-y\right) \cdot \color{blue}{\left(\left(-y\right) + x\right)} + x \cdot \left(x - y\right)\right)}^{0.25} \]

      6. distribute-lft-in 100.0%

         \[\leadsto {\left(\color{blue}{\left(\left(-y\right) \cdot \left(-y\right) + \left(-y\right) \cdot x\right)} + x \cdot \left(x - y\right)\right)}^{0.25} \]

      7. sqr-neg 100.0%

         \[\leadsto {\left(\left(\color{blue}{y \cdot y} + \left(-y\right) \cdot x\right) + x \cdot \left(x - y\right)\right)}^{0.25} \]

      8. cancel-sign-sub-inv 100.0%

         \[\leadsto {\left(\color{blue}{\left(y \cdot y - y \cdot x\right)} + x \cdot \left(x - y\right)\right)}^{0.25} \]

      9. distribute-lft-out-- 100.0%

         \[\leadsto {\left(\color{blue}{y \cdot \left(y - x\right)} + x \cdot \left(x - y\right)\right)}^{0.25} \]

   7. Applied egg-rr 100.0%

      \[\leadsto {\color{blue}{\left(y \cdot \left(y - x\right) + x \cdot \left(x - y\right)\right)}}^{0.25} \]

   8. Taylor expanded in y around 0 63.8%

      \[\leadsto {\color{blue}{\left(-2 \cdot \left(x \cdot y\right) + {x}^{2}\right)}}^{0.25} \]
   9. Step-by-step derivation

      1. +-commutative 63.8%

         \[\leadsto {\color{blue}{\left({x}^{2} + -2 \cdot \left(x \cdot y\right)\right)}}^{0.25} \]

      2. unpow2 63.8%

         \[\leadsto {\left(\color{blue}{x \cdot x} + -2 \cdot \left(x \cdot y\right)\right)}^{0.25} \]

      3. *-commutative 63.8%

         \[\leadsto {\left(x \cdot x + \color{blue}{\left(x \cdot y\right) \cdot -2}\right)}^{0.25} \]

      4. associate-*l* 63.8%

         \[\leadsto {\left(x \cdot x + \color{blue}{x \cdot \left(y \cdot -2\right)}\right)}^{0.25} \]

      5. metadata-eval 63.8%

         \[\leadsto {\left(x \cdot x + x \cdot \left(y \cdot \color{blue}{\left(-1 + -1\right)}\right)\right)}^{0.25} \]

      6. distribute-rgt-out 63.8%

         \[\leadsto {\left(x \cdot x + x \cdot \color{blue}{\left(-1 \cdot y + -1 \cdot y\right)}\right)}^{0.25} \]

      7. mul-1-neg 63.8%

         \[\leadsto {\left(x \cdot x + x \cdot \left(-1 \cdot y + \color{blue}{\left(-y\right)}\right)\right)}^{0.25} \]

      8. fma-def 63.8%

         \[\leadsto {\left(x \cdot x + x \cdot \color{blue}{\mathsf{fma}\left(-1, y, -y\right)}\right)}^{0.25} \]

      9. fma-neg 63.8%

         \[\leadsto {\left(x \cdot x + x \cdot \color{blue}{\left(-1 \cdot y - y\right)}\right)}^{0.25} \]

      10. distribute-lft-out 63.8%

          \[\leadsto {\color{blue}{\left(x \cdot \left(x + \left(-1 \cdot y - y\right)\right)\right)}}^{0.25} \]

      11. fma-neg 63.8%

          \[\leadsto {\left(x \cdot \left(x + \color{blue}{\mathsf{fma}\left(-1, y, -y\right)}\right)\right)}^{0.25} \]

      12. fma-def 63.8%

          \[\leadsto {\left(x \cdot \left(x + \color{blue}{\left(-1 \cdot y + \left(-y\right)\right)}\right)\right)}^{0.25} \]

      13. mul-1-neg 63.8%

          \[\leadsto {\left(x \cdot \left(x + \left(-1 \cdot y + \color{blue}{-1 \cdot y}\right)\right)\right)}^{0.25} \]

      14. distribute-rgt-out 63.8%

          \[\leadsto {\left(x \cdot \left(x + \color{blue}{y \cdot \left(-1 + -1\right)}\right)\right)}^{0.25} \]

      15. metadata-eval 63.8%

          \[\leadsto {\left(x \cdot \left(x + y \cdot \color{blue}{-2}\right)\right)}^{0.25} \]

   10. Simplified 63.8%

       \[\leadsto {\color{blue}{\left(x \cdot \left(x + y \cdot -2\right)\right)}}^{0.25} \]

   if -5e5 < (-.f64 x y) < -5e-53

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

   3. Simplified 100.0%

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

      1. pow1/2 100.0%

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

      2. sqr-pow 99.5%

         \[\leadsto \color{blue}{{\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      3. pow-prod-down 100.0%

         \[\leadsto \color{blue}{{\left(\left|y - x\right| \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      4. neg-fabs 100.0%

         \[\leadsto {\left(\color{blue}{\left|-\left(y - x\right)\right|} \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \]

      5. neg-fabs 100.0%

         \[\leadsto {\left(\left|-\left(y - x\right)\right| \cdot \color{blue}{\left|-\left(y - x\right)\right|}\right)}^{\left(\frac{0.5}{2}\right)} \]

      6. sqr-abs 100.0%

         \[\leadsto {\color{blue}{\left(\left(-\left(y - x\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)} \]

      7. sub-neg 100.0%

         \[\leadsto {\left(\left(-\color{blue}{\left(y + \left(-x\right)\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      8. +-commutative 100.0%

         \[\leadsto {\left(\left(-\color{blue}{\left(\left(-x\right) + y\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      9. distribute-neg-in 100.0%

         \[\leadsto {\left(\color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      10. remove-double-neg 100.0%

          \[\leadsto {\left(\left(\color{blue}{x} + \left(-y\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      11. sub-neg 100.0%

          \[\leadsto {\left(\color{blue}{\left(x - y\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      12. sub-neg 100.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(y + \left(-x\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      13. +-commutative 100.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(\left(-x\right) + y\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      14. distribute-neg-in 100.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      15. remove-double-neg 100.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(\color{blue}{x} + \left(-y\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      16. sub-neg 100.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(x - y\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      17. metadata-eval 100.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{\color{blue}{0.25}} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}} \]
   6. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(x + \left(-y\right)\right)}\right)}^{0.25} \]

      2. +-commutative 100.0%

         \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(\left(-y\right) + x\right)}\right)}^{0.25} \]

      3. distribute-rgt-in 100.0%

         \[\leadsto {\color{blue}{\left(\left(-y\right) \cdot \left(x - y\right) + x \cdot \left(x - y\right)\right)}}^{0.25} \]

      4. sub-neg 100.0%

         \[\leadsto {\left(\left(-y\right) \cdot \color{blue}{\left(x + \left(-y\right)\right)} + x \cdot \left(x - y\right)\right)}^{0.25} \]

      5. +-commutative 100.0%

         \[\leadsto {\left(\left(-y\right) \cdot \color{blue}{\left(\left(-y\right) + x\right)} + x \cdot \left(x - y\right)\right)}^{0.25} \]

      6. distribute-lft-in 100.0%

         \[\leadsto {\left(\color{blue}{\left(\left(-y\right) \cdot \left(-y\right) + \left(-y\right) \cdot x\right)} + x \cdot \left(x - y\right)\right)}^{0.25} \]

      7. sqr-neg 100.0%

         \[\leadsto {\left(\left(\color{blue}{y \cdot y} + \left(-y\right) \cdot x\right) + x \cdot \left(x - y\right)\right)}^{0.25} \]

      8. cancel-sign-sub-inv 100.0%

         \[\leadsto {\left(\color{blue}{\left(y \cdot y - y \cdot x\right)} + x \cdot \left(x - y\right)\right)}^{0.25} \]

      9. distribute-lft-out-- 100.0%

         \[\leadsto {\left(\color{blue}{y \cdot \left(y - x\right)} + x \cdot \left(x - y\right)\right)}^{0.25} \]

   7. Applied egg-rr 100.0%

      \[\leadsto {\color{blue}{\left(y \cdot \left(y - x\right) + x \cdot \left(x - y\right)\right)}}^{0.25} \]

   8. Taylor expanded in y around inf 81.1%

      \[\leadsto {\color{blue}{\left({y}^{2}\right)}}^{0.25} \]
   9. Step-by-step derivation

      1. unpow2 81.1%

         \[\leadsto {\color{blue}{\left(y \cdot y\right)}}^{0.25} \]

   10. Simplified 81.1%

       \[\leadsto {\color{blue}{\left(y \cdot y\right)}}^{0.25} \]

   if -2.00000000000000001e-183 < (-.f64 x y)

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

   3. Simplified 100.0%

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

      1. pow1/2 100.0%

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

      2. sqr-pow 99.1%

         \[\leadsto \color{blue}{{\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      3. pow-prod-down 54.6%

         \[\leadsto \color{blue}{{\left(\left|y - x\right| \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      4. neg-fabs 54.6%

         \[\leadsto {\left(\color{blue}{\left|-\left(y - x\right)\right|} \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \]

      5. neg-fabs 54.6%

         \[\leadsto {\left(\left|-\left(y - x\right)\right| \cdot \color{blue}{\left|-\left(y - x\right)\right|}\right)}^{\left(\frac{0.5}{2}\right)} \]

      6. sqr-abs 54.6%

         \[\leadsto {\color{blue}{\left(\left(-\left(y - x\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)} \]

      7. sub-neg 54.6%

         \[\leadsto {\left(\left(-\color{blue}{\left(y + \left(-x\right)\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      8. +-commutative 54.6%

         \[\leadsto {\left(\left(-\color{blue}{\left(\left(-x\right) + y\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      9. distribute-neg-in 54.6%

         \[\leadsto {\left(\color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      10. remove-double-neg 54.6%

          \[\leadsto {\left(\left(\color{blue}{x} + \left(-y\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      11. sub-neg 54.6%

          \[\leadsto {\left(\color{blue}{\left(x - y\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      12. sub-neg 54.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(y + \left(-x\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      13. +-commutative 54.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(\left(-x\right) + y\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      14. distribute-neg-in 54.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      15. remove-double-neg 54.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(\color{blue}{x} + \left(-y\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      16. sub-neg 54.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(x - y\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      17. metadata-eval 54.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{\color{blue}{0.25}} \]

   5. Applied egg-rr 54.6%

      \[\leadsto \color{blue}{{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}} \]
   6. Step-by-step derivation

      1. unpow-prod-down 98.3%

         \[\leadsto \color{blue}{{\left(x - y\right)}^{0.25} \cdot {\left(x - y\right)}^{0.25}} \]

      2. pow-sqr 99.2%

         \[\leadsto \color{blue}{{\left(x - y\right)}^{\left(2 \cdot 0.25\right)}} \]

      3. metadata-eval 99.2%

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

      4. unpow1/2 99.2%

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

   7. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\sqrt{x - y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - y \leq -6 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{elif}\;x - y \leq -500000:\\ \;\;\;\;{\left(x \cdot \left(x + y \cdot -2\right)\right)}^{0.25}\\ \mathbf{elif}\;x - y \leq -5 \cdot 10^{-53}:\\ \;\;\;\;{\left(y \cdot y\right)}^{0.25}\\ \mathbf{elif}\;x - y \leq -5 \cdot 10^{-130}:\\ \;\;\;\;{\left(x \cdot \left(x + y \cdot -2\right)\right)}^{0.25}\\ \mathbf{elif}\;x - y \leq -2 \cdot 10^{-183}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x - y}\\ \end{array} \]

Alternative 3: 86.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - y \leq -1 \cdot 10^{+155}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{elif}\;x - y \leq -1 \cdot 10^{-189}:\\ \;\;\;\;{\left(y \cdot \left(y - x\right) + x \cdot \left(x - y\right)\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- x y) -1e+155)
   (sqrt y)
   (if (<= (- x y) -1e-189)
     (pow (+ (* y (- y x)) (* x (- x y))) 0.25)
     (sqrt (- x y)))))

C

double code(double x, double y) {
	double tmp;
	if ((x - y) <= -1e+155) {
		tmp = sqrt(y);
	} else if ((x - y) <= -1e-189) {
		tmp = pow(((y * (y - x)) + (x * (x - y))), 0.25);
	} 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) <= (-1d+155)) then
        tmp = sqrt(y)
    else if ((x - y) <= (-1d-189)) then
        tmp = ((y * (y - x)) + (x * (x - y))) ** 0.25d0
    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) <= -1e+155) {
		tmp = Math.sqrt(y);
	} else if ((x - y) <= -1e-189) {
		tmp = Math.pow(((y * (y - x)) + (x * (x - y))), 0.25);
	} else {
		tmp = Math.sqrt((x - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x - y) <= -1e+155:
		tmp = math.sqrt(y)
	elif (x - y) <= -1e-189:
		tmp = math.pow(((y * (y - x)) + (x * (x - y))), 0.25)
	else:
		tmp = math.sqrt((x - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x - y) <= -1e+155)
		tmp = sqrt(y);
	elseif (Float64(x - y) <= -1e-189)
		tmp = Float64(Float64(y * Float64(y - x)) + Float64(x * Float64(x - y))) ^ 0.25;
	else
		tmp = sqrt(Float64(x - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x - y) <= -1e+155)
		tmp = sqrt(y);
	elseif ((x - y) <= -1e-189)
		tmp = ((y * (y - x)) + (x * (x - y))) ^ 0.25;
	else
		tmp = sqrt((x - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x - y), $MachinePrecision], -1e+155], N[Sqrt[y], $MachinePrecision], If[LessEqual[N[(x - y), $MachinePrecision], -1e-189], N[Power[N[(N[(y * N[(y - x), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], N[Sqrt[N[(x - y), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x y) < -1.00000000000000001e155

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

   3. Simplified 100.0%

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

      1. pow1/2 100.0%

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

      2. sqr-pow 99.3%

         \[\leadsto \color{blue}{{\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      3. pow-prod-down 4.3%

         \[\leadsto \color{blue}{{\left(\left|y - x\right| \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      4. neg-fabs 4.3%

         \[\leadsto {\left(\color{blue}{\left|-\left(y - x\right)\right|} \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \]

      5. neg-fabs 4.3%

         \[\leadsto {\left(\left|-\left(y - x\right)\right| \cdot \color{blue}{\left|-\left(y - x\right)\right|}\right)}^{\left(\frac{0.5}{2}\right)} \]

      6. sqr-abs 4.3%

         \[\leadsto {\color{blue}{\left(\left(-\left(y - x\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)} \]

      7. sub-neg 4.3%

         \[\leadsto {\left(\left(-\color{blue}{\left(y + \left(-x\right)\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      8. +-commutative 4.3%

         \[\leadsto {\left(\left(-\color{blue}{\left(\left(-x\right) + y\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      9. distribute-neg-in 4.3%

         \[\leadsto {\left(\color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      10. remove-double-neg 4.3%

          \[\leadsto {\left(\left(\color{blue}{x} + \left(-y\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      11. sub-neg 4.3%

          \[\leadsto {\left(\color{blue}{\left(x - y\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      12. sub-neg 4.3%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(y + \left(-x\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      13. +-commutative 4.3%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(\left(-x\right) + y\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      14. distribute-neg-in 4.3%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      15. remove-double-neg 4.3%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(\color{blue}{x} + \left(-y\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      16. sub-neg 4.3%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(x - y\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      17. metadata-eval 4.3%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{\color{blue}{0.25}} \]

   5. Applied egg-rr 4.3%

      \[\leadsto \color{blue}{{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}} \]

   6. Taylor expanded in x around 0 57.1%

      \[\leadsto \color{blue}{{1}^{0.25} \cdot \sqrt{y}} \]
   7. Step-by-step derivation

      1. pow-base-1 57.1%

         \[\leadsto \color{blue}{1} \cdot \sqrt{y} \]

      2. *-lft-identity 57.1%

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

   8. Simplified 57.1%

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

   if -1.00000000000000001e155 < (-.f64 x y) < -1.00000000000000007e-189

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

   3. Simplified 100.0%

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

      1. pow1/2 100.0%

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

      2. sqr-pow 99.2%

         \[\leadsto \color{blue}{{\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      3. pow-prod-down 94.8%

         \[\leadsto \color{blue}{{\left(\left|y - x\right| \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      4. neg-fabs 94.8%

         \[\leadsto {\left(\color{blue}{\left|-\left(y - x\right)\right|} \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \]

      5. neg-fabs 94.8%

         \[\leadsto {\left(\left|-\left(y - x\right)\right| \cdot \color{blue}{\left|-\left(y - x\right)\right|}\right)}^{\left(\frac{0.5}{2}\right)} \]

      6. sqr-abs 94.8%

         \[\leadsto {\color{blue}{\left(\left(-\left(y - x\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)} \]

      7. sub-neg 94.8%

         \[\leadsto {\left(\left(-\color{blue}{\left(y + \left(-x\right)\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      8. +-commutative 94.8%

         \[\leadsto {\left(\left(-\color{blue}{\left(\left(-x\right) + y\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      9. distribute-neg-in 94.8%

         \[\leadsto {\left(\color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      10. remove-double-neg 94.8%

          \[\leadsto {\left(\left(\color{blue}{x} + \left(-y\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      11. sub-neg 94.8%

          \[\leadsto {\left(\color{blue}{\left(x - y\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      12. sub-neg 94.8%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(y + \left(-x\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      13. +-commutative 94.8%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(\left(-x\right) + y\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      14. distribute-neg-in 94.8%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      15. remove-double-neg 94.8%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(\color{blue}{x} + \left(-y\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      16. sub-neg 94.8%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(x - y\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      17. metadata-eval 94.8%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{\color{blue}{0.25}} \]

   5. Applied egg-rr 94.8%

      \[\leadsto \color{blue}{{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}} \]
   6. Step-by-step derivation

      1. sub-neg 94.8%

         \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(x + \left(-y\right)\right)}\right)}^{0.25} \]

      2. +-commutative 94.8%

         \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(\left(-y\right) + x\right)}\right)}^{0.25} \]

      3. distribute-rgt-in 94.8%

         \[\leadsto {\color{blue}{\left(\left(-y\right) \cdot \left(x - y\right) + x \cdot \left(x - y\right)\right)}}^{0.25} \]

      4. sub-neg 94.8%

         \[\leadsto {\left(\left(-y\right) \cdot \color{blue}{\left(x + \left(-y\right)\right)} + x \cdot \left(x - y\right)\right)}^{0.25} \]

      5. +-commutative 94.8%

         \[\leadsto {\left(\left(-y\right) \cdot \color{blue}{\left(\left(-y\right) + x\right)} + x \cdot \left(x - y\right)\right)}^{0.25} \]

      6. distribute-lft-in 94.8%

         \[\leadsto {\left(\color{blue}{\left(\left(-y\right) \cdot \left(-y\right) + \left(-y\right) \cdot x\right)} + x \cdot \left(x - y\right)\right)}^{0.25} \]

      7. sqr-neg 94.8%

         \[\leadsto {\left(\left(\color{blue}{y \cdot y} + \left(-y\right) \cdot x\right) + x \cdot \left(x - y\right)\right)}^{0.25} \]

      8. cancel-sign-sub-inv 94.8%

         \[\leadsto {\left(\color{blue}{\left(y \cdot y - y \cdot x\right)} + x \cdot \left(x - y\right)\right)}^{0.25} \]

      9. distribute-lft-out-- 94.8%

         \[\leadsto {\left(\color{blue}{y \cdot \left(y - x\right)} + x \cdot \left(x - y\right)\right)}^{0.25} \]

   7. Applied egg-rr 94.8%

      \[\leadsto {\color{blue}{\left(y \cdot \left(y - x\right) + x \cdot \left(x - y\right)\right)}}^{0.25} \]

   if -1.00000000000000007e-189 < (-.f64 x y)

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

   3. Simplified 100.0%

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

      1. pow1/2 100.0%

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

      2. sqr-pow 99.1%

         \[\leadsto \color{blue}{{\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      3. pow-prod-down 55.0%

         \[\leadsto \color{blue}{{\left(\left|y - x\right| \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      4. neg-fabs 55.0%

         \[\leadsto {\left(\color{blue}{\left|-\left(y - x\right)\right|} \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \]

      5. neg-fabs 55.0%

         \[\leadsto {\left(\left|-\left(y - x\right)\right| \cdot \color{blue}{\left|-\left(y - x\right)\right|}\right)}^{\left(\frac{0.5}{2}\right)} \]

      6. sqr-abs 55.0%

         \[\leadsto {\color{blue}{\left(\left(-\left(y - x\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)} \]

      7. sub-neg 55.0%

         \[\leadsto {\left(\left(-\color{blue}{\left(y + \left(-x\right)\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      8. +-commutative 55.0%

         \[\leadsto {\left(\left(-\color{blue}{\left(\left(-x\right) + y\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      9. distribute-neg-in 55.0%

         \[\leadsto {\left(\color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      10. remove-double-neg 55.0%

          \[\leadsto {\left(\left(\color{blue}{x} + \left(-y\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      11. sub-neg 55.0%

          \[\leadsto {\left(\color{blue}{\left(x - y\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      12. sub-neg 55.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(y + \left(-x\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      13. +-commutative 55.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(\left(-x\right) + y\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      14. distribute-neg-in 55.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      15. remove-double-neg 55.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(\color{blue}{x} + \left(-y\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      16. sub-neg 55.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(x - y\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      17. metadata-eval 55.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{\color{blue}{0.25}} \]

   5. Applied egg-rr 55.0%

      \[\leadsto \color{blue}{{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}} \]
   6. Step-by-step derivation

      1. unpow-prod-down 99.1%

         \[\leadsto \color{blue}{{\left(x - y\right)}^{0.25} \cdot {\left(x - y\right)}^{0.25}} \]

      2. pow-sqr 100.0%

         \[\leadsto \color{blue}{{\left(x - y\right)}^{\left(2 \cdot 0.25\right)}} \]

      3. metadata-eval 100.0%

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

      4. unpow1/2 100.0%

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - y \leq -1 \cdot 10^{+155}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{elif}\;x - y \leq -1 \cdot 10^{-189}:\\ \;\;\;\;{\left(y \cdot \left(y - x\right) + x \cdot \left(x - y\right)\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x - y}\\ \end{array} \]

Alternative 4: 86.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - y \leq -1 \cdot 10^{+155}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{elif}\;x - y \leq -1 \cdot 10^{-189}:\\ \;\;\;\;{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- x y) -1e+155)
   (sqrt y)
   (if (<= (- x y) -1e-189) (pow (* (- x y) (- x y)) 0.25) (sqrt (- x y)))))

C

double code(double x, double y) {
	double tmp;
	if ((x - y) <= -1e+155) {
		tmp = sqrt(y);
	} else if ((x - y) <= -1e-189) {
		tmp = pow(((x - y) * (x - y)), 0.25);
	} 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) <= (-1d+155)) then
        tmp = sqrt(y)
    else if ((x - y) <= (-1d-189)) then
        tmp = ((x - y) * (x - y)) ** 0.25d0
    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) <= -1e+155) {
		tmp = Math.sqrt(y);
	} else if ((x - y) <= -1e-189) {
		tmp = Math.pow(((x - y) * (x - y)), 0.25);
	} else {
		tmp = Math.sqrt((x - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x - y) <= -1e+155:
		tmp = math.sqrt(y)
	elif (x - y) <= -1e-189:
		tmp = math.pow(((x - y) * (x - y)), 0.25)
	else:
		tmp = math.sqrt((x - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x - y) <= -1e+155)
		tmp = sqrt(y);
	elseif (Float64(x - y) <= -1e-189)
		tmp = Float64(Float64(x - y) * Float64(x - y)) ^ 0.25;
	else
		tmp = sqrt(Float64(x - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x - y) <= -1e+155)
		tmp = sqrt(y);
	elseif ((x - y) <= -1e-189)
		tmp = ((x - y) * (x - y)) ^ 0.25;
	else
		tmp = sqrt((x - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x - y), $MachinePrecision], -1e+155], N[Sqrt[y], $MachinePrecision], If[LessEqual[N[(x - y), $MachinePrecision], -1e-189], N[Power[N[(N[(x - y), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], N[Sqrt[N[(x - y), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x y) < -1.00000000000000001e155

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

   3. Simplified 100.0%

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

      1. pow1/2 100.0%

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

      2. sqr-pow 99.3%

         \[\leadsto \color{blue}{{\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      3. pow-prod-down 4.3%

         \[\leadsto \color{blue}{{\left(\left|y - x\right| \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      4. neg-fabs 4.3%

         \[\leadsto {\left(\color{blue}{\left|-\left(y - x\right)\right|} \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \]

      5. neg-fabs 4.3%

         \[\leadsto {\left(\left|-\left(y - x\right)\right| \cdot \color{blue}{\left|-\left(y - x\right)\right|}\right)}^{\left(\frac{0.5}{2}\right)} \]

      6. sqr-abs 4.3%

         \[\leadsto {\color{blue}{\left(\left(-\left(y - x\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)} \]

      7. sub-neg 4.3%

         \[\leadsto {\left(\left(-\color{blue}{\left(y + \left(-x\right)\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      8. +-commutative 4.3%

         \[\leadsto {\left(\left(-\color{blue}{\left(\left(-x\right) + y\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      9. distribute-neg-in 4.3%

         \[\leadsto {\left(\color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      10. remove-double-neg 4.3%

          \[\leadsto {\left(\left(\color{blue}{x} + \left(-y\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      11. sub-neg 4.3%

          \[\leadsto {\left(\color{blue}{\left(x - y\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      12. sub-neg 4.3%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(y + \left(-x\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      13. +-commutative 4.3%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(\left(-x\right) + y\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      14. distribute-neg-in 4.3%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      15. remove-double-neg 4.3%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(\color{blue}{x} + \left(-y\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      16. sub-neg 4.3%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(x - y\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      17. metadata-eval 4.3%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{\color{blue}{0.25}} \]

   5. Applied egg-rr 4.3%

      \[\leadsto \color{blue}{{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}} \]

   6. Taylor expanded in x around 0 57.1%

      \[\leadsto \color{blue}{{1}^{0.25} \cdot \sqrt{y}} \]
   7. Step-by-step derivation

      1. pow-base-1 57.1%

         \[\leadsto \color{blue}{1} \cdot \sqrt{y} \]

      2. *-lft-identity 57.1%

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

   8. Simplified 57.1%

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

   if -1.00000000000000001e155 < (-.f64 x y) < -1.00000000000000007e-189

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

   3. Simplified 100.0%

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

      1. pow1/2 100.0%

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

      2. sqr-pow 99.2%

         \[\leadsto \color{blue}{{\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      3. pow-prod-down 94.8%

         \[\leadsto \color{blue}{{\left(\left|y - x\right| \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      4. neg-fabs 94.8%

         \[\leadsto {\left(\color{blue}{\left|-\left(y - x\right)\right|} \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \]

      5. neg-fabs 94.8%

         \[\leadsto {\left(\left|-\left(y - x\right)\right| \cdot \color{blue}{\left|-\left(y - x\right)\right|}\right)}^{\left(\frac{0.5}{2}\right)} \]

      6. sqr-abs 94.8%

         \[\leadsto {\color{blue}{\left(\left(-\left(y - x\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)} \]

      7. sub-neg 94.8%

         \[\leadsto {\left(\left(-\color{blue}{\left(y + \left(-x\right)\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      8. +-commutative 94.8%

         \[\leadsto {\left(\left(-\color{blue}{\left(\left(-x\right) + y\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      9. distribute-neg-in 94.8%

         \[\leadsto {\left(\color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      10. remove-double-neg 94.8%

          \[\leadsto {\left(\left(\color{blue}{x} + \left(-y\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      11. sub-neg 94.8%

          \[\leadsto {\left(\color{blue}{\left(x - y\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      12. sub-neg 94.8%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(y + \left(-x\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      13. +-commutative 94.8%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(\left(-x\right) + y\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      14. distribute-neg-in 94.8%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      15. remove-double-neg 94.8%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(\color{blue}{x} + \left(-y\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      16. sub-neg 94.8%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(x - y\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      17. metadata-eval 94.8%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{\color{blue}{0.25}} \]

   5. Applied egg-rr 94.8%

      \[\leadsto \color{blue}{{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}} \]

   if -1.00000000000000007e-189 < (-.f64 x y)

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

   3. Simplified 100.0%

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

      1. pow1/2 100.0%

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

      2. sqr-pow 99.1%

         \[\leadsto \color{blue}{{\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      3. pow-prod-down 55.0%

         \[\leadsto \color{blue}{{\left(\left|y - x\right| \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      4. neg-fabs 55.0%

         \[\leadsto {\left(\color{blue}{\left|-\left(y - x\right)\right|} \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \]

      5. neg-fabs 55.0%

         \[\leadsto {\left(\left|-\left(y - x\right)\right| \cdot \color{blue}{\left|-\left(y - x\right)\right|}\right)}^{\left(\frac{0.5}{2}\right)} \]

      6. sqr-abs 55.0%

         \[\leadsto {\color{blue}{\left(\left(-\left(y - x\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)} \]

      7. sub-neg 55.0%

         \[\leadsto {\left(\left(-\color{blue}{\left(y + \left(-x\right)\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      8. +-commutative 55.0%

         \[\leadsto {\left(\left(-\color{blue}{\left(\left(-x\right) + y\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      9. distribute-neg-in 55.0%

         \[\leadsto {\left(\color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      10. remove-double-neg 55.0%

          \[\leadsto {\left(\left(\color{blue}{x} + \left(-y\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      11. sub-neg 55.0%

          \[\leadsto {\left(\color{blue}{\left(x - y\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      12. sub-neg 55.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(y + \left(-x\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      13. +-commutative 55.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(\left(-x\right) + y\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      14. distribute-neg-in 55.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      15. remove-double-neg 55.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(\color{blue}{x} + \left(-y\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      16. sub-neg 55.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(x - y\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      17. metadata-eval 55.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{\color{blue}{0.25}} \]

   5. Applied egg-rr 55.0%

      \[\leadsto \color{blue}{{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}} \]
   6. Step-by-step derivation

      1. unpow-prod-down 99.1%

         \[\leadsto \color{blue}{{\left(x - y\right)}^{0.25} \cdot {\left(x - y\right)}^{0.25}} \]

      2. pow-sqr 100.0%

         \[\leadsto \color{blue}{{\left(x - y\right)}^{\left(2 \cdot 0.25\right)}} \]

      3. metadata-eval 100.0%

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

      4. unpow1/2 100.0%

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - y \leq -1 \cdot 10^{+155}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{elif}\;x - y \leq -1 \cdot 10^{-189}:\\ \;\;\;\;{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x - y}\\ \end{array} \]

Alternative 5: 74.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - y \leq -6 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{elif}\;x - y \leq -500000:\\ \;\;\;\;{\left(x \cdot x\right)}^{0.25}\\ \mathbf{elif}\;x - y \leq -2 \cdot 10^{-183}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- x y) -6e+147)
   (sqrt y)
   (if (<= (- x y) -500000.0)
     (pow (* x x) 0.25)
     (if (<= (- x y) -2e-183) (sqrt y) (sqrt (- x y))))))

C

double code(double x, double y) {
	double tmp;
	if ((x - y) <= -6e+147) {
		tmp = sqrt(y);
	} else if ((x - y) <= -500000.0) {
		tmp = pow((x * x), 0.25);
	} else if ((x - y) <= -2e-183) {
		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) <= (-6d+147)) then
        tmp = sqrt(y)
    else if ((x - y) <= (-500000.0d0)) then
        tmp = (x * x) ** 0.25d0
    else if ((x - y) <= (-2d-183)) 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) <= -6e+147) {
		tmp = Math.sqrt(y);
	} else if ((x - y) <= -500000.0) {
		tmp = Math.pow((x * x), 0.25);
	} else if ((x - y) <= -2e-183) {
		tmp = Math.sqrt(y);
	} else {
		tmp = Math.sqrt((x - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x - y) <= -6e+147:
		tmp = math.sqrt(y)
	elif (x - y) <= -500000.0:
		tmp = math.pow((x * x), 0.25)
	elif (x - y) <= -2e-183:
		tmp = math.sqrt(y)
	else:
		tmp = math.sqrt((x - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x - y) <= -6e+147)
		tmp = sqrt(y);
	elseif (Float64(x - y) <= -500000.0)
		tmp = Float64(x * x) ^ 0.25;
	elseif (Float64(x - y) <= -2e-183)
		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) <= -6e+147)
		tmp = sqrt(y);
	elseif ((x - y) <= -500000.0)
		tmp = (x * x) ^ 0.25;
	elseif ((x - y) <= -2e-183)
		tmp = sqrt(y);
	else
		tmp = sqrt((x - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x - y), $MachinePrecision], -6e+147], N[Sqrt[y], $MachinePrecision], If[LessEqual[N[(x - y), $MachinePrecision], -500000.0], N[Power[N[(x * x), $MachinePrecision], 0.25], $MachinePrecision], If[LessEqual[N[(x - y), $MachinePrecision], -2e-183], N[Sqrt[y], $MachinePrecision], N[Sqrt[N[(x - y), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x y) < -5.99999999999999987e147 or -5e5 < (-.f64 x y) < -2.00000000000000001e-183

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

   3. Simplified 100.0%

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

      1. pow1/2 100.0%

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

      2. sqr-pow 99.3%

         \[\leadsto \color{blue}{{\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      3. pow-prod-down 32.6%

         \[\leadsto \color{blue}{{\left(\left|y - x\right| \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      4. neg-fabs 32.6%

         \[\leadsto {\left(\color{blue}{\left|-\left(y - x\right)\right|} \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \]

      5. neg-fabs 32.6%

         \[\leadsto {\left(\left|-\left(y - x\right)\right| \cdot \color{blue}{\left|-\left(y - x\right)\right|}\right)}^{\left(\frac{0.5}{2}\right)} \]

      6. sqr-abs 32.6%

         \[\leadsto {\color{blue}{\left(\left(-\left(y - x\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)} \]

      7. sub-neg 32.6%

         \[\leadsto {\left(\left(-\color{blue}{\left(y + \left(-x\right)\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      8. +-commutative 32.6%

         \[\leadsto {\left(\left(-\color{blue}{\left(\left(-x\right) + y\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      9. distribute-neg-in 32.6%

         \[\leadsto {\left(\color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      10. remove-double-neg 32.6%

          \[\leadsto {\left(\left(\color{blue}{x} + \left(-y\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      11. sub-neg 32.6%

          \[\leadsto {\left(\color{blue}{\left(x - y\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      12. sub-neg 32.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(y + \left(-x\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      13. +-commutative 32.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(\left(-x\right) + y\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      14. distribute-neg-in 32.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      15. remove-double-neg 32.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(\color{blue}{x} + \left(-y\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      16. sub-neg 32.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(x - y\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      17. metadata-eval 32.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{\color{blue}{0.25}} \]

   5. Applied egg-rr 32.6%

      \[\leadsto \color{blue}{{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}} \]

   6. Taylor expanded in x around 0 57.5%

      \[\leadsto \color{blue}{{1}^{0.25} \cdot \sqrt{y}} \]
   7. Step-by-step derivation

      1. pow-base-1 57.5%

         \[\leadsto \color{blue}{1} \cdot \sqrt{y} \]

      2. *-lft-identity 57.5%

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

   8. Simplified 57.5%

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

   if -5.99999999999999987e147 < (-.f64 x y) < -5e5

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

   3. Simplified 100.0%

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

      1. pow1/2 100.0%

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

      2. sqr-pow 99.1%

         \[\leadsto \color{blue}{{\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      3. pow-prod-down 100.0%

         \[\leadsto \color{blue}{{\left(\left|y - x\right| \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      4. neg-fabs 100.0%

         \[\leadsto {\left(\color{blue}{\left|-\left(y - x\right)\right|} \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \]

      5. neg-fabs 100.0%

         \[\leadsto {\left(\left|-\left(y - x\right)\right| \cdot \color{blue}{\left|-\left(y - x\right)\right|}\right)}^{\left(\frac{0.5}{2}\right)} \]

      6. sqr-abs 100.0%

         \[\leadsto {\color{blue}{\left(\left(-\left(y - x\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)} \]

      7. sub-neg 100.0%

         \[\leadsto {\left(\left(-\color{blue}{\left(y + \left(-x\right)\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      8. +-commutative 100.0%

         \[\leadsto {\left(\left(-\color{blue}{\left(\left(-x\right) + y\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      9. distribute-neg-in 100.0%

         \[\leadsto {\left(\color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      10. remove-double-neg 100.0%

          \[\leadsto {\left(\left(\color{blue}{x} + \left(-y\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      11. sub-neg 100.0%

          \[\leadsto {\left(\color{blue}{\left(x - y\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      12. sub-neg 100.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(y + \left(-x\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      13. +-commutative 100.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(\left(-x\right) + y\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      14. distribute-neg-in 100.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      15. remove-double-neg 100.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(\color{blue}{x} + \left(-y\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      16. sub-neg 100.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(x - y\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      17. metadata-eval 100.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{\color{blue}{0.25}} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}} \]

   6. Taylor expanded in x around inf 63.1%

      \[\leadsto {\color{blue}{\left({x}^{2}\right)}}^{0.25} \]
   7. Step-by-step derivation

      1. unpow2 63.1%

         \[\leadsto {\color{blue}{\left(x \cdot x\right)}}^{0.25} \]

   8. Simplified 63.1%

      \[\leadsto {\color{blue}{\left(x \cdot x\right)}}^{0.25} \]

   if -2.00000000000000001e-183 < (-.f64 x y)

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

   3. Simplified 100.0%

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

      1. pow1/2 100.0%

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

      2. sqr-pow 99.1%

         \[\leadsto \color{blue}{{\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      3. pow-prod-down 54.6%

         \[\leadsto \color{blue}{{\left(\left|y - x\right| \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      4. neg-fabs 54.6%

         \[\leadsto {\left(\color{blue}{\left|-\left(y - x\right)\right|} \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \]

      5. neg-fabs 54.6%

         \[\leadsto {\left(\left|-\left(y - x\right)\right| \cdot \color{blue}{\left|-\left(y - x\right)\right|}\right)}^{\left(\frac{0.5}{2}\right)} \]

      6. sqr-abs 54.6%

         \[\leadsto {\color{blue}{\left(\left(-\left(y - x\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)} \]

      7. sub-neg 54.6%

         \[\leadsto {\left(\left(-\color{blue}{\left(y + \left(-x\right)\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      8. +-commutative 54.6%

         \[\leadsto {\left(\left(-\color{blue}{\left(\left(-x\right) + y\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      9. distribute-neg-in 54.6%

         \[\leadsto {\left(\color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      10. remove-double-neg 54.6%

          \[\leadsto {\left(\left(\color{blue}{x} + \left(-y\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      11. sub-neg 54.6%

          \[\leadsto {\left(\color{blue}{\left(x - y\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      12. sub-neg 54.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(y + \left(-x\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      13. +-commutative 54.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(\left(-x\right) + y\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      14. distribute-neg-in 54.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      15. remove-double-neg 54.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(\color{blue}{x} + \left(-y\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      16. sub-neg 54.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(x - y\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      17. metadata-eval 54.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{\color{blue}{0.25}} \]

   5. Applied egg-rr 54.6%

      \[\leadsto \color{blue}{{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}} \]
   6. Step-by-step derivation

      1. unpow-prod-down 98.3%

         \[\leadsto \color{blue}{{\left(x - y\right)}^{0.25} \cdot {\left(x - y\right)}^{0.25}} \]

      2. pow-sqr 99.2%

         \[\leadsto \color{blue}{{\left(x - y\right)}^{\left(2 \cdot 0.25\right)}} \]

      3. metadata-eval 99.2%

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

      4. unpow1/2 99.2%

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

   7. Applied egg-rr 99.2%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - y \leq -6 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{elif}\;x - y \leq -500000:\\ \;\;\;\;{\left(x \cdot x\right)}^{0.25}\\ \mathbf{elif}\;x - y \leq -2 \cdot 10^{-183}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x - y}\\ \end{array} \]

Alternative 6: 74.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - y \leq -2 \cdot 10^{-183}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- x y) -2e-183) (sqrt y) (sqrt (- x y))))

C

double code(double x, double y) {
	double tmp;
	if ((x - y) <= -2e-183) {
		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-183)) 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-183) {
		tmp = Math.sqrt(y);
	} else {
		tmp = Math.sqrt((x - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x - y) <= -2e-183:
		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-183)
		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-183)
		tmp = sqrt(y);
	else
		tmp = sqrt((x - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x - y), $MachinePrecision], -2e-183], N[Sqrt[y], $MachinePrecision], N[Sqrt[N[(x - y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x y) < -2.00000000000000001e-183

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

   3. Simplified 100.0%

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

      1. pow1/2 100.0%

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

      2. sqr-pow 99.2%

         \[\leadsto \color{blue}{{\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      3. pow-prod-down 54.2%

         \[\leadsto \color{blue}{{\left(\left|y - x\right| \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      4. neg-fabs 54.2%

         \[\leadsto {\left(\color{blue}{\left|-\left(y - x\right)\right|} \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \]

      5. neg-fabs 54.2%

         \[\leadsto {\left(\left|-\left(y - x\right)\right| \cdot \color{blue}{\left|-\left(y - x\right)\right|}\right)}^{\left(\frac{0.5}{2}\right)} \]

      6. sqr-abs 54.2%

         \[\leadsto {\color{blue}{\left(\left(-\left(y - x\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)} \]

      7. sub-neg 54.2%

         \[\leadsto {\left(\left(-\color{blue}{\left(y + \left(-x\right)\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      8. +-commutative 54.2%

         \[\leadsto {\left(\left(-\color{blue}{\left(\left(-x\right) + y\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      9. distribute-neg-in 54.2%

         \[\leadsto {\left(\color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      10. remove-double-neg 54.2%

          \[\leadsto {\left(\left(\color{blue}{x} + \left(-y\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      11. sub-neg 54.2%

          \[\leadsto {\left(\color{blue}{\left(x - y\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      12. sub-neg 54.2%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(y + \left(-x\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      13. +-commutative 54.2%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(\left(-x\right) + y\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      14. distribute-neg-in 54.2%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      15. remove-double-neg 54.2%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(\color{blue}{x} + \left(-y\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      16. sub-neg 54.2%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(x - y\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      17. metadata-eval 54.2%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{\color{blue}{0.25}} \]

   5. Applied egg-rr 54.2%

      \[\leadsto \color{blue}{{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}} \]

   6. Taylor expanded in x around 0 51.3%

      \[\leadsto \color{blue}{{1}^{0.25} \cdot \sqrt{y}} \]
   7. Step-by-step derivation

      1. pow-base-1 51.3%

         \[\leadsto \color{blue}{1} \cdot \sqrt{y} \]

      2. *-lft-identity 51.3%

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

   8. Simplified 51.3%

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

   if -2.00000000000000001e-183 < (-.f64 x y)

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

   3. Simplified 100.0%

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

      1. pow1/2 100.0%

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

      2. sqr-pow 99.1%

         \[\leadsto \color{blue}{{\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      3. pow-prod-down 54.6%

         \[\leadsto \color{blue}{{\left(\left|y - x\right| \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      4. neg-fabs 54.6%

         \[\leadsto {\left(\color{blue}{\left|-\left(y - x\right)\right|} \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \]

      5. neg-fabs 54.6%

         \[\leadsto {\left(\left|-\left(y - x\right)\right| \cdot \color{blue}{\left|-\left(y - x\right)\right|}\right)}^{\left(\frac{0.5}{2}\right)} \]

      6. sqr-abs 54.6%

         \[\leadsto {\color{blue}{\left(\left(-\left(y - x\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)} \]

      7. sub-neg 54.6%

         \[\leadsto {\left(\left(-\color{blue}{\left(y + \left(-x\right)\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      8. +-commutative 54.6%

         \[\leadsto {\left(\left(-\color{blue}{\left(\left(-x\right) + y\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      9. distribute-neg-in 54.6%

         \[\leadsto {\left(\color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      10. remove-double-neg 54.6%

          \[\leadsto {\left(\left(\color{blue}{x} + \left(-y\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      11. sub-neg 54.6%

          \[\leadsto {\left(\color{blue}{\left(x - y\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      12. sub-neg 54.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(y + \left(-x\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      13. +-commutative 54.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(\left(-x\right) + y\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      14. distribute-neg-in 54.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      15. remove-double-neg 54.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(\color{blue}{x} + \left(-y\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      16. sub-neg 54.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(x - y\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      17. metadata-eval 54.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{\color{blue}{0.25}} \]

   5. Applied egg-rr 54.6%

      \[\leadsto \color{blue}{{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}} \]
   6. Step-by-step derivation

      1. unpow-prod-down 98.3%

         \[\leadsto \color{blue}{{\left(x - y\right)}^{0.25} \cdot {\left(x - y\right)}^{0.25}} \]

      2. pow-sqr 99.2%

         \[\leadsto \color{blue}{{\left(x - y\right)}^{\left(2 \cdot 0.25\right)}} \]

      3. metadata-eval 99.2%

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

      4. unpow1/2 99.2%

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

   7. Applied egg-rr 99.2%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - y \leq -2 \cdot 10^{-183}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x - y}\\ \end{array} \]

Alternative 7: 29.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 2.4e-164) 1.0 (sqrt x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.4e-164) {
		tmp = 1.0;
	} else {
		tmp = sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2.4d-164) then
        tmp = 1.0d0
    else
        tmp = sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.4e-164) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.4e-164:
		tmp = 1.0
	else:
		tmp = math.sqrt(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.4e-164)
		tmp = 1.0;
	else
		tmp = sqrt(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.4e-164)
		tmp = 1.0;
	else
		tmp = sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.4e-164], 1.0, N[Sqrt[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999983e-164

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

   3. Simplified 100.0%

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

      1. pow1/2 100.0%

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

      2. sqr-pow 99.2%

         \[\leadsto \color{blue}{{\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      3. pow-prod-down 52.4%

         \[\leadsto \color{blue}{{\left(\left|y - x\right| \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      4. neg-fabs 52.4%

         \[\leadsto {\left(\color{blue}{\left|-\left(y - x\right)\right|} \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \]

      5. neg-fabs 52.4%

         \[\leadsto {\left(\left|-\left(y - x\right)\right| \cdot \color{blue}{\left|-\left(y - x\right)\right|}\right)}^{\left(\frac{0.5}{2}\right)} \]

      6. sqr-abs 52.4%

         \[\leadsto {\color{blue}{\left(\left(-\left(y - x\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)} \]

      7. sub-neg 52.4%

         \[\leadsto {\left(\left(-\color{blue}{\left(y + \left(-x\right)\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      8. +-commutative 52.4%

         \[\leadsto {\left(\left(-\color{blue}{\left(\left(-x\right) + y\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      9. distribute-neg-in 52.4%

         \[\leadsto {\left(\color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      10. remove-double-neg 52.4%

          \[\leadsto {\left(\left(\color{blue}{x} + \left(-y\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      11. sub-neg 52.4%

          \[\leadsto {\left(\color{blue}{\left(x - y\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      12. sub-neg 52.4%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(y + \left(-x\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      13. +-commutative 52.4%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(\left(-x\right) + y\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      14. distribute-neg-in 52.4%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      15. remove-double-neg 52.4%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(\color{blue}{x} + \left(-y\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      16. sub-neg 52.4%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(x - y\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      17. metadata-eval 52.4%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{\color{blue}{0.25}} \]

   5. Applied egg-rr 52.4%

      \[\leadsto \color{blue}{{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}} \]

   6. Taylor expanded in x around inf 25.2%

      \[\leadsto {\color{blue}{\left({x}^{2}\right)}}^{0.25} \]
   7. Step-by-step derivation

      1. unpow2 25.2%

         \[\leadsto {\color{blue}{\left(x \cdot x\right)}}^{0.25} \]

   8. Simplified 25.2%

      \[\leadsto {\color{blue}{\left(x \cdot x\right)}}^{0.25} \]

   10. Applied egg-rr 6.9%

       \[\leadsto \color{blue}{\frac{-x}{-x}} \]

   12. Simplified 6.9%

       \[\leadsto \color{blue}{1} \]

   if 2.39999999999999983e-164 < x

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

   3. Simplified 100.0%

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

      1. pow1/2 100.0%

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

      2. sqr-pow 99.2%

         \[\leadsto \color{blue}{{\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      3. pow-prod-down 58.1%

         \[\leadsto \color{blue}{{\left(\left|y - x\right| \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      4. neg-fabs 58.1%

         \[\leadsto {\left(\color{blue}{\left|-\left(y - x\right)\right|} \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \]

      5. neg-fabs 58.1%

         \[\leadsto {\left(\left|-\left(y - x\right)\right| \cdot \color{blue}{\left|-\left(y - x\right)\right|}\right)}^{\left(\frac{0.5}{2}\right)} \]

      6. sqr-abs 58.1%

         \[\leadsto {\color{blue}{\left(\left(-\left(y - x\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)} \]

      7. sub-neg 58.1%

         \[\leadsto {\left(\left(-\color{blue}{\left(y + \left(-x\right)\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      8. +-commutative 58.1%

         \[\leadsto {\left(\left(-\color{blue}{\left(\left(-x\right) + y\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      9. distribute-neg-in 58.1%

         \[\leadsto {\left(\color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      10. remove-double-neg 58.1%

          \[\leadsto {\left(\left(\color{blue}{x} + \left(-y\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      11. sub-neg 58.1%

          \[\leadsto {\left(\color{blue}{\left(x - y\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      12. sub-neg 58.1%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(y + \left(-x\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      13. +-commutative 58.1%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(\left(-x\right) + y\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      14. distribute-neg-in 58.1%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      15. remove-double-neg 58.1%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(\color{blue}{x} + \left(-y\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      16. sub-neg 58.1%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(x - y\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      17. metadata-eval 58.1%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{\color{blue}{0.25}} \]

   5. Applied egg-rr 58.1%

      \[\leadsto \color{blue}{{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}} \]

   6. Taylor expanded in x around inf 67.4%

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

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \]

Alternative 8: 42.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 3.2e-156) (sqrt x) (sqrt y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-156) {
		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 <= 3.2d-156) 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 <= 3.2e-156) {
		tmp = Math.sqrt(x);
	} else {
		tmp = Math.sqrt(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.2e-156:
		tmp = math.sqrt(x)
	else:
		tmp = math.sqrt(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.2e-156)
		tmp = sqrt(x);
	else
		tmp = sqrt(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.2e-156)
		tmp = sqrt(x);
	else
		tmp = sqrt(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.2e-156], N[Sqrt[x], $MachinePrecision], N[Sqrt[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.19999999999999982e-156

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

   3. Simplified 100.0%

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

      1. pow1/2 100.0%

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

      2. sqr-pow 99.1%

         \[\leadsto \color{blue}{{\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      3. pow-prod-down 59.0%

         \[\leadsto \color{blue}{{\left(\left|y - x\right| \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      4. neg-fabs 59.0%

         \[\leadsto {\left(\color{blue}{\left|-\left(y - x\right)\right|} \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \]

      5. neg-fabs 59.0%

         \[\leadsto {\left(\left|-\left(y - x\right)\right| \cdot \color{blue}{\left|-\left(y - x\right)\right|}\right)}^{\left(\frac{0.5}{2}\right)} \]

      6. sqr-abs 59.0%

         \[\leadsto {\color{blue}{\left(\left(-\left(y - x\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)} \]

      7. sub-neg 59.0%

         \[\leadsto {\left(\left(-\color{blue}{\left(y + \left(-x\right)\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      8. +-commutative 59.0%

         \[\leadsto {\left(\left(-\color{blue}{\left(\left(-x\right) + y\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      9. distribute-neg-in 59.0%

         \[\leadsto {\left(\color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      10. remove-double-neg 59.0%

          \[\leadsto {\left(\left(\color{blue}{x} + \left(-y\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      11. sub-neg 59.0%

          \[\leadsto {\left(\color{blue}{\left(x - y\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      12. sub-neg 59.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(y + \left(-x\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      13. +-commutative 59.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(\left(-x\right) + y\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      14. distribute-neg-in 59.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      15. remove-double-neg 59.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(\color{blue}{x} + \left(-y\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      16. sub-neg 59.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(x - y\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      17. metadata-eval 59.0%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{\color{blue}{0.25}} \]

   5. Applied egg-rr 59.0%

      \[\leadsto \color{blue}{{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}} \]

   6. Taylor expanded in x around inf 32.0%

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

   if 3.19999999999999982e-156 < y

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

   3. Simplified 100.0%

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

      1. pow1/2 100.0%

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

      2. sqr-pow 99.3%

         \[\leadsto \color{blue}{{\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      3. pow-prod-down 45.6%

         \[\leadsto \color{blue}{{\left(\left|y - x\right| \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

      4. neg-fabs 45.6%

         \[\leadsto {\left(\color{blue}{\left|-\left(y - x\right)\right|} \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \]

      5. neg-fabs 45.6%

         \[\leadsto {\left(\left|-\left(y - x\right)\right| \cdot \color{blue}{\left|-\left(y - x\right)\right|}\right)}^{\left(\frac{0.5}{2}\right)} \]

      6. sqr-abs 45.6%

         \[\leadsto {\color{blue}{\left(\left(-\left(y - x\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)} \]

      7. sub-neg 45.6%

         \[\leadsto {\left(\left(-\color{blue}{\left(y + \left(-x\right)\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      8. +-commutative 45.6%

         \[\leadsto {\left(\left(-\color{blue}{\left(\left(-x\right) + y\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      9. distribute-neg-in 45.6%

         \[\leadsto {\left(\color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      10. remove-double-neg 45.6%

          \[\leadsto {\left(\left(\color{blue}{x} + \left(-y\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      11. sub-neg 45.6%

          \[\leadsto {\left(\color{blue}{\left(x - y\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      12. sub-neg 45.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(y + \left(-x\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      13. +-commutative 45.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(\left(-x\right) + y\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      14. distribute-neg-in 45.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      15. remove-double-neg 45.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(\color{blue}{x} + \left(-y\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      16. sub-neg 45.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(x - y\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      17. metadata-eval 45.6%

          \[\leadsto {\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{\color{blue}{0.25}} \]

   5. Applied egg-rr 45.6%

      \[\leadsto \color{blue}{{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}} \]

   6. Taylor expanded in x around 0 75.8%

      \[\leadsto \color{blue}{{1}^{0.25} \cdot \sqrt{y}} \]
   7. Step-by-step derivation

      1. pow-base-1 75.8%

         \[\leadsto \color{blue}{1} \cdot \sqrt{y} \]

      2. *-lft-identity 75.8%

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

   8. Simplified 75.8%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \]

Alternative 9: 6.9% accurate, 203.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

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

   1. fabs-sub 100.0%

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

3. Simplified 100.0%

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

   1. pow1/2 100.0%

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

   2. sqr-pow 99.2%

      \[\leadsto \color{blue}{{\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

   3. pow-prod-down 54.4%

      \[\leadsto \color{blue}{{\left(\left|y - x\right| \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)}} \]

   4. neg-fabs 54.4%

      \[\leadsto {\left(\color{blue}{\left|-\left(y - x\right)\right|} \cdot \left|y - x\right|\right)}^{\left(\frac{0.5}{2}\right)} \]

   5. neg-fabs 54.4%

      \[\leadsto {\left(\left|-\left(y - x\right)\right| \cdot \color{blue}{\left|-\left(y - x\right)\right|}\right)}^{\left(\frac{0.5}{2}\right)} \]

   6. sqr-abs 54.4%

      \[\leadsto {\color{blue}{\left(\left(-\left(y - x\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)} \]

   7. sub-neg 54.4%

      \[\leadsto {\left(\left(-\color{blue}{\left(y + \left(-x\right)\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

   8. +-commutative 54.4%

      \[\leadsto {\left(\left(-\color{blue}{\left(\left(-x\right) + y\right)}\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

   9. distribute-neg-in 54.4%

      \[\leadsto {\left(\color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

   10. remove-double-neg 54.4%

       \[\leadsto {\left(\left(\color{blue}{x} + \left(-y\right)\right) \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

   11. sub-neg 54.4%

       \[\leadsto {\left(\color{blue}{\left(x - y\right)} \cdot \left(-\left(y - x\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

   12. sub-neg 54.4%

       \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(y + \left(-x\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

   13. +-commutative 54.4%

       \[\leadsto {\left(\left(x - y\right) \cdot \left(-\color{blue}{\left(\left(-x\right) + y\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

   14. distribute-neg-in 54.4%

       \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

   15. remove-double-neg 54.4%

       \[\leadsto {\left(\left(x - y\right) \cdot \left(\color{blue}{x} + \left(-y\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

   16. sub-neg 54.4%

       \[\leadsto {\left(\left(x - y\right) \cdot \color{blue}{\left(x - y\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

   17. metadata-eval 54.4%

       \[\leadsto {\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{\color{blue}{0.25}} \]

5. Applied egg-rr 54.4%

   \[\leadsto \color{blue}{{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}} \]

6. Taylor expanded in x around inf 31.3%

   \[\leadsto {\color{blue}{\left({x}^{2}\right)}}^{0.25} \]
7. Step-by-step derivation

   1. unpow2 31.3%

      \[\leadsto {\color{blue}{\left(x \cdot x\right)}}^{0.25} \]

8. Simplified 31.3%

   \[\leadsto {\color{blue}{\left(x \cdot x\right)}}^{0.25} \]

10. Applied egg-rr 7.0%

    \[\leadsto \color{blue}{\frac{-x}{-x}} \]

12. Simplified 7.0%

    \[\leadsto \color{blue}{1} \]

13. Final simplification 7.0%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023297 
(FPCore (x y)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, C"
  :precision binary64
  (sqrt (fabs (- x y))))