Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3

Percentage Accurate: 99.9% → 99.9%

Time: 5.0s

Alternatives: 10

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ x + \frac{\left|y - x\right|}{2} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (/ (fabs (- y x)) 2.0)))

C

double code(double x, double y) {
	return x + (fabs((y - x)) / 2.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + (abs((y - x)) / 2.0d0)
end function

Java

public static double code(double x, double y) {
	return x + (Math.abs((y - x)) / 2.0);
}

Python

def code(x, y):
	return x + (math.fabs((y - x)) / 2.0)

Julia

function code(x, y)
	return Float64(x + Float64(abs(Float64(y - x)) / 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = x + (abs((y - x)) / 2.0);
end

Wolfram

code[x_, y_] := N[(x + N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{\left|y - x\right|}{2} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (/ (fabs (- y x)) 2.0)))

C

double code(double x, double y) {
	return x + (fabs((y - x)) / 2.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + (abs((y - x)) / 2.0d0)
end function

Java

public static double code(double x, double y) {
	return x + (Math.abs((y - x)) / 2.0);
}

Python

def code(x, y):
	return x + (math.fabs((y - x)) / 2.0)

Julia

function code(x, y)
	return Float64(x + Float64(abs(Float64(y - x)) / 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = x + (abs((y - x)) / 2.0);
end

Wolfram

code[x_, y_] := N[(x + N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{\left|y - x\right|}{2} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (/ (fabs (- y x)) 2.0)))

C

double code(double x, double y) {
	return x + (fabs((y - x)) / 2.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + (abs((y - x)) / 2.0d0)
end function

Java

public static double code(double x, double y) {
	return x + (Math.abs((y - x)) / 2.0);
}

Python

def code(x, y):
	return x + (math.fabs((y - x)) / 2.0)

Julia

function code(x, y)
	return Float64(x + Float64(abs(Float64(y - x)) / 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = x + (abs((y - x)) / 2.0);
end

Wolfram

code[x_, y_] := N[(x + N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x + \frac{\left|y - x\right|}{2} \]

2. Final simplification 99.9%

   \[\leadsto x + \frac{\left|y - x\right|}{2} \]

Alternative 2: 82.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-72}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-147}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \left(x - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.5e-72)
   (* 0.5 (+ x y))
   (if (<= x 2.8e-147) (* (fabs (- y x)) 0.5) (+ x (* 0.5 (- x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.5e-72) {
		tmp = 0.5 * (x + y);
	} else if (x <= 2.8e-147) {
		tmp = fabs((y - x)) * 0.5;
	} else {
		tmp = x + (0.5 * (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 <= (-5.5d-72)) then
        tmp = 0.5d0 * (x + y)
    else if (x <= 2.8d-147) then
        tmp = abs((y - x)) * 0.5d0
    else
        tmp = x + (0.5d0 * (x - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.5e-72) {
		tmp = 0.5 * (x + y);
	} else if (x <= 2.8e-147) {
		tmp = Math.abs((y - x)) * 0.5;
	} else {
		tmp = x + (0.5 * (x - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.5e-72:
		tmp = 0.5 * (x + y)
	elif x <= 2.8e-147:
		tmp = math.fabs((y - x)) * 0.5
	else:
		tmp = x + (0.5 * (x - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.5e-72)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 2.8e-147)
		tmp = Float64(abs(Float64(y - x)) * 0.5);
	else
		tmp = Float64(x + Float64(0.5 * Float64(x - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.5e-72)
		tmp = 0.5 * (x + y);
	elseif (x <= 2.8e-147)
		tmp = abs((y - x)) * 0.5;
	else
		tmp = x + (0.5 * (x - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.5e-72], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.8e-147], N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(x + N[(0.5 * N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.49999999999999994e-72

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\frac{\left|y - x\right|}{2} + x} \]

      2. div-inv 100.0%

         \[\leadsto \color{blue}{\left|y - x\right| \cdot \frac{1}{2}} + x \]

      3. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left|y - x\right|, \frac{1}{2}, x\right)} \]

      4. add-sqr-sqrt 86.0%

         \[\leadsto \mathsf{fma}\left(\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|, \frac{1}{2}, x\right) \]

      5. fabs-sqr 86.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right) \]

      6. add-sqr-sqrt 86.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{1}{2}, x\right) \]

      7. metadata-eval 86.8%

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

   3. Applied egg-rr 86.8%

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

   4. Taylor expanded in y around 0 86.8%

      \[\leadsto \color{blue}{-0.5 \cdot x + \left(0.5 \cdot y + x\right)} \]
   5. Step-by-step derivation

      1. +-commutative 86.8%

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

      2. associate-+r+ 86.8%

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

      3. distribute-lft1-in 86.8%

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

      4. metadata-eval 86.8%

         \[\leadsto \color{blue}{0.5} \cdot x + 0.5 \cdot y \]

      5. distribute-lft-out 86.8%

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

   6. Simplified 86.8%

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

   if -5.49999999999999994e-72 < x < 2.8e-147

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]

   2. Taylor expanded in x around 0 87.6%

      \[\leadsto \color{blue}{0.5 \cdot \left|y - x\right|} \]

   if 2.8e-147 < x

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 99.2%

         \[\leadsto \color{blue}{\sqrt{x + \frac{\left|y - x\right|}{2}} \cdot \sqrt{x + \frac{\left|y - x\right|}{2}}} \]

      2. pow2 99.2%

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

      3. +-commutative 99.2%

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

      4. div-inv 99.2%

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

      5. fma-def 99.2%

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

      6. add-sqr-sqrt 16.6%

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

      7. fabs-sqr 16.6%

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right)}\right)}^{2} \]

      8. add-sqr-sqrt 28.2%

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

      9. metadata-eval 28.2%

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(y - x, \color{blue}{0.5}, x\right)}\right)}^{2} \]

   3. Applied egg-rr 28.2%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)}\right)}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 28.2%

         \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)} \cdot \sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)}} \]

      2. add-sqr-sqrt 28.4%

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

      3. fma-udef 28.4%

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

      4. add-sqr-sqrt 16.5%

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

      5. sqrt-prod 16.5%

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

      6. associate-*r* 16.5%

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

      7. fma-def 16.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot 0.5} \cdot \sqrt{y - x}, \sqrt{0.5}, x\right)} \]

      8. pow1/2 16.5%

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

      9. pow1/2 16.5%

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

      10. pow-prod-down 61.4%

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

   5. Applied egg-rr 61.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(y - x\right)\right)}^{0.5}, \sqrt{0.5}, x\right)} \]
   6. Step-by-step derivation

      1. unpow1/2 61.4%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(y - x\right)}}, \sqrt{0.5}, x\right) \]

      2. associate-*l* 61.4%

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\left(y - x\right) \cdot \left(0.5 \cdot \left(y - x\right)\right)}}, \sqrt{0.5}, x\right) \]

   7. Simplified 61.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot \left(0.5 \cdot \left(y - x\right)\right)}, \sqrt{0.5}, x\right)} \]

   8. Taylor expanded in y around -inf 83.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot \left(\sqrt{0.5} \cdot y\right) + \sqrt{0.5} \cdot x}, \sqrt{0.5}, x\right) \]
   9. Step-by-step derivation

      1. +-commutative 83.0%

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

      2. mul-1-neg 83.0%

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

      3. unsub-neg 83.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{0.5} \cdot x - \sqrt{0.5} \cdot y}, \sqrt{0.5}, x\right) \]

      4. distribute-lft-out-- 83.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{0.5} \cdot \left(x - y\right)}, \sqrt{0.5}, x\right) \]

   10. Simplified 83.0%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{0.5} \cdot \left(x - y\right)}, \sqrt{0.5}, x\right) \]
   11. Step-by-step derivation

       1. fma-udef 82.9%

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

       2. *-commutative 82.9%

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

       3. associate-*l* 82.7%

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

       4. add-sqr-sqrt 83.3%

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

   12. Applied egg-rr 83.3%

       \[\leadsto \color{blue}{\left(x - y\right) \cdot 0.5 + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-72}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-147}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \left(x - y\right)\\ \end{array} \]

Alternative 3: 82.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-73}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-147}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1.5, x, y \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.2e-73)
   (* 0.5 (+ x y))
   (if (<= x 4.3e-147) (* (fabs (- y x)) 0.5) (fma 1.5 x (* y -0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.2e-73) {
		tmp = 0.5 * (x + y);
	} else if (x <= 4.3e-147) {
		tmp = fabs((y - x)) * 0.5;
	} else {
		tmp = fma(1.5, x, (y * -0.5));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.2e-73)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 4.3e-147)
		tmp = Float64(abs(Float64(y - x)) * 0.5);
	else
		tmp = fma(1.5, x, Float64(y * -0.5));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.2e-73], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.3e-147], N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(1.5 * x + N[(y * -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.19999999999999938e-73

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\frac{\left|y - x\right|}{2} + x} \]

      2. div-inv 100.0%

         \[\leadsto \color{blue}{\left|y - x\right| \cdot \frac{1}{2}} + x \]

      3. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left|y - x\right|, \frac{1}{2}, x\right)} \]

      4. add-sqr-sqrt 86.0%

         \[\leadsto \mathsf{fma}\left(\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|, \frac{1}{2}, x\right) \]

      5. fabs-sqr 86.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right) \]

      6. add-sqr-sqrt 86.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{1}{2}, x\right) \]

      7. metadata-eval 86.8%

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

   3. Applied egg-rr 86.8%

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

   4. Taylor expanded in y around 0 86.8%

      \[\leadsto \color{blue}{-0.5 \cdot x + \left(0.5 \cdot y + x\right)} \]
   5. Step-by-step derivation

      1. +-commutative 86.8%

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

      2. associate-+r+ 86.8%

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

      3. distribute-lft1-in 86.8%

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

      4. metadata-eval 86.8%

         \[\leadsto \color{blue}{0.5} \cdot x + 0.5 \cdot y \]

      5. distribute-lft-out 86.8%

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

   6. Simplified 86.8%

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

   if -6.19999999999999938e-73 < x < 4.3000000000000001e-147

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]

   2. Taylor expanded in x around 0 87.6%

      \[\leadsto \color{blue}{0.5 \cdot \left|y - x\right|} \]

   if 4.3000000000000001e-147 < x

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 99.2%

         \[\leadsto \color{blue}{\sqrt{x + \frac{\left|y - x\right|}{2}} \cdot \sqrt{x + \frac{\left|y - x\right|}{2}}} \]

      2. pow2 99.2%

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

      3. +-commutative 99.2%

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

      4. div-inv 99.2%

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

      5. fma-def 99.2%

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

      6. add-sqr-sqrt 16.6%

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

      7. fabs-sqr 16.6%

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right)}\right)}^{2} \]

      8. add-sqr-sqrt 28.2%

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

      9. metadata-eval 28.2%

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(y - x, \color{blue}{0.5}, x\right)}\right)}^{2} \]

   3. Applied egg-rr 28.2%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)}\right)}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 28.2%

         \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)} \cdot \sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)}} \]

      2. add-sqr-sqrt 28.4%

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

      3. fma-udef 28.4%

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

      4. add-sqr-sqrt 16.5%

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

      5. sqrt-prod 16.5%

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

      6. associate-*r* 16.5%

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

      7. fma-def 16.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot 0.5} \cdot \sqrt{y - x}, \sqrt{0.5}, x\right)} \]

      8. pow1/2 16.5%

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

      9. pow1/2 16.5%

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

      10. pow-prod-down 61.4%

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

   5. Applied egg-rr 61.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(y - x\right)\right)}^{0.5}, \sqrt{0.5}, x\right)} \]
   6. Step-by-step derivation

      1. unpow1/2 61.4%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(y - x\right)}}, \sqrt{0.5}, x\right) \]

      2. associate-*l* 61.4%

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\left(y - x\right) \cdot \left(0.5 \cdot \left(y - x\right)\right)}}, \sqrt{0.5}, x\right) \]

   7. Simplified 61.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot \left(0.5 \cdot \left(y - x\right)\right)}, \sqrt{0.5}, x\right)} \]

   8. Taylor expanded in y around -inf 82.7%

      \[\leadsto \color{blue}{{\left(\sqrt{0.5}\right)}^{2} \cdot x + \left(-1 \cdot \left({\left(\sqrt{0.5}\right)}^{2} \cdot y\right) + x\right)} \]
   9. Step-by-step derivation

      1. +-commutative 82.7%

         \[\leadsto \color{blue}{\left(-1 \cdot \left({\left(\sqrt{0.5}\right)}^{2} \cdot y\right) + x\right) + {\left(\sqrt{0.5}\right)}^{2} \cdot x} \]

      2. associate-+l+ 82.7%

         \[\leadsto \color{blue}{-1 \cdot \left({\left(\sqrt{0.5}\right)}^{2} \cdot y\right) + \left(x + {\left(\sqrt{0.5}\right)}^{2} \cdot x\right)} \]

      3. mul-1-neg 82.7%

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

      4. *-commutative 82.7%

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

      5. unpow2 82.7%

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

      6. rem-square-sqrt 83.1%

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

      7. distribute-rgt-neg-in 83.1%

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

      8. fma-def 83.1%

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

      9. metadata-eval 83.1%

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

      10. unpow2 83.1%

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

      11. rem-square-sqrt 83.3%

          \[\leadsto \mathsf{fma}\left(y, -0.5, x + \color{blue}{0.5} \cdot x\right) \]

      12. distribute-rgt1-in 83.3%

          \[\leadsto \mathsf{fma}\left(y, -0.5, \color{blue}{\left(0.5 + 1\right) \cdot x}\right) \]

      13. metadata-eval 83.3%

          \[\leadsto \mathsf{fma}\left(y, -0.5, \color{blue}{1.5} \cdot x\right) \]

   10. Simplified 83.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.5, 1.5 \cdot x\right)} \]

   11. Taylor expanded in y around 0 83.3%

       \[\leadsto \color{blue}{-0.5 \cdot y + 1.5 \cdot x} \]
   12. Step-by-step derivation

       1. +-commutative 83.3%

          \[\leadsto \color{blue}{1.5 \cdot x + -0.5 \cdot y} \]

       2. *-commutative 83.3%

          \[\leadsto 1.5 \cdot x + \color{blue}{y \cdot -0.5} \]

       3. fma-def 83.5%

          \[\leadsto \color{blue}{\mathsf{fma}\left(1.5, x, y \cdot -0.5\right)} \]

       4. *-commutative 83.5%

          \[\leadsto \mathsf{fma}\left(1.5, x, \color{blue}{-0.5 \cdot y}\right) \]

   13. Simplified 83.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-73}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-147}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1.5, x, y \cdot -0.5\right)\\ \end{array} \]

Alternative 4: 58.8% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-72}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-237}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-268}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-114}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -8.5e-72)
   (* x 0.5)
   (if (<= x -2.7e-237)
     (* y -0.5)
     (if (<= x 8.5e-268)
       (* y 0.5)
       (if (<= x 3.2e-114) (* y -0.5) (* x 1.5))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8.5e-72) {
		tmp = x * 0.5;
	} else if (x <= -2.7e-237) {
		tmp = y * -0.5;
	} else if (x <= 8.5e-268) {
		tmp = y * 0.5;
	} else if (x <= 3.2e-114) {
		tmp = y * -0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-8.5d-72)) then
        tmp = x * 0.5d0
    else if (x <= (-2.7d-237)) then
        tmp = y * (-0.5d0)
    else if (x <= 8.5d-268) then
        tmp = y * 0.5d0
    else if (x <= 3.2d-114) then
        tmp = y * (-0.5d0)
    else
        tmp = x * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -8.5e-72) {
		tmp = x * 0.5;
	} else if (x <= -2.7e-237) {
		tmp = y * -0.5;
	} else if (x <= 8.5e-268) {
		tmp = y * 0.5;
	} else if (x <= 3.2e-114) {
		tmp = y * -0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -8.5e-72:
		tmp = x * 0.5
	elif x <= -2.7e-237:
		tmp = y * -0.5
	elif x <= 8.5e-268:
		tmp = y * 0.5
	elif x <= 3.2e-114:
		tmp = y * -0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8.5e-72)
		tmp = Float64(x * 0.5);
	elseif (x <= -2.7e-237)
		tmp = Float64(y * -0.5);
	elseif (x <= 8.5e-268)
		tmp = Float64(y * 0.5);
	elseif (x <= 3.2e-114)
		tmp = Float64(y * -0.5);
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8.5e-72)
		tmp = x * 0.5;
	elseif (x <= -2.7e-237)
		tmp = y * -0.5;
	elseif (x <= 8.5e-268)
		tmp = y * 0.5;
	elseif (x <= 3.2e-114)
		tmp = y * -0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8.5e-72], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, -2.7e-237], N[(y * -0.5), $MachinePrecision], If[LessEqual[x, 8.5e-268], N[(y * 0.5), $MachinePrecision], If[LessEqual[x, 3.2e-114], N[(y * -0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.50000000000000008e-72

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\frac{\left|y - x\right|}{2} + x} \]

      2. div-inv 100.0%

         \[\leadsto \color{blue}{\left|y - x\right| \cdot \frac{1}{2}} + x \]

      3. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left|y - x\right|, \frac{1}{2}, x\right)} \]

      4. add-sqr-sqrt 86.0%

         \[\leadsto \mathsf{fma}\left(\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|, \frac{1}{2}, x\right) \]

      5. fabs-sqr 86.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right) \]

      6. add-sqr-sqrt 86.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{1}{2}, x\right) \]

      7. metadata-eval 86.8%

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

   3. Applied egg-rr 86.8%

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

   4. Taylor expanded in y around 0 68.1%

      \[\leadsto \color{blue}{-0.5 \cdot x + x} \]
   5. Step-by-step derivation

      1. distribute-lft1-in 68.1%

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

      2. metadata-eval 68.1%

         \[\leadsto \color{blue}{0.5} \cdot x \]

      3. *-commutative 68.1%

         \[\leadsto \color{blue}{x \cdot 0.5} \]

   6. Simplified 68.1%

      \[\leadsto \color{blue}{x \cdot 0.5} \]

   if -8.50000000000000008e-72 < x < -2.69999999999999984e-237 or 8.50000000000000052e-268 < x < 3.2000000000000002e-114

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 87.5%

         \[\leadsto \color{blue}{\sqrt{x + \frac{\left|y - x\right|}{2}} \cdot \sqrt{x + \frac{\left|y - x\right|}{2}}} \]

      2. pow2 87.5%

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

      3. +-commutative 87.5%

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

      4. div-inv 87.5%

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

      5. fma-def 87.5%

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

      6. add-sqr-sqrt 29.1%

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

      7. fabs-sqr 29.1%

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right)}\right)}^{2} \]

      8. add-sqr-sqrt 30.5%

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

      9. metadata-eval 30.5%

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(y - x, \color{blue}{0.5}, x\right)}\right)}^{2} \]

   3. Applied egg-rr 30.5%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)}\right)}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 30.5%

         \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)} \cdot \sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)}} \]

      2. add-sqr-sqrt 43.1%

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

      3. fma-udef 43.1%

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

      4. add-sqr-sqrt 40.8%

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

      5. sqrt-prod 40.8%

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

      6. associate-*r* 40.9%

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

      7. fma-def 40.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot 0.5} \cdot \sqrt{y - x}, \sqrt{0.5}, x\right)} \]

      8. pow1/2 40.9%

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

      9. pow1/2 40.9%

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

      10. pow-prod-down 63.6%

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

   5. Applied egg-rr 63.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(y - x\right)\right)}^{0.5}, \sqrt{0.5}, x\right)} \]
   6. Step-by-step derivation

      1. unpow1/2 63.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(y - x\right)}}, \sqrt{0.5}, x\right) \]

      2. associate-*l* 63.6%

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\left(y - x\right) \cdot \left(0.5 \cdot \left(y - x\right)\right)}}, \sqrt{0.5}, x\right) \]

   7. Simplified 63.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot \left(0.5 \cdot \left(y - x\right)\right)}, \sqrt{0.5}, x\right)} \]

   8. Taylor expanded in y around -inf 51.9%

      \[\leadsto \color{blue}{-1 \cdot \left({\left(\sqrt{0.5}\right)}^{2} \cdot y\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 51.9%

         \[\leadsto \color{blue}{\left(-1 \cdot {\left(\sqrt{0.5}\right)}^{2}\right) \cdot y} \]

      2. *-commutative 51.9%

         \[\leadsto \color{blue}{y \cdot \left(-1 \cdot {\left(\sqrt{0.5}\right)}^{2}\right)} \]

      3. unpow2 51.9%

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

      4. rem-square-sqrt 52.9%

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

      5. metadata-eval 52.9%

         \[\leadsto y \cdot \color{blue}{-0.5} \]

   10. Simplified 52.9%

       \[\leadsto \color{blue}{y \cdot -0.5} \]

   if -2.69999999999999984e-237 < x < 8.50000000000000052e-268

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

         \[\leadsto \color{blue}{\frac{\left|y - x\right|}{2} + x} \]

      2. div-inv 99.9%

         \[\leadsto \color{blue}{\left|y - x\right| \cdot \frac{1}{2}} + x \]

      3. fma-def 99.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left|y - x\right|, \frac{1}{2}, x\right)} \]

      4. add-sqr-sqrt 57.9%

         \[\leadsto \mathsf{fma}\left(\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|, \frac{1}{2}, x\right) \]

      5. fabs-sqr 57.9%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right) \]

      6. add-sqr-sqrt 59.9%

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{1}{2}, x\right) \]

      7. metadata-eval 59.9%

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

   3. Applied egg-rr 59.9%

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

   4. Taylor expanded in y around inf 59.4%

      \[\leadsto \color{blue}{0.5 \cdot y} \]

   if 3.2000000000000002e-114 < x

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 99.2%

         \[\leadsto \color{blue}{\sqrt{x + \frac{\left|y - x\right|}{2}} \cdot \sqrt{x + \frac{\left|y - x\right|}{2}}} \]

      2. pow2 99.2%

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

      3. +-commutative 99.2%

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

      4. div-inv 99.2%

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

      5. fma-def 99.2%

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

      6. add-sqr-sqrt 16.0%

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

      7. fabs-sqr 16.0%

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right)}\right)}^{2} \]

      8. add-sqr-sqrt 28.2%

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

      9. metadata-eval 28.2%

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(y - x, \color{blue}{0.5}, x\right)}\right)}^{2} \]

   3. Applied egg-rr 28.2%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)}\right)}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 28.2%

         \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)} \cdot \sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)}} \]

      2. add-sqr-sqrt 28.4%

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

      3. fma-udef 28.4%

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

      4. add-sqr-sqrt 15.9%

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

      5. sqrt-prod 15.9%

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

      6. associate-*r* 15.9%

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

      7. fma-def 15.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot 0.5} \cdot \sqrt{y - x}, \sqrt{0.5}, x\right)} \]

      8. pow1/2 15.9%

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

      9. pow1/2 15.9%

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

      10. pow-prod-down 58.3%

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

   5. Applied egg-rr 58.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(y - x\right)\right)}^{0.5}, \sqrt{0.5}, x\right)} \]
   6. Step-by-step derivation

      1. unpow1/2 58.3%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(y - x\right)}}, \sqrt{0.5}, x\right) \]

      2. associate-*l* 58.3%

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\left(y - x\right) \cdot \left(0.5 \cdot \left(y - x\right)\right)}}, \sqrt{0.5}, x\right) \]

   7. Simplified 58.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot \left(0.5 \cdot \left(y - x\right)\right)}, \sqrt{0.5}, x\right)} \]

   8. Taylor expanded in y around -inf 83.4%

      \[\leadsto \color{blue}{{\left(\sqrt{0.5}\right)}^{2} \cdot x + \left(-1 \cdot \left({\left(\sqrt{0.5}\right)}^{2} \cdot y\right) + x\right)} \]
   9. Step-by-step derivation

      1. +-commutative 83.4%

         \[\leadsto \color{blue}{\left(-1 \cdot \left({\left(\sqrt{0.5}\right)}^{2} \cdot y\right) + x\right) + {\left(\sqrt{0.5}\right)}^{2} \cdot x} \]

      2. associate-+l+ 83.3%

         \[\leadsto \color{blue}{-1 \cdot \left({\left(\sqrt{0.5}\right)}^{2} \cdot y\right) + \left(x + {\left(\sqrt{0.5}\right)}^{2} \cdot x\right)} \]

      3. mul-1-neg 83.3%

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

      4. *-commutative 83.3%

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

      5. unpow2 83.3%

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

      6. rem-square-sqrt 83.6%

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

      7. distribute-rgt-neg-in 83.6%

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

      8. fma-def 83.6%

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

      9. metadata-eval 83.6%

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

      10. unpow2 83.6%

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

      11. rem-square-sqrt 83.9%

          \[\leadsto \mathsf{fma}\left(y, -0.5, x + \color{blue}{0.5} \cdot x\right) \]

      12. distribute-rgt1-in 83.9%

          \[\leadsto \mathsf{fma}\left(y, -0.5, \color{blue}{\left(0.5 + 1\right) \cdot x}\right) \]

      13. metadata-eval 83.9%

          \[\leadsto \mathsf{fma}\left(y, -0.5, \color{blue}{1.5} \cdot x\right) \]

   10. Simplified 83.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.5, 1.5 \cdot x\right)} \]

   11. Taylor expanded in y around 0 68.0%

       \[\leadsto \color{blue}{1.5 \cdot x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-72}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-237}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-268}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-114}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]

Alternative 5: 69.4% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+25}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-35}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{-89}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7e+25)
   (* y -0.5)
   (if (<= y -1.6e-35)
     (* x 0.5)
     (if (<= y -1.55e-89) (* y -0.5) (* 0.5 (+ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7e+25) {
		tmp = y * -0.5;
	} else if (y <= -1.6e-35) {
		tmp = x * 0.5;
	} else if (y <= -1.55e-89) {
		tmp = y * -0.5;
	} else {
		tmp = 0.5 * (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 (y <= (-7d+25)) then
        tmp = y * (-0.5d0)
    else if (y <= (-1.6d-35)) then
        tmp = x * 0.5d0
    else if (y <= (-1.55d-89)) then
        tmp = y * (-0.5d0)
    else
        tmp = 0.5d0 * (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7e+25) {
		tmp = y * -0.5;
	} else if (y <= -1.6e-35) {
		tmp = x * 0.5;
	} else if (y <= -1.55e-89) {
		tmp = y * -0.5;
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7e+25:
		tmp = y * -0.5
	elif y <= -1.6e-35:
		tmp = x * 0.5
	elif y <= -1.55e-89:
		tmp = y * -0.5
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7e+25)
		tmp = Float64(y * -0.5);
	elseif (y <= -1.6e-35)
		tmp = Float64(x * 0.5);
	elseif (y <= -1.55e-89)
		tmp = Float64(y * -0.5);
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7e+25)
		tmp = y * -0.5;
	elseif (y <= -1.6e-35)
		tmp = x * 0.5;
	elseif (y <= -1.55e-89)
		tmp = y * -0.5;
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7e+25], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, -1.6e-35], N[(x * 0.5), $MachinePrecision], If[LessEqual[y, -1.55e-89], N[(y * -0.5), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.99999999999999999e25 or -1.5999999999999999e-35 < y < -1.54999999999999998e-89

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 89.8%

         \[\leadsto \color{blue}{\sqrt{x + \frac{\left|y - x\right|}{2}} \cdot \sqrt{x + \frac{\left|y - x\right|}{2}}} \]

      2. pow2 89.8%

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

      3. +-commutative 89.8%

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

      4. div-inv 89.8%

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

      5. fma-def 89.8%

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

      6. add-sqr-sqrt 0.0%

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

      7. fabs-sqr 0.0%

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right)}\right)}^{2} \]

      8. add-sqr-sqrt 2.4%

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

      9. metadata-eval 2.4%

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(y - x, \color{blue}{0.5}, x\right)}\right)}^{2} \]

   3. Applied egg-rr 2.4%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)}\right)}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 2.4%

         \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)} \cdot \sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)}} \]

      2. add-sqr-sqrt 12.5%

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

      3. fma-udef 12.5%

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

      4. add-sqr-sqrt 9.4%

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

      5. sqrt-prod 9.4%

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

      6. associate-*r* 9.4%

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

      7. fma-def 9.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot 0.5} \cdot \sqrt{y - x}, \sqrt{0.5}, x\right)} \]

      8. pow1/2 9.4%

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

      9. pow1/2 9.4%

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

      10. pow-prod-down 61.8%

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

   5. Applied egg-rr 61.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(y - x\right)\right)}^{0.5}, \sqrt{0.5}, x\right)} \]
   6. Step-by-step derivation

      1. unpow1/2 61.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(y - x\right)}}, \sqrt{0.5}, x\right) \]

      2. associate-*l* 61.8%

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\left(y - x\right) \cdot \left(0.5 \cdot \left(y - x\right)\right)}}, \sqrt{0.5}, x\right) \]

   7. Simplified 61.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot \left(0.5 \cdot \left(y - x\right)\right)}, \sqrt{0.5}, x\right)} \]

   8. Taylor expanded in y around -inf 74.7%

      \[\leadsto \color{blue}{-1 \cdot \left({\left(\sqrt{0.5}\right)}^{2} \cdot y\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 74.7%

         \[\leadsto \color{blue}{\left(-1 \cdot {\left(\sqrt{0.5}\right)}^{2}\right) \cdot y} \]

      2. *-commutative 74.7%

         \[\leadsto \color{blue}{y \cdot \left(-1 \cdot {\left(\sqrt{0.5}\right)}^{2}\right)} \]

      3. unpow2 74.7%

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

      4. rem-square-sqrt 76.2%

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

      5. metadata-eval 76.2%

         \[\leadsto y \cdot \color{blue}{-0.5} \]

   10. Simplified 76.2%

       \[\leadsto \color{blue}{y \cdot -0.5} \]

   if -6.99999999999999999e25 < y < -1.5999999999999999e-35

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

         \[\leadsto \color{blue}{\frac{\left|y - x\right|}{2} + x} \]

      2. div-inv 99.9%

         \[\leadsto \color{blue}{\left|y - x\right| \cdot \frac{1}{2}} + x \]

      3. fma-def 99.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left|y - x\right|, \frac{1}{2}, x\right)} \]

      4. add-sqr-sqrt 52.1%

         \[\leadsto \mathsf{fma}\left(\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|, \frac{1}{2}, x\right) \]

      5. fabs-sqr 52.1%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right) \]

      6. add-sqr-sqrt 58.5%

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{1}{2}, x\right) \]

      7. metadata-eval 58.5%

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

   3. Applied egg-rr 58.5%

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

   4. Taylor expanded in y around 0 59.1%

      \[\leadsto \color{blue}{-0.5 \cdot x + x} \]
   5. Step-by-step derivation

      1. distribute-lft1-in 59.1%

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

      2. metadata-eval 59.1%

         \[\leadsto \color{blue}{0.5} \cdot x \]

      3. *-commutative 59.1%

         \[\leadsto \color{blue}{x \cdot 0.5} \]

   6. Simplified 59.1%

      \[\leadsto \color{blue}{x \cdot 0.5} \]

   if -1.54999999999999998e-89 < y

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

         \[\leadsto \color{blue}{\frac{\left|y - x\right|}{2} + x} \]

      2. div-inv 99.9%

         \[\leadsto \color{blue}{\left|y - x\right| \cdot \frac{1}{2}} + x \]

      3. fma-def 99.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left|y - x\right|, \frac{1}{2}, x\right)} \]

      4. add-sqr-sqrt 67.0%

         \[\leadsto \mathsf{fma}\left(\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|, \frac{1}{2}, x\right) \]

      5. fabs-sqr 67.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right) \]

      6. add-sqr-sqrt 72.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{1}{2}, x\right) \]

      7. metadata-eval 72.6%

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

   3. Applied egg-rr 72.6%

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

   4. Taylor expanded in y around 0 72.6%

      \[\leadsto \color{blue}{-0.5 \cdot x + \left(0.5 \cdot y + x\right)} \]
   5. Step-by-step derivation

      1. +-commutative 72.6%

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

      2. associate-+r+ 72.6%

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

      3. distribute-lft1-in 72.6%

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

      4. metadata-eval 72.6%

         \[\leadsto \color{blue}{0.5} \cdot x + 0.5 \cdot y \]

      5. distribute-lft-out 72.6%

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

   6. Simplified 72.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(x + y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+25}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-35}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{-89}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]

Alternative 6: 79.3% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-282}:\\ \;\;\;\;x + 0.5 \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.4e-282) (+ x (* 0.5 (- x y))) (* 0.5 (+ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.4e-282) {
		tmp = x + (0.5 * (x - y));
	} else {
		tmp = 0.5 * (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 (y <= 1.4d-282) then
        tmp = x + (0.5d0 * (x - y))
    else
        tmp = 0.5d0 * (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.4e-282) {
		tmp = x + (0.5 * (x - y));
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.4e-282:
		tmp = x + (0.5 * (x - y))
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.4e-282)
		tmp = Float64(x + Float64(0.5 * Float64(x - y)));
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.4e-282)
		tmp = x + (0.5 * (x - y));
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.4e-282], N[(x + N[(0.5 * N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3999999999999999e-282

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 72.9%

         \[\leadsto \color{blue}{\sqrt{x + \frac{\left|y - x\right|}{2}} \cdot \sqrt{x + \frac{\left|y - x\right|}{2}}} \]

      2. pow2 72.9%

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

      3. +-commutative 72.9%

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

      4. div-inv 72.9%

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

      5. fma-def 72.9%

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

      6. add-sqr-sqrt 0.0%

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

      7. fabs-sqr 0.0%

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right)}\right)}^{2} \]

      8. add-sqr-sqrt 4.6%

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

      9. metadata-eval 4.6%

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(y - x, \color{blue}{0.5}, x\right)}\right)}^{2} \]

   3. Applied egg-rr 4.6%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)}\right)}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 4.6%

         \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)} \cdot \sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)}} \]

      2. add-sqr-sqrt 31.5%

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

      3. fma-udef 31.5%

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

      4. add-sqr-sqrt 26.2%

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

      5. sqrt-prod 26.2%

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

      6. associate-*r* 26.2%

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

      7. fma-def 26.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot 0.5} \cdot \sqrt{y - x}, \sqrt{0.5}, x\right)} \]

      8. pow1/2 26.2%

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

      9. pow1/2 26.2%

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

      10. pow-prod-down 61.4%

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

   5. Applied egg-rr 61.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(y - x\right)\right)}^{0.5}, \sqrt{0.5}, x\right)} \]
   6. Step-by-step derivation

      1. unpow1/2 61.4%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(y - x\right)}}, \sqrt{0.5}, x\right) \]

      2. associate-*l* 61.4%

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\left(y - x\right) \cdot \left(0.5 \cdot \left(y - x\right)\right)}}, \sqrt{0.5}, x\right) \]

   7. Simplified 61.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot \left(0.5 \cdot \left(y - x\right)\right)}, \sqrt{0.5}, x\right)} \]

   8. Taylor expanded in y around -inf 77.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot \left(\sqrt{0.5} \cdot y\right) + \sqrt{0.5} \cdot x}, \sqrt{0.5}, x\right) \]
   9. Step-by-step derivation

      1. +-commutative 77.5%

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

      2. mul-1-neg 77.5%

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

      3. unsub-neg 77.5%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{0.5} \cdot x - \sqrt{0.5} \cdot y}, \sqrt{0.5}, x\right) \]

      4. distribute-lft-out-- 77.5%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{0.5} \cdot \left(x - y\right)}, \sqrt{0.5}, x\right) \]

   10. Simplified 77.5%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{0.5} \cdot \left(x - y\right)}, \sqrt{0.5}, x\right) \]
   11. Step-by-step derivation

       1. fma-udef 77.5%

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

       2. *-commutative 77.5%

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

       3. associate-*l* 77.2%

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

       4. add-sqr-sqrt 78.2%

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

   12. Applied egg-rr 78.2%

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

   if 1.3999999999999999e-282 < y

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\frac{\left|y - x\right|}{2} + x} \]

      2. div-inv 100.0%

         \[\leadsto \color{blue}{\left|y - x\right| \cdot \frac{1}{2}} + x \]

      3. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left|y - x\right|, \frac{1}{2}, x\right)} \]

      4. add-sqr-sqrt 77.1%

         \[\leadsto \mathsf{fma}\left(\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|, \frac{1}{2}, x\right) \]

      5. fabs-sqr 77.1%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right) \]

      6. add-sqr-sqrt 81.5%

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{1}{2}, x\right) \]

      7. metadata-eval 81.5%

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

   3. Applied egg-rr 81.5%

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

   4. Taylor expanded in y around 0 81.5%

      \[\leadsto \color{blue}{-0.5 \cdot x + \left(0.5 \cdot y + x\right)} \]
   5. Step-by-step derivation

      1. +-commutative 81.5%

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

      2. associate-+r+ 81.6%

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

      3. distribute-lft1-in 81.6%

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

      4. metadata-eval 81.6%

         \[\leadsto \color{blue}{0.5} \cdot x + 0.5 \cdot y \]

      5. distribute-lft-out 81.6%

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

   6. Simplified 81.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(x + y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-282}:\\ \;\;\;\;x + 0.5 \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]

Alternative 7: 58.5% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-106}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-120}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3e-106) (* x 0.5) (if (<= x 3.3e-120) (* y 0.5) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3e-106) {
		tmp = x * 0.5;
	} else if (x <= 3.3e-120) {
		tmp = y * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3d-106)) then
        tmp = x * 0.5d0
    else if (x <= 3.3d-120) then
        tmp = y * 0.5d0
    else
        tmp = x * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3e-106) {
		tmp = x * 0.5;
	} else if (x <= 3.3e-120) {
		tmp = y * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3e-106:
		tmp = x * 0.5
	elif x <= 3.3e-120:
		tmp = y * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3e-106)
		tmp = Float64(x * 0.5);
	elseif (x <= 3.3e-120)
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3e-106)
		tmp = x * 0.5;
	elseif (x <= 3.3e-120)
		tmp = y * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3e-106], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 3.3e-120], N[(y * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.00000000000000019e-106

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\frac{\left|y - x\right|}{2} + x} \]

      2. div-inv 100.0%

         \[\leadsto \color{blue}{\left|y - x\right| \cdot \frac{1}{2}} + x \]

      3. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left|y - x\right|, \frac{1}{2}, x\right)} \]

      4. add-sqr-sqrt 82.8%

         \[\leadsto \mathsf{fma}\left(\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|, \frac{1}{2}, x\right) \]

      5. fabs-sqr 82.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right) \]

      6. add-sqr-sqrt 83.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{1}{2}, x\right) \]

      7. metadata-eval 83.6%

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

   3. Applied egg-rr 83.6%

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

   4. Taylor expanded in y around 0 64.6%

      \[\leadsto \color{blue}{-0.5 \cdot x + x} \]
   5. Step-by-step derivation

      1. distribute-lft1-in 64.6%

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

      2. metadata-eval 64.6%

         \[\leadsto \color{blue}{0.5} \cdot x \]

      3. *-commutative 64.6%

         \[\leadsto \color{blue}{x \cdot 0.5} \]

   6. Simplified 64.6%

      \[\leadsto \color{blue}{x \cdot 0.5} \]

   if -3.00000000000000019e-106 < x < 3.29999999999999967e-120

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\frac{\left|y - x\right|}{2} + x} \]

      2. div-inv 100.0%

         \[\leadsto \color{blue}{\left|y - x\right| \cdot \frac{1}{2}} + x \]

      3. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left|y - x\right|, \frac{1}{2}, x\right)} \]

      4. add-sqr-sqrt 45.5%

         \[\leadsto \mathsf{fma}\left(\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|, \frac{1}{2}, x\right) \]

      5. fabs-sqr 45.5%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right) \]

      6. add-sqr-sqrt 47.7%

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{1}{2}, x\right) \]

      7. metadata-eval 47.7%

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

   3. Applied egg-rr 47.7%

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

   4. Taylor expanded in y around inf 41.3%

      \[\leadsto \color{blue}{0.5 \cdot y} \]

   if 3.29999999999999967e-120 < x

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 99.2%

         \[\leadsto \color{blue}{\sqrt{x + \frac{\left|y - x\right|}{2}} \cdot \sqrt{x + \frac{\left|y - x\right|}{2}}} \]

      2. pow2 99.2%

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

      3. +-commutative 99.2%

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

      4. div-inv 99.2%

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

      5. fma-def 99.2%

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

      6. add-sqr-sqrt 15.8%

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

      7. fabs-sqr 15.8%

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right)}\right)}^{2} \]

      8. add-sqr-sqrt 27.9%

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

      9. metadata-eval 27.9%

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(y - x, \color{blue}{0.5}, x\right)}\right)}^{2} \]

   3. Applied egg-rr 27.9%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)}\right)}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 27.9%

         \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)} \cdot \sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)}} \]

      2. add-sqr-sqrt 28.1%

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

      3. fma-udef 28.1%

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

      4. add-sqr-sqrt 15.7%

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

      5. sqrt-prod 15.7%

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

      6. associate-*r* 15.7%

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

      7. fma-def 15.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot 0.5} \cdot \sqrt{y - x}, \sqrt{0.5}, x\right)} \]

      8. pow1/2 15.7%

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

      9. pow1/2 15.7%

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

      10. pow-prod-down 58.8%

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

   5. Applied egg-rr 58.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(y - x\right)\right)}^{0.5}, \sqrt{0.5}, x\right)} \]
   6. Step-by-step derivation

      1. unpow1/2 58.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(y - x\right)}}, \sqrt{0.5}, x\right) \]

      2. associate-*l* 58.8%

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\left(y - x\right) \cdot \left(0.5 \cdot \left(y - x\right)\right)}}, \sqrt{0.5}, x\right) \]

   7. Simplified 58.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot \left(0.5 \cdot \left(y - x\right)\right)}, \sqrt{0.5}, x\right)} \]

   8. Taylor expanded in y around -inf 83.5%

      \[\leadsto \color{blue}{{\left(\sqrt{0.5}\right)}^{2} \cdot x + \left(-1 \cdot \left({\left(\sqrt{0.5}\right)}^{2} \cdot y\right) + x\right)} \]
   9. Step-by-step derivation

      1. +-commutative 83.5%

         \[\leadsto \color{blue}{\left(-1 \cdot \left({\left(\sqrt{0.5}\right)}^{2} \cdot y\right) + x\right) + {\left(\sqrt{0.5}\right)}^{2} \cdot x} \]

      2. associate-+l+ 83.5%

         \[\leadsto \color{blue}{-1 \cdot \left({\left(\sqrt{0.5}\right)}^{2} \cdot y\right) + \left(x + {\left(\sqrt{0.5}\right)}^{2} \cdot x\right)} \]

      3. mul-1-neg 83.5%

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

      4. *-commutative 83.5%

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

      5. unpow2 83.5%

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

      6. rem-square-sqrt 83.8%

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

      7. distribute-rgt-neg-in 83.8%

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

      8. fma-def 83.8%

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

      9. metadata-eval 83.8%

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

      10. unpow2 83.8%

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

      11. rem-square-sqrt 84.1%

          \[\leadsto \mathsf{fma}\left(y, -0.5, x + \color{blue}{0.5} \cdot x\right) \]

      12. distribute-rgt1-in 84.1%

          \[\leadsto \mathsf{fma}\left(y, -0.5, \color{blue}{\left(0.5 + 1\right) \cdot x}\right) \]

      13. metadata-eval 84.1%

          \[\leadsto \mathsf{fma}\left(y, -0.5, \color{blue}{1.5} \cdot x\right) \]

   10. Simplified 84.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.5, 1.5 \cdot x\right)} \]

   11. Taylor expanded in y around 0 67.3%

       \[\leadsto \color{blue}{1.5 \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-106}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-120}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]

Alternative 8: 31.8% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-199}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.3e-199) x (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.3e-199) {
		tmp = x;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.3d-199) then
        tmp = x
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.3e-199) {
		tmp = x;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.3e-199:
		tmp = x
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.3e-199)
		tmp = x;
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.3e-199)
		tmp = x;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.3e-199], x, N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3e-199

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]

   2. Taylor expanded in x around inf 12.3%

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

   if 1.3e-199 < y

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\frac{\left|y - x\right|}{2} + x} \]

      2. div-inv 100.0%

         \[\leadsto \color{blue}{\left|y - x\right| \cdot \frac{1}{2}} + x \]

      3. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left|y - x\right|, \frac{1}{2}, x\right)} \]

      4. add-sqr-sqrt 78.5%

         \[\leadsto \mathsf{fma}\left(\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|, \frac{1}{2}, x\right) \]

      5. fabs-sqr 78.5%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right) \]

      6. add-sqr-sqrt 82.7%

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{1}{2}, x\right) \]

      7. metadata-eval 82.7%

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

   3. Applied egg-rr 82.7%

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

   4. Taylor expanded in y around inf 60.5%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-199}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 9: 46.0% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-92}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.1e-92) (* x 1.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.1e-92) {
		tmp = x * 1.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.1d-92) then
        tmp = x * 1.5d0
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.1e-92) {
		tmp = x * 1.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.1e-92:
		tmp = x * 1.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.1e-92)
		tmp = Float64(x * 1.5);
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.1e-92)
		tmp = x * 1.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.1e-92], N[(x * 1.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.1e-92

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 69.9%

         \[\leadsto \color{blue}{\sqrt{x + \frac{\left|y - x\right|}{2}} \cdot \sqrt{x + \frac{\left|y - x\right|}{2}}} \]

      2. pow2 69.9%

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

      3. +-commutative 69.9%

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

      4. div-inv 69.9%

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

      5. fma-def 69.9%

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

      6. add-sqr-sqrt 2.3%

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

      7. fabs-sqr 2.3%

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right)}\right)}^{2} \]

      8. add-sqr-sqrt 7.5%

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

      9. metadata-eval 7.5%

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(y - x, \color{blue}{0.5}, x\right)}\right)}^{2} \]

   3. Applied egg-rr 7.5%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)}\right)}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 7.5%

         \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)} \cdot \sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)}} \]

      2. add-sqr-sqrt 37.5%

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

      3. fma-udef 37.5%

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

      4. add-sqr-sqrt 31.6%

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

      5. sqrt-prod 31.5%

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

      6. associate-*r* 31.5%

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

      7. fma-def 31.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot 0.5} \cdot \sqrt{y - x}, \sqrt{0.5}, x\right)} \]

      8. pow1/2 31.5%

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

      9. pow1/2 31.5%

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

      10. pow-prod-down 61.7%

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

   5. Applied egg-rr 61.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(y - x\right)\right)}^{0.5}, \sqrt{0.5}, x\right)} \]
   6. Step-by-step derivation

      1. unpow1/2 61.7%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(y - x\right)}}, \sqrt{0.5}, x\right) \]

      2. associate-*l* 61.7%

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\left(y - x\right) \cdot \left(0.5 \cdot \left(y - x\right)\right)}}, \sqrt{0.5}, x\right) \]

   7. Simplified 61.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot \left(0.5 \cdot \left(y - x\right)\right)}, \sqrt{0.5}, x\right)} \]

   8. Taylor expanded in y around -inf 72.6%

      \[\leadsto \color{blue}{{\left(\sqrt{0.5}\right)}^{2} \cdot x + \left(-1 \cdot \left({\left(\sqrt{0.5}\right)}^{2} \cdot y\right) + x\right)} \]
   9. Step-by-step derivation

      1. +-commutative 72.6%

         \[\leadsto \color{blue}{\left(-1 \cdot \left({\left(\sqrt{0.5}\right)}^{2} \cdot y\right) + x\right) + {\left(\sqrt{0.5}\right)}^{2} \cdot x} \]

      2. associate-+l+ 72.6%

         \[\leadsto \color{blue}{-1 \cdot \left({\left(\sqrt{0.5}\right)}^{2} \cdot y\right) + \left(x + {\left(\sqrt{0.5}\right)}^{2} \cdot x\right)} \]

      3. mul-1-neg 72.6%

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

      4. *-commutative 72.6%

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

      5. unpow2 72.6%

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

      6. rem-square-sqrt 73.3%

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

      7. distribute-rgt-neg-in 73.3%

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

      8. fma-def 73.3%

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

      9. metadata-eval 73.3%

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

      10. unpow2 73.3%

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

      11. rem-square-sqrt 73.4%

          \[\leadsto \mathsf{fma}\left(y, -0.5, x + \color{blue}{0.5} \cdot x\right) \]

      12. distribute-rgt1-in 73.4%

          \[\leadsto \mathsf{fma}\left(y, -0.5, \color{blue}{\left(0.5 + 1\right) \cdot x}\right) \]

      13. metadata-eval 73.4%

          \[\leadsto \mathsf{fma}\left(y, -0.5, \color{blue}{1.5} \cdot x\right) \]

   10. Simplified 73.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.5, 1.5 \cdot x\right)} \]

   11. Taylor expanded in y around 0 34.0%

       \[\leadsto \color{blue}{1.5 \cdot x} \]

   if 2.1e-92 < y

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\frac{\left|y - x\right|}{2} + x} \]

      2. div-inv 100.0%

         \[\leadsto \color{blue}{\left|y - x\right| \cdot \frac{1}{2}} + x \]

      3. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left|y - x\right|, \frac{1}{2}, x\right)} \]

      4. add-sqr-sqrt 86.2%

         \[\leadsto \mathsf{fma}\left(\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|, \frac{1}{2}, x\right) \]

      5. fabs-sqr 86.2%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right) \]

      6. add-sqr-sqrt 89.1%

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{1}{2}, x\right) \]

      7. metadata-eval 89.1%

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

   3. Applied egg-rr 89.1%

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

   4. Taylor expanded in y around inf 68.5%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-92}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 10: 11.4% accurate, 107.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.9%

   \[x + \frac{\left|y - x\right|}{2} \]

2. Taylor expanded in x around inf 11.2%

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

3. Final simplification 11.2%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023257 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))