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

Percentage Accurate: 99.9% → 99.9%

Time: 9.5s

Alternatives: 11

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. Add Preprocessing
3. Add Preprocessing

Alternative 2: 83.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-93}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-57}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x - y}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.65e-93)
   (* 0.5 (+ x y))
   (if (<= x 4.1e-57) (+ x (/ (fabs y) 2.0)) (+ x (/ (- x y) 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.65e-93) {
		tmp = 0.5 * (x + y);
	} else if (x <= 4.1e-57) {
		tmp = x + (fabs(y) / 2.0);
	} else {
		tmp = x + ((x - y) / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.65d-93)) then
        tmp = 0.5d0 * (x + y)
    else if (x <= 4.1d-57) then
        tmp = x + (abs(y) / 2.0d0)
    else
        tmp = x + ((x - y) / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.65e-93) {
		tmp = 0.5 * (x + y);
	} else if (x <= 4.1e-57) {
		tmp = x + (Math.abs(y) / 2.0);
	} else {
		tmp = x + ((x - y) / 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.65e-93:
		tmp = 0.5 * (x + y)
	elif x <= 4.1e-57:
		tmp = x + (math.fabs(y) / 2.0)
	else:
		tmp = x + ((x - y) / 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.65e-93)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 4.1e-57)
		tmp = Float64(x + Float64(abs(y) / 2.0));
	else
		tmp = Float64(x + Float64(Float64(x - y) / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.65e-93)
		tmp = 0.5 * (x + y);
	elseif (x <= 4.1e-57)
		tmp = x + (abs(y) / 2.0);
	else
		tmp = x + ((x - y) / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.65e-93], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.1e-57], N[(x + N[(N[Abs[y], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(x - y), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6500000000000001e-93

   1. Initial program 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. fma-define 100.0%

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

      3. add-sqr-sqrt 90.5%

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

      4. fabs-sqr 90.5%

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

      5. add-sqr-sqrt 91.3%

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

      6. frac-2neg 91.3%

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

      7. distribute-frac-neg 91.3%

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

      8. fmm-undef 91.3%

         \[\leadsto \color{blue}{1 \cdot x - \frac{y - x}{-2}} \]

      9. *-un-lft-identity 91.3%

         \[\leadsto \color{blue}{x} - \frac{y - x}{-2} \]

      10. metadata-eval 91.3%

          \[\leadsto x - \frac{y - x}{\color{blue}{-2}} \]

   4. Applied egg-rr 91.3%

      \[\leadsto \color{blue}{x - \frac{y - x}{-2}} \]

   5. Taylor expanded in x around 0 91.3%

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

      1. cancel-sign-sub-inv 91.3%

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

      2. metadata-eval 91.3%

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

      3. distribute-lft-in 91.3%

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

      4. +-commutative 91.3%

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

   7. Simplified 91.3%

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

   if -1.6500000000000001e-93 < x < 4.1000000000000001e-57

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 90.5%

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

   if 4.1000000000000001e-57 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 72.0%

      \[\leadsto x + \frac{\left|\color{blue}{y \cdot \left(1 + -1 \cdot \frac{x}{y}\right)}\right|}{2} \]
   4. Step-by-step derivation

      1. mul-1-neg 72.0%

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

      2. unsub-neg 72.0%

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

   5. Simplified 72.0%

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

   6. Taylor expanded in x around inf 71.8%

      \[\leadsto x + \frac{\left|y \cdot \color{blue}{\left(x \cdot \left(\frac{1}{x} - \frac{1}{y}\right)\right)}\right|}{2} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 9.8%

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

      2. fabs-sqr 9.8%

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

      3. add-sqr-sqrt 18.1%

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

      4. associate-*r* 18.2%

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

      5. add-sqr-sqrt 18.2%

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

      6. sqrt-unprod 11.9%

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

      7. sqr-neg 11.9%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 80.8%

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

      10. distribute-rgt-neg-out 80.8%

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

      11. distribute-lft-neg-in 80.8%

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

      12. associate-*r* 61.9%

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

      13. add-sqr-sqrt 0.0%

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

      14. fabs-sqr 0.0%

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

      15. add-sqr-sqrt 8.3%

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

   8. Applied egg-rr 89.8%

      \[\leadsto x + \frac{\color{blue}{0 - x \cdot \left(\left(\frac{1}{x} + \frac{-1}{y}\right) \cdot y\right)}}{2} \]
   9. Step-by-step derivation

      1. associate-*r* 61.9%

         \[\leadsto x + \frac{0 - \color{blue}{\left(x \cdot \left(\frac{1}{x} + \frac{-1}{y}\right)\right) \cdot y}}{2} \]

      2. *-commutative 61.9%

         \[\leadsto x + \frac{0 - \color{blue}{y \cdot \left(x \cdot \left(\frac{1}{x} + \frac{-1}{y}\right)\right)}}{2} \]

      3. distribute-rgt-in 62.0%

         \[\leadsto x + \frac{0 - y \cdot \color{blue}{\left(\frac{1}{x} \cdot x + \frac{-1}{y} \cdot x\right)}}{2} \]

      4. lft-mult-inverse 62.0%

         \[\leadsto x + \frac{0 - y \cdot \left(\color{blue}{1} + \frac{-1}{y} \cdot x\right)}{2} \]

      5. associate-*l/ 62.0%

         \[\leadsto x + \frac{0 - y \cdot \left(1 + \color{blue}{\frac{-1 \cdot x}{y}}\right)}{2} \]

      6. associate-*r/ 62.0%

         \[\leadsto x + \frac{0 - y \cdot \left(1 + \color{blue}{-1 \cdot \frac{x}{y}}\right)}{2} \]

      7. distribute-lft-in 62.0%

         \[\leadsto x + \frac{0 - \color{blue}{\left(y \cdot 1 + y \cdot \left(-1 \cdot \frac{x}{y}\right)\right)}}{2} \]

      8. neg-mul-1 62.0%

         \[\leadsto x + \frac{0 - \left(y \cdot 1 + y \cdot \color{blue}{\left(-\frac{x}{y}\right)}\right)}{2} \]

      9. distribute-rgt-neg-in 62.0%

         \[\leadsto x + \frac{0 - \left(y \cdot 1 + \color{blue}{\left(-y \cdot \frac{x}{y}\right)}\right)}{2} \]

      10. unsub-neg 62.0%

          \[\leadsto x + \frac{0 - \color{blue}{\left(y \cdot 1 - y \cdot \frac{x}{y}\right)}}{2} \]

      11. associate-*r/ 81.1%

          \[\leadsto x + \frac{0 - \left(y \cdot 1 - \color{blue}{\frac{y \cdot x}{y}}\right)}{2} \]

      12. associate-*l/ 89.8%

          \[\leadsto x + \frac{0 - \left(y \cdot 1 - \color{blue}{\frac{y}{y} \cdot x}\right)}{2} \]

      13. *-inverses 89.8%

          \[\leadsto x + \frac{0 - \left(y \cdot 1 - \color{blue}{1} \cdot x\right)}{2} \]

      14. *-lft-identity 89.8%

          \[\leadsto x + \frac{0 - \left(y \cdot 1 - \color{blue}{x}\right)}{2} \]

      15. *-rgt-identity 89.8%

          \[\leadsto x + \frac{0 - \left(\color{blue}{y} - x\right)}{2} \]

      16. associate-+l- 89.8%

          \[\leadsto x + \frac{\color{blue}{\left(0 - y\right) + x}}{2} \]

      17. neg-sub0 89.8%

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

      18. +-commutative 89.8%

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

      19. sub-neg 89.8%

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

   10. Simplified 89.8%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-93}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-57}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x - y}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.5% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(x + y\right)\\ \mathbf{if}\;x \leq -2.85 \cdot 10^{-154}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -6.1 \cdot 10^{-232}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-280}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-57}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ x y))))
   (if (<= x -2.85e-154)
     t_0
     (if (<= x -6.1e-232)
       (* y -0.5)
       (if (<= x 2.4e-280) t_0 (if (<= x 3e-57) (* y -0.5) (* x 1.5)))))))

C

double code(double x, double y) {
	double t_0 = 0.5 * (x + y);
	double tmp;
	if (x <= -2.85e-154) {
		tmp = t_0;
	} else if (x <= -6.1e-232) {
		tmp = y * -0.5;
	} else if (x <= 2.4e-280) {
		tmp = t_0;
	} else if (x <= 3e-57) {
		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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (x + y)
    if (x <= (-2.85d-154)) then
        tmp = t_0
    else if (x <= (-6.1d-232)) then
        tmp = y * (-0.5d0)
    else if (x <= 2.4d-280) then
        tmp = t_0
    else if (x <= 3d-57) 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 t_0 = 0.5 * (x + y);
	double tmp;
	if (x <= -2.85e-154) {
		tmp = t_0;
	} else if (x <= -6.1e-232) {
		tmp = y * -0.5;
	} else if (x <= 2.4e-280) {
		tmp = t_0;
	} else if (x <= 3e-57) {
		tmp = y * -0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.5 * (x + y)
	tmp = 0
	if x <= -2.85e-154:
		tmp = t_0
	elif x <= -6.1e-232:
		tmp = y * -0.5
	elif x <= 2.4e-280:
		tmp = t_0
	elif x <= 3e-57:
		tmp = y * -0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.5 * Float64(x + y))
	tmp = 0.0
	if (x <= -2.85e-154)
		tmp = t_0;
	elseif (x <= -6.1e-232)
		tmp = Float64(y * -0.5);
	elseif (x <= 2.4e-280)
		tmp = t_0;
	elseif (x <= 3e-57)
		tmp = Float64(y * -0.5);
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.5 * (x + y);
	tmp = 0.0;
	if (x <= -2.85e-154)
		tmp = t_0;
	elseif (x <= -6.1e-232)
		tmp = y * -0.5;
	elseif (x <= 2.4e-280)
		tmp = t_0;
	elseif (x <= 3e-57)
		tmp = y * -0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.85e-154], t$95$0, If[LessEqual[x, -6.1e-232], N[(y * -0.5), $MachinePrecision], If[LessEqual[x, 2.4e-280], t$95$0, If[LessEqual[x, 3e-57], N[(y * -0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.8499999999999999e-154 or -6.1000000000000001e-232 < x < 2.3999999999999998e-280

   1. Initial program 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. fma-define 100.0%

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

      3. add-sqr-sqrt 82.3%

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

      4. fabs-sqr 82.3%

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

      5. add-sqr-sqrt 83.4%

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

      6. frac-2neg 83.4%

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

      7. distribute-frac-neg 83.4%

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

      8. fmm-undef 83.4%

         \[\leadsto \color{blue}{1 \cdot x - \frac{y - x}{-2}} \]

      9. *-un-lft-identity 83.4%

         \[\leadsto \color{blue}{x} - \frac{y - x}{-2} \]

      10. metadata-eval 83.4%

          \[\leadsto x - \frac{y - x}{\color{blue}{-2}} \]

   4. Applied egg-rr 83.4%

      \[\leadsto \color{blue}{x - \frac{y - x}{-2}} \]

   5. Taylor expanded in x around 0 83.4%

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

      1. cancel-sign-sub-inv 83.4%

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

      2. metadata-eval 83.4%

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

      3. distribute-lft-in 83.4%

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

      4. +-commutative 83.4%

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

   7. Simplified 83.4%

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

   if -2.8499999999999999e-154 < x < -6.1000000000000001e-232 or 2.3999999999999998e-280 < x < 3.00000000000000001e-57

   1. Initial program 99.9%

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

      1. add-cube-cbrt 98.1%

         \[\leadsto x + \frac{\left|\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \sqrt[3]{y - x}}\right|}{2} \]

      2. fabs-mul 98.1%

         \[\leadsto x + \frac{\color{blue}{\left|\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]

      3. pow2 98.1%

         \[\leadsto x + \frac{\left|\color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2}}\right| \cdot \left|\sqrt[3]{y - x}\right|}{2} \]

   4. Applied egg-rr 98.1%

      \[\leadsto x + \frac{\color{blue}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]

   6. Applied egg-rr 62.1%

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

   7. Taylor expanded in x around 0 60.3%

      \[\leadsto \color{blue}{-0.5 \cdot y} \]
   8. Step-by-step derivation

      1. *-commutative 60.3%

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

   9. Simplified 60.3%

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

   if 3.00000000000000001e-57 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 74.3%

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

      1. neg-mul-1 74.3%

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

   5. Simplified 74.3%

      \[\leadsto x + \frac{\left|\color{blue}{-x}\right|}{2} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 74.2%

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

      2. associate-/l* 74.2%

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

      3. fabs-neg 74.2%

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

      4. add-sqr-sqrt 74.2%

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

      5. fabs-sqr 74.2%

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

      6. add-sqr-sqrt 74.2%

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

      7. fabs-neg 74.2%

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

      8. add-sqr-sqrt 74.2%

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

      9. fabs-sqr 74.2%

         \[\leadsto x + \sqrt{x} \cdot \frac{\sqrt{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}}{2} \]

      10. add-sqr-sqrt 74.2%

          \[\leadsto x + \sqrt{x} \cdot \frac{\sqrt{\color{blue}{x}}}{2} \]

   7. Applied egg-rr 74.2%

      \[\leadsto x + \color{blue}{\sqrt{x} \cdot \frac{\sqrt{x}}{2}} \]
   8. Step-by-step derivation

      1. associate-*r/ 74.2%

         \[\leadsto x + \color{blue}{\frac{\sqrt{x} \cdot \sqrt{x}}{2}} \]

      2. rem-square-sqrt 74.3%

         \[\leadsto x + \frac{\color{blue}{x}}{2} \]

   9. Simplified 74.3%

      \[\leadsto x + \color{blue}{\frac{x}{2}} \]

   10. Taylor expanded in x around 0 74.3%

       \[\leadsto \color{blue}{1.5 \cdot x} \]
   11. Step-by-step derivation

       1. *-commutative 74.3%

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

   12. Simplified 74.3%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{-154}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq -6.1 \cdot 10^{-232}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-280}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-57}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.1% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-103}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-252}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{-281}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-57}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9.5e-103)
   (* x 0.5)
   (if (<= x -3e-252)
     (* y -0.5)
     (if (<= x 7.4e-281) (* y 0.5) (if (<= x 3e-57) (* y -0.5) (* x 1.5))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9.5e-103) {
		tmp = x * 0.5;
	} else if (x <= -3e-252) {
		tmp = y * -0.5;
	} else if (x <= 7.4e-281) {
		tmp = y * 0.5;
	} else if (x <= 3e-57) {
		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 <= (-9.5d-103)) then
        tmp = x * 0.5d0
    else if (x <= (-3d-252)) then
        tmp = y * (-0.5d0)
    else if (x <= 7.4d-281) then
        tmp = y * 0.5d0
    else if (x <= 3d-57) 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 <= -9.5e-103) {
		tmp = x * 0.5;
	} else if (x <= -3e-252) {
		tmp = y * -0.5;
	} else if (x <= 7.4e-281) {
		tmp = y * 0.5;
	} else if (x <= 3e-57) {
		tmp = y * -0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9.5e-103:
		tmp = x * 0.5
	elif x <= -3e-252:
		tmp = y * -0.5
	elif x <= 7.4e-281:
		tmp = y * 0.5
	elif x <= 3e-57:
		tmp = y * -0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9.5e-103)
		tmp = Float64(x * 0.5);
	elseif (x <= -3e-252)
		tmp = Float64(y * -0.5);
	elseif (x <= 7.4e-281)
		tmp = Float64(y * 0.5);
	elseif (x <= 3e-57)
		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 <= -9.5e-103)
		tmp = x * 0.5;
	elseif (x <= -3e-252)
		tmp = y * -0.5;
	elseif (x <= 7.4e-281)
		tmp = y * 0.5;
	elseif (x <= 3e-57)
		tmp = y * -0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9.5e-103], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, -3e-252], N[(y * -0.5), $MachinePrecision], If[LessEqual[x, 7.4e-281], N[(y * 0.5), $MachinePrecision], If[LessEqual[x, 3e-57], N[(y * -0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -9.50000000000000065e-103

   1. Initial program 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. fma-define 100.0%

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

      3. add-sqr-sqrt 90.8%

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

      4. fabs-sqr 90.8%

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

      5. add-sqr-sqrt 91.6%

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

      6. frac-2neg 91.6%

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

      7. distribute-frac-neg 91.6%

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

      8. fmm-undef 91.6%

         \[\leadsto \color{blue}{1 \cdot x - \frac{y - x}{-2}} \]

      9. *-un-lft-identity 91.6%

         \[\leadsto \color{blue}{x} - \frac{y - x}{-2} \]

      10. metadata-eval 91.6%

          \[\leadsto x - \frac{y - x}{\color{blue}{-2}} \]

   4. Applied egg-rr 91.6%

      \[\leadsto \color{blue}{x - \frac{y - x}{-2}} \]

   5. Taylor expanded in x around inf 73.0%

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

   if -9.50000000000000065e-103 < x < -2.99999999999999995e-252 or 7.39999999999999983e-281 < x < 3.00000000000000001e-57

   1. Initial program 100.0%

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

      1. add-cube-cbrt 98.0%

         \[\leadsto x + \frac{\left|\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \sqrt[3]{y - x}}\right|}{2} \]

      2. fabs-mul 98.0%

         \[\leadsto x + \frac{\color{blue}{\left|\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]

      3. pow2 98.0%

         \[\leadsto x + \frac{\left|\color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2}}\right| \cdot \left|\sqrt[3]{y - x}\right|}{2} \]

   4. Applied egg-rr 98.0%

      \[\leadsto x + \frac{\color{blue}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]

   6. Applied egg-rr 63.9%

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

   7. Taylor expanded in x around 0 57.5%

      \[\leadsto \color{blue}{-0.5 \cdot y} \]
   8. Step-by-step derivation

      1. *-commutative 57.5%

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

   9. Simplified 57.5%

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

   if -2.99999999999999995e-252 < x < 7.39999999999999983e-281

   1. Initial program 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. fma-define 100.0%

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

      3. add-sqr-sqrt 69.6%

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

      4. fabs-sqr 69.6%

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

      5. add-sqr-sqrt 71.3%

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

      6. frac-2neg 71.3%

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

      7. distribute-frac-neg 71.3%

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

      8. fmm-undef 71.3%

         \[\leadsto \color{blue}{1 \cdot x - \frac{y - x}{-2}} \]

      9. *-un-lft-identity 71.3%

         \[\leadsto \color{blue}{x} - \frac{y - x}{-2} \]

      10. metadata-eval 71.3%

          \[\leadsto x - \frac{y - x}{\color{blue}{-2}} \]

   4. Applied egg-rr 71.3%

      \[\leadsto \color{blue}{x - \frac{y - x}{-2}} \]

   5. Taylor expanded in x around 0 67.9%

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

   if 3.00000000000000001e-57 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 74.3%

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

      1. neg-mul-1 74.3%

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

   5. Simplified 74.3%

      \[\leadsto x + \frac{\left|\color{blue}{-x}\right|}{2} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 74.2%

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

      2. associate-/l* 74.2%

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

      3. fabs-neg 74.2%

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

      4. add-sqr-sqrt 74.2%

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

      5. fabs-sqr 74.2%

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

      6. add-sqr-sqrt 74.2%

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

      7. fabs-neg 74.2%

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

      8. add-sqr-sqrt 74.2%

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

      9. fabs-sqr 74.2%

         \[\leadsto x + \sqrt{x} \cdot \frac{\sqrt{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}}{2} \]

      10. add-sqr-sqrt 74.2%

          \[\leadsto x + \sqrt{x} \cdot \frac{\sqrt{\color{blue}{x}}}{2} \]

   7. Applied egg-rr 74.2%

      \[\leadsto x + \color{blue}{\sqrt{x} \cdot \frac{\sqrt{x}}{2}} \]
   8. Step-by-step derivation

      1. associate-*r/ 74.2%

         \[\leadsto x + \color{blue}{\frac{\sqrt{x} \cdot \sqrt{x}}{2}} \]

      2. rem-square-sqrt 74.3%

         \[\leadsto x + \frac{\color{blue}{x}}{2} \]

   9. Simplified 74.3%

      \[\leadsto x + \color{blue}{\frac{x}{2}} \]

   10. Taylor expanded in x around 0 74.3%

       \[\leadsto \color{blue}{1.5 \cdot x} \]
   11. Step-by-step derivation

       1. *-commutative 74.3%

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

   12. Simplified 74.3%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-103}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-252}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{-281}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-57}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(x + y\right)\\ \mathbf{if}\;y \leq -2.65 \cdot 10^{-133}:\\ \;\;\;\;x - t\_0\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{-279}:\\ \;\;\;\;x + \frac{x - y}{2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ x y))))
   (if (<= y -2.65e-133)
     (- x t_0)
     (if (<= y -2.4e-279) (+ x (/ (- x y) 2.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = 0.5 * (x + y);
	double tmp;
	if (y <= -2.65e-133) {
		tmp = x - t_0;
	} else if (y <= -2.4e-279) {
		tmp = x + ((x - y) / 2.0);
	} else {
		tmp = t_0;
	}
	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 = 0.5d0 * (x + y)
    if (y <= (-2.65d-133)) then
        tmp = x - t_0
    else if (y <= (-2.4d-279)) then
        tmp = x + ((x - y) / 2.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.5 * (x + y);
	double tmp;
	if (y <= -2.65e-133) {
		tmp = x - t_0;
	} else if (y <= -2.4e-279) {
		tmp = x + ((x - y) / 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.5 * (x + y)
	tmp = 0
	if y <= -2.65e-133:
		tmp = x - t_0
	elif y <= -2.4e-279:
		tmp = x + ((x - y) / 2.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.5 * Float64(x + y))
	tmp = 0.0
	if (y <= -2.65e-133)
		tmp = Float64(x - t_0);
	elseif (y <= -2.4e-279)
		tmp = Float64(x + Float64(Float64(x - y) / 2.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.5 * (x + y);
	tmp = 0.0;
	if (y <= -2.65e-133)
		tmp = x - t_0;
	elseif (y <= -2.4e-279)
		tmp = x + ((x - y) / 2.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.65e-133], N[(x - t$95$0), $MachinePrecision], If[LessEqual[y, -2.4e-279], N[(x + N[(N[(x - y), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.64999999999999992e-133

   1. Initial program 100.0%

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

      1. add-cube-cbrt 98.3%

         \[\leadsto x + \frac{\left|\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \sqrt[3]{y - x}}\right|}{2} \]

      2. fabs-mul 98.3%

         \[\leadsto x + \frac{\color{blue}{\left|\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]

      3. pow2 98.3%

         \[\leadsto x + \frac{\left|\color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2}}\right| \cdot \left|\sqrt[3]{y - x}\right|}{2} \]

   4. Applied egg-rr 98.3%

      \[\leadsto x + \frac{\color{blue}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]

   6. Applied egg-rr 88.4%

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

   if -2.64999999999999992e-133 < y < -2.3999999999999999e-279

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 67.6%

      \[\leadsto x + \frac{\left|\color{blue}{y \cdot \left(1 + -1 \cdot \frac{x}{y}\right)}\right|}{2} \]
   4. Step-by-step derivation

      1. mul-1-neg 67.6%

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

      2. unsub-neg 67.6%

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

   5. Simplified 67.6%

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

   6. Taylor expanded in x around inf 67.3%

      \[\leadsto x + \frac{\left|y \cdot \color{blue}{\left(x \cdot \left(\frac{1}{x} - \frac{1}{y}\right)\right)}\right|}{2} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 16.9%

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

      2. fabs-sqr 16.9%

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

      3. add-sqr-sqrt 21.9%

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

      4. associate-*r* 26.8%

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

      5. add-sqr-sqrt 10.0%

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

      6. sqrt-unprod 8.9%

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

      7. sqr-neg 8.9%

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

      8. sqrt-unprod 4.7%

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

      9. add-sqr-sqrt 51.9%

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

      10. distribute-rgt-neg-out 51.9%

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

      11. distribute-lft-neg-in 51.9%

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

      12. associate-*r* 53.7%

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

      13. add-sqr-sqrt 3.3%

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

      14. fabs-sqr 3.3%

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

      15. add-sqr-sqrt 8.2%

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

   8. Applied egg-rr 81.6%

      \[\leadsto x + \frac{\color{blue}{0 - x \cdot \left(\left(\frac{1}{x} + \frac{-1}{y}\right) \cdot y\right)}}{2} \]
   9. Step-by-step derivation

      1. associate-*r* 53.7%

         \[\leadsto x + \frac{0 - \color{blue}{\left(x \cdot \left(\frac{1}{x} + \frac{-1}{y}\right)\right) \cdot y}}{2} \]

      2. *-commutative 53.7%

         \[\leadsto x + \frac{0 - \color{blue}{y \cdot \left(x \cdot \left(\frac{1}{x} + \frac{-1}{y}\right)\right)}}{2} \]

      3. distribute-rgt-in 53.7%

         \[\leadsto x + \frac{0 - y \cdot \color{blue}{\left(\frac{1}{x} \cdot x + \frac{-1}{y} \cdot x\right)}}{2} \]

      4. lft-mult-inverse 53.8%

         \[\leadsto x + \frac{0 - y \cdot \left(\color{blue}{1} + \frac{-1}{y} \cdot x\right)}{2} \]

      5. associate-*l/ 53.9%

         \[\leadsto x + \frac{0 - y \cdot \left(1 + \color{blue}{\frac{-1 \cdot x}{y}}\right)}{2} \]

      6. associate-*r/ 53.9%

         \[\leadsto x + \frac{0 - y \cdot \left(1 + \color{blue}{-1 \cdot \frac{x}{y}}\right)}{2} \]

      7. distribute-lft-in 53.9%

         \[\leadsto x + \frac{0 - \color{blue}{\left(y \cdot 1 + y \cdot \left(-1 \cdot \frac{x}{y}\right)\right)}}{2} \]

      8. neg-mul-1 53.9%

         \[\leadsto x + \frac{0 - \left(y \cdot 1 + y \cdot \color{blue}{\left(-\frac{x}{y}\right)}\right)}{2} \]

      9. distribute-rgt-neg-in 53.9%

         \[\leadsto x + \frac{0 - \left(y \cdot 1 + \color{blue}{\left(-y \cdot \frac{x}{y}\right)}\right)}{2} \]

      10. unsub-neg 53.9%

          \[\leadsto x + \frac{0 - \color{blue}{\left(y \cdot 1 - y \cdot \frac{x}{y}\right)}}{2} \]

      11. associate-*r/ 74.3%

          \[\leadsto x + \frac{0 - \left(y \cdot 1 - \color{blue}{\frac{y \cdot x}{y}}\right)}{2} \]

      12. associate-*l/ 81.7%

          \[\leadsto x + \frac{0 - \left(y \cdot 1 - \color{blue}{\frac{y}{y} \cdot x}\right)}{2} \]

      13. *-inverses 81.7%

          \[\leadsto x + \frac{0 - \left(y \cdot 1 - \color{blue}{1} \cdot x\right)}{2} \]

      14. *-lft-identity 81.7%

          \[\leadsto x + \frac{0 - \left(y \cdot 1 - \color{blue}{x}\right)}{2} \]

      15. *-rgt-identity 81.7%

          \[\leadsto x + \frac{0 - \left(\color{blue}{y} - x\right)}{2} \]

      16. associate-+l- 81.7%

          \[\leadsto x + \frac{\color{blue}{\left(0 - y\right) + x}}{2} \]

      17. neg-sub0 81.7%

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

      18. +-commutative 81.7%

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

      19. sub-neg 81.7%

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

   10. Simplified 81.7%

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

   if -2.3999999999999999e-279 < y

   1. Initial program 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. fma-define 99.9%

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

      3. add-sqr-sqrt 78.8%

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

      4. fabs-sqr 78.8%

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

      5. add-sqr-sqrt 83.1%

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

      6. frac-2neg 83.1%

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

      7. distribute-frac-neg 83.1%

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

      8. fmm-undef 83.1%

         \[\leadsto \color{blue}{1 \cdot x - \frac{y - x}{-2}} \]

      9. *-un-lft-identity 83.1%

         \[\leadsto \color{blue}{x} - \frac{y - x}{-2} \]

      10. metadata-eval 83.1%

          \[\leadsto x - \frac{y - x}{\color{blue}{-2}} \]

   4. Applied egg-rr 83.1%

      \[\leadsto \color{blue}{x - \frac{y - x}{-2}} \]

   5. Taylor expanded in x around 0 83.1%

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

      1. cancel-sign-sub-inv 83.1%

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

      2. metadata-eval 83.1%

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

      3. distribute-lft-in 83.1%

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

      4. +-commutative 83.1%

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

   7. Simplified 83.1%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{-133}:\\ \;\;\;\;x - 0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{-279}:\\ \;\;\;\;x + \frac{x - y}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.8% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-133}:\\ \;\;\;\;0.5 \cdot \left(x - y\right)\\ \mathbf{elif}\;y \leq -1.22 \cdot 10^{-281}:\\ \;\;\;\;x + \frac{x - y}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.8e-133)
   (* 0.5 (- x y))
   (if (<= y -1.22e-281) (+ x (/ (- x y) 2.0)) (* 0.5 (+ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.8e-133) {
		tmp = 0.5 * (x - y);
	} else if (y <= -1.22e-281) {
		tmp = x + ((x - y) / 2.0);
	} 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 <= (-2.8d-133)) then
        tmp = 0.5d0 * (x - y)
    else if (y <= (-1.22d-281)) then
        tmp = x + ((x - y) / 2.0d0)
    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 <= -2.8e-133) {
		tmp = 0.5 * (x - y);
	} else if (y <= -1.22e-281) {
		tmp = x + ((x - y) / 2.0);
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.8e-133:
		tmp = 0.5 * (x - y)
	elif y <= -1.22e-281:
		tmp = x + ((x - y) / 2.0)
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.8e-133)
		tmp = Float64(0.5 * Float64(x - y));
	elseif (y <= -1.22e-281)
		tmp = Float64(x + Float64(Float64(x - y) / 2.0));
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.8e-133)
		tmp = 0.5 * (x - y);
	elseif (y <= -1.22e-281)
		tmp = x + ((x - y) / 2.0);
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.8e-133], N[(0.5 * N[(x - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.22e-281], N[(x + N[(N[(x - y), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.7999999999999999e-133

   1. Initial program 100.0%

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

      1. add-cube-cbrt 98.3%

         \[\leadsto x + \frac{\left|\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \sqrt[3]{y - x}}\right|}{2} \]

      2. fabs-mul 98.3%

         \[\leadsto x + \frac{\color{blue}{\left|\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]

      3. pow2 98.3%

         \[\leadsto x + \frac{\left|\color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2}}\right| \cdot \left|\sqrt[3]{y - x}\right|}{2} \]

   4. Applied egg-rr 98.3%

      \[\leadsto x + \frac{\color{blue}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]

   6. Applied egg-rr 88.4%

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

   7. Taylor expanded in x around 0 88.4%

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

      1. distribute-lft-out-- 88.4%

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

   9. Simplified 88.4%

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

   if -2.7999999999999999e-133 < y < -1.21999999999999996e-281

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 67.6%

      \[\leadsto x + \frac{\left|\color{blue}{y \cdot \left(1 + -1 \cdot \frac{x}{y}\right)}\right|}{2} \]
   4. Step-by-step derivation

      1. mul-1-neg 67.6%

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

      2. unsub-neg 67.6%

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

   5. Simplified 67.6%

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

   6. Taylor expanded in x around inf 67.3%

      \[\leadsto x + \frac{\left|y \cdot \color{blue}{\left(x \cdot \left(\frac{1}{x} - \frac{1}{y}\right)\right)}\right|}{2} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 16.9%

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

      2. fabs-sqr 16.9%

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

      3. add-sqr-sqrt 21.9%

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

      4. associate-*r* 26.8%

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

      5. add-sqr-sqrt 10.0%

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

      6. sqrt-unprod 8.9%

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

      7. sqr-neg 8.9%

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

      8. sqrt-unprod 4.7%

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

      9. add-sqr-sqrt 51.9%

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

      10. distribute-rgt-neg-out 51.9%

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

      11. distribute-lft-neg-in 51.9%

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

      12. associate-*r* 53.7%

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

      13. add-sqr-sqrt 3.3%

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

      14. fabs-sqr 3.3%

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

      15. add-sqr-sqrt 8.2%

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

   8. Applied egg-rr 81.6%

      \[\leadsto x + \frac{\color{blue}{0 - x \cdot \left(\left(\frac{1}{x} + \frac{-1}{y}\right) \cdot y\right)}}{2} \]
   9. Step-by-step derivation

      1. associate-*r* 53.7%

         \[\leadsto x + \frac{0 - \color{blue}{\left(x \cdot \left(\frac{1}{x} + \frac{-1}{y}\right)\right) \cdot y}}{2} \]

      2. *-commutative 53.7%

         \[\leadsto x + \frac{0 - \color{blue}{y \cdot \left(x \cdot \left(\frac{1}{x} + \frac{-1}{y}\right)\right)}}{2} \]

      3. distribute-rgt-in 53.7%

         \[\leadsto x + \frac{0 - y \cdot \color{blue}{\left(\frac{1}{x} \cdot x + \frac{-1}{y} \cdot x\right)}}{2} \]

      4. lft-mult-inverse 53.8%

         \[\leadsto x + \frac{0 - y \cdot \left(\color{blue}{1} + \frac{-1}{y} \cdot x\right)}{2} \]

      5. associate-*l/ 53.9%

         \[\leadsto x + \frac{0 - y \cdot \left(1 + \color{blue}{\frac{-1 \cdot x}{y}}\right)}{2} \]

      6. associate-*r/ 53.9%

         \[\leadsto x + \frac{0 - y \cdot \left(1 + \color{blue}{-1 \cdot \frac{x}{y}}\right)}{2} \]

      7. distribute-lft-in 53.9%

         \[\leadsto x + \frac{0 - \color{blue}{\left(y \cdot 1 + y \cdot \left(-1 \cdot \frac{x}{y}\right)\right)}}{2} \]

      8. neg-mul-1 53.9%

         \[\leadsto x + \frac{0 - \left(y \cdot 1 + y \cdot \color{blue}{\left(-\frac{x}{y}\right)}\right)}{2} \]

      9. distribute-rgt-neg-in 53.9%

         \[\leadsto x + \frac{0 - \left(y \cdot 1 + \color{blue}{\left(-y \cdot \frac{x}{y}\right)}\right)}{2} \]

      10. unsub-neg 53.9%

          \[\leadsto x + \frac{0 - \color{blue}{\left(y \cdot 1 - y \cdot \frac{x}{y}\right)}}{2} \]

      11. associate-*r/ 74.3%

          \[\leadsto x + \frac{0 - \left(y \cdot 1 - \color{blue}{\frac{y \cdot x}{y}}\right)}{2} \]

      12. associate-*l/ 81.7%

          \[\leadsto x + \frac{0 - \left(y \cdot 1 - \color{blue}{\frac{y}{y} \cdot x}\right)}{2} \]

      13. *-inverses 81.7%

          \[\leadsto x + \frac{0 - \left(y \cdot 1 - \color{blue}{1} \cdot x\right)}{2} \]

      14. *-lft-identity 81.7%

          \[\leadsto x + \frac{0 - \left(y \cdot 1 - \color{blue}{x}\right)}{2} \]

      15. *-rgt-identity 81.7%

          \[\leadsto x + \frac{0 - \left(\color{blue}{y} - x\right)}{2} \]

      16. associate-+l- 81.7%

          \[\leadsto x + \frac{\color{blue}{\left(0 - y\right) + x}}{2} \]

      17. neg-sub0 81.7%

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

      18. +-commutative 81.7%

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

      19. sub-neg 81.7%

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

   10. Simplified 81.7%

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

   if -1.21999999999999996e-281 < y

   1. Initial program 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. fma-define 99.9%

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

      3. add-sqr-sqrt 78.8%

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

      4. fabs-sqr 78.8%

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

      5. add-sqr-sqrt 83.1%

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

      6. frac-2neg 83.1%

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

      7. distribute-frac-neg 83.1%

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

      8. fmm-undef 83.1%

         \[\leadsto \color{blue}{1 \cdot x - \frac{y - x}{-2}} \]

      9. *-un-lft-identity 83.1%

         \[\leadsto \color{blue}{x} - \frac{y - x}{-2} \]

      10. metadata-eval 83.1%

          \[\leadsto x - \frac{y - x}{\color{blue}{-2}} \]

   4. Applied egg-rr 83.1%

      \[\leadsto \color{blue}{x - \frac{y - x}{-2}} \]

   5. Taylor expanded in x around 0 83.1%

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

      1. cancel-sign-sub-inv 83.1%

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

      2. metadata-eval 83.1%

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

      3. distribute-lft-in 83.1%

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

      4. +-commutative 83.1%

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

   7. Simplified 83.1%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-133}:\\ \;\;\;\;0.5 \cdot \left(x - y\right)\\ \mathbf{elif}\;y \leq -1.22 \cdot 10^{-281}:\\ \;\;\;\;x + \frac{x - y}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.9% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-184}:\\ \;\;\;\;0.5 \cdot \left(x - y\right)\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{-280}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.05e-184)
   (* 0.5 (- x y))
   (if (<= y -3.8e-280) (* x 1.5) (* 0.5 (+ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.05e-184) {
		tmp = 0.5 * (x - y);
	} else if (y <= -3.8e-280) {
		tmp = x * 1.5;
	} 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 <= (-1.05d-184)) then
        tmp = 0.5d0 * (x - y)
    else if (y <= (-3.8d-280)) then
        tmp = x * 1.5d0
    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 <= -1.05e-184) {
		tmp = 0.5 * (x - y);
	} else if (y <= -3.8e-280) {
		tmp = x * 1.5;
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.05e-184:
		tmp = 0.5 * (x - y)
	elif y <= -3.8e-280:
		tmp = x * 1.5
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.05e-184)
		tmp = Float64(0.5 * Float64(x - y));
	elseif (y <= -3.8e-280)
		tmp = Float64(x * 1.5);
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.05e-184)
		tmp = 0.5 * (x - y);
	elseif (y <= -3.8e-280)
		tmp = x * 1.5;
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.05e-184], N[(0.5 * N[(x - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -3.8e-280], N[(x * 1.5), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.0499999999999999e-184

   1. Initial program 100.0%

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

      1. add-cube-cbrt 98.2%

         \[\leadsto x + \frac{\left|\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \sqrt[3]{y - x}}\right|}{2} \]

      2. fabs-mul 98.2%

         \[\leadsto x + \frac{\color{blue}{\left|\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]

      3. pow2 98.2%

         \[\leadsto x + \frac{\left|\color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2}}\right| \cdot \left|\sqrt[3]{y - x}\right|}{2} \]

   4. Applied egg-rr 98.2%

      \[\leadsto x + \frac{\color{blue}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]

   6. Applied egg-rr 85.2%

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

   7. Taylor expanded in x around 0 85.2%

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

      1. distribute-lft-out-- 85.2%

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

   9. Simplified 85.2%

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

   if -1.0499999999999999e-184 < y < -3.8000000000000001e-280

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 84.3%

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

      1. neg-mul-1 84.3%

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

   5. Simplified 84.3%

      \[\leadsto x + \frac{\left|\color{blue}{-x}\right|}{2} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 84.2%

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

      2. associate-/l* 84.2%

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

      3. fabs-neg 84.2%

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

      4. add-sqr-sqrt 63.0%

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

      5. fabs-sqr 63.0%

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

      6. add-sqr-sqrt 63.0%

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

      7. fabs-neg 63.0%

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

      8. add-sqr-sqrt 63.0%

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

      9. fabs-sqr 63.0%

         \[\leadsto x + \sqrt{x} \cdot \frac{\sqrt{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}}{2} \]

      10. add-sqr-sqrt 63.0%

          \[\leadsto x + \sqrt{x} \cdot \frac{\sqrt{\color{blue}{x}}}{2} \]

   7. Applied egg-rr 63.0%

      \[\leadsto x + \color{blue}{\sqrt{x} \cdot \frac{\sqrt{x}}{2}} \]
   8. Step-by-step derivation

      1. associate-*r/ 63.0%

         \[\leadsto x + \color{blue}{\frac{\sqrt{x} \cdot \sqrt{x}}{2}} \]

      2. rem-square-sqrt 67.2%

         \[\leadsto x + \frac{\color{blue}{x}}{2} \]

   9. Simplified 67.2%

      \[\leadsto x + \color{blue}{\frac{x}{2}} \]

   10. Taylor expanded in x around 0 67.2%

       \[\leadsto \color{blue}{1.5 \cdot x} \]
   11. Step-by-step derivation

       1. *-commutative 67.2%

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

   12. Simplified 67.2%

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

   if -3.8000000000000001e-280 < y

   1. Initial program 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. fma-define 99.9%

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

      3. add-sqr-sqrt 78.8%

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

      4. fabs-sqr 78.8%

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

      5. add-sqr-sqrt 83.1%

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

      6. frac-2neg 83.1%

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

      7. distribute-frac-neg 83.1%

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

      8. fmm-undef 83.1%

         \[\leadsto \color{blue}{1 \cdot x - \frac{y - x}{-2}} \]

      9. *-un-lft-identity 83.1%

         \[\leadsto \color{blue}{x} - \frac{y - x}{-2} \]

      10. metadata-eval 83.1%

          \[\leadsto x - \frac{y - x}{\color{blue}{-2}} \]

   4. Applied egg-rr 83.1%

      \[\leadsto \color{blue}{x - \frac{y - x}{-2}} \]

   5. Taylor expanded in x around 0 83.1%

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

      1. cancel-sign-sub-inv 83.1%

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

      2. metadata-eval 83.1%

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

      3. distribute-lft-in 83.1%

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

      4. +-commutative 83.1%

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

   7. Simplified 83.1%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-184}:\\ \;\;\;\;0.5 \cdot \left(x - y\right)\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{-280}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.7% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-69}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-57}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.85e-69) (* x 0.5) (if (<= x 4.3e-57) (* y 0.5) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.85e-69) {
		tmp = x * 0.5;
	} else if (x <= 4.3e-57) {
		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 <= (-1.85d-69)) then
        tmp = x * 0.5d0
    else if (x <= 4.3d-57) 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 <= -1.85e-69) {
		tmp = x * 0.5;
	} else if (x <= 4.3e-57) {
		tmp = y * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.85e-69:
		tmp = x * 0.5
	elif x <= 4.3e-57:
		tmp = y * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.85e-69)
		tmp = Float64(x * 0.5);
	elseif (x <= 4.3e-57)
		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 <= -1.85e-69)
		tmp = x * 0.5;
	elseif (x <= 4.3e-57)
		tmp = y * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.85e-69], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 4.3e-57], N[(y * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.8500000000000001e-69

   1. Initial program 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. fma-define 100.0%

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

      3. add-sqr-sqrt 90.8%

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

      4. fabs-sqr 90.8%

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

      5. add-sqr-sqrt 91.6%

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

      6. frac-2neg 91.6%

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

      7. distribute-frac-neg 91.6%

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

      8. fmm-undef 91.6%

         \[\leadsto \color{blue}{1 \cdot x - \frac{y - x}{-2}} \]

      9. *-un-lft-identity 91.6%

         \[\leadsto \color{blue}{x} - \frac{y - x}{-2} \]

      10. metadata-eval 91.6%

          \[\leadsto x - \frac{y - x}{\color{blue}{-2}} \]

   4. Applied egg-rr 91.6%

      \[\leadsto \color{blue}{x - \frac{y - x}{-2}} \]

   5. Taylor expanded in x around inf 79.5%

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

   if -1.8500000000000001e-69 < x < 4.30000000000000022e-57

   1. Initial program 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. fma-define 100.0%

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

      3. add-sqr-sqrt 50.0%

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

      4. fabs-sqr 50.0%

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

      5. add-sqr-sqrt 52.0%

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

      6. frac-2neg 52.0%

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

      7. distribute-frac-neg 52.0%

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

      8. fmm-undef 52.0%

         \[\leadsto \color{blue}{1 \cdot x - \frac{y - x}{-2}} \]

      9. *-un-lft-identity 52.0%

         \[\leadsto \color{blue}{x} - \frac{y - x}{-2} \]

      10. metadata-eval 52.0%

          \[\leadsto x - \frac{y - x}{\color{blue}{-2}} \]

   4. Applied egg-rr 52.0%

      \[\leadsto \color{blue}{x - \frac{y - x}{-2}} \]

   5. Taylor expanded in x around 0 43.2%

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

   if 4.30000000000000022e-57 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 75.3%

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

      1. neg-mul-1 75.3%

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

   5. Simplified 75.3%

      \[\leadsto x + \frac{\left|\color{blue}{-x}\right|}{2} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 75.2%

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

      2. associate-/l* 75.2%

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

      3. fabs-neg 75.2%

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

      4. add-sqr-sqrt 75.2%

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

      5. fabs-sqr 75.2%

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

      6. add-sqr-sqrt 75.2%

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

      7. fabs-neg 75.2%

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

      8. add-sqr-sqrt 75.2%

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

      9. fabs-sqr 75.2%

         \[\leadsto x + \sqrt{x} \cdot \frac{\sqrt{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}}{2} \]

      10. add-sqr-sqrt 75.2%

          \[\leadsto x + \sqrt{x} \cdot \frac{\sqrt{\color{blue}{x}}}{2} \]

   7. Applied egg-rr 75.2%

      \[\leadsto x + \color{blue}{\sqrt{x} \cdot \frac{\sqrt{x}}{2}} \]
   8. Step-by-step derivation

      1. associate-*r/ 75.2%

         \[\leadsto x + \color{blue}{\frac{\sqrt{x} \cdot \sqrt{x}}{2}} \]

      2. rem-square-sqrt 75.3%

         \[\leadsto x + \frac{\color{blue}{x}}{2} \]

   9. Simplified 75.3%

      \[\leadsto x + \color{blue}{\frac{x}{2}} \]

   10. Taylor expanded in x around 0 75.3%

       \[\leadsto \color{blue}{1.5 \cdot x} \]
   11. Step-by-step derivation

       1. *-commutative 75.3%

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

   12. Simplified 75.3%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-69}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-57}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.3% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{-31}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 5.6e-31) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.6e-31) {
		tmp = x * 0.5;
	} 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 <= 5.6d-31) then
        tmp = x * 0.5d0
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.6e-31) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.6e-31:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.6e-31)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.6e-31)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.6e-31], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.5999999999999998e-31

   1. Initial program 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. fma-define 99.9%

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

      3. add-sqr-sqrt 38.9%

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

      4. fabs-sqr 38.9%

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

      5. add-sqr-sqrt 44.4%

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

      6. frac-2neg 44.4%

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

      7. distribute-frac-neg 44.4%

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

      8. fmm-undef 44.4%

         \[\leadsto \color{blue}{1 \cdot x - \frac{y - x}{-2}} \]

      9. *-un-lft-identity 44.4%

         \[\leadsto \color{blue}{x} - \frac{y - x}{-2} \]

      10. metadata-eval 44.4%

          \[\leadsto x - \frac{y - x}{\color{blue}{-2}} \]

   4. Applied egg-rr 44.4%

      \[\leadsto \color{blue}{x - \frac{y - x}{-2}} \]

   5. Taylor expanded in x around inf 37.6%

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

   if 5.5999999999999998e-31 < y

   1. Initial program 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. fma-define 99.9%

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

      3. add-sqr-sqrt 87.6%

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

      4. fabs-sqr 87.6%

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

      5. add-sqr-sqrt 90.4%

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

      6. frac-2neg 90.4%

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

      7. distribute-frac-neg 90.4%

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

      8. fmm-undef 90.4%

         \[\leadsto \color{blue}{1 \cdot x - \frac{y - x}{-2}} \]

      9. *-un-lft-identity 90.4%

         \[\leadsto \color{blue}{x} - \frac{y - x}{-2} \]

      10. metadata-eval 90.4%

          \[\leadsto x - \frac{y - x}{\color{blue}{-2}} \]

   4. Applied egg-rr 90.4%

      \[\leadsto \color{blue}{x - \frac{y - x}{-2}} \]

   5. Taylor expanded in x around 0 80.7%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{-31}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 30.2% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x 0.5))

C

double code(double x, double y) {
	return x * 0.5;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * 0.5d0
end function

Java

public static double code(double x, double y) {
	return x * 0.5;
}

Python

def code(x, y):
	return x * 0.5

Julia

function code(x, y)
	return Float64(x * 0.5)
end

MATLAB

function tmp = code(x, y)
	tmp = x * 0.5;
end

Wolfram

code[x_, y_] := N[(x * 0.5), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. fma-define 99.9%

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

   3. add-sqr-sqrt 50.3%

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

   4. fabs-sqr 50.3%

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

   5. add-sqr-sqrt 55.2%

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

   6. frac-2neg 55.2%

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

   7. distribute-frac-neg 55.2%

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

   8. fmm-undef 55.2%

      \[\leadsto \color{blue}{1 \cdot x - \frac{y - x}{-2}} \]

   9. *-un-lft-identity 55.2%

      \[\leadsto \color{blue}{x} - \frac{y - x}{-2} \]

   10. metadata-eval 55.2%

       \[\leadsto x - \frac{y - x}{\color{blue}{-2}} \]

4. Applied egg-rr 55.2%

   \[\leadsto \color{blue}{x - \frac{y - x}{-2}} \]

5. Taylor expanded in x around inf 31.5%

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

6. Final simplification 31.5%

   \[\leadsto x \cdot 0.5 \]
7. Add Preprocessing

Alternative 11: 11.3% 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. Add Preprocessing

3. Taylor expanded in x around inf 11.2%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024180 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))