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

Percentage Accurate: 99.9% → 99.9%

Time: 5.8s

Alternatives: 8

Speedup: 1.7×

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, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left|y - x\right|, 0.5, x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma (fabs (- y x)) 0.5 x))

C

double code(double x, double y) {
	return fma(fabs((y - x)), 0.5, x);
}

Julia

function code(x, y)
	return fma(abs(Float64(y - x)), 0.5, x)
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. *-lft-identity N/A

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

   5. metadata-eval N/A

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

   6. fabs-sub N/A

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

   7. fp-cancel-sign-sub-inv N/A

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

   8. lower-fabs.f64 N/A

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

   9. fp-cancel-sign-sub-inv N/A

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

   10. metadata-eval N/A

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

   11. *-lft-identity N/A

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

   12. lower--.f64 99.9

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

5. Applied rewrites 99.9%

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

6. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(\left|y - x\right|, 0.5, x\right) \]
7. Add Preprocessing

Alternative 2: 49.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \frac{\left|y - x\right|}{2} \leq 4.8 \cdot 10^{-246}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ x (/ (fabs (- y x)) 2.0)) 4.8e-246) (* 0.5 x) (* 0.5 y)))

C

double code(double x, double y) {
	double tmp;
	if ((x + (fabs((y - x)) / 2.0)) <= 4.8e-246) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * 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 + (abs((y - x)) / 2.0d0)) <= 4.8d-246) then
        tmp = 0.5d0 * x
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x + (Math.abs((y - x)) / 2.0)) <= 4.8e-246) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x + (math.fabs((y - x)) / 2.0)) <= 4.8e-246:
		tmp = 0.5 * x
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x + Float64(abs(Float64(y - x)) / 2.0)) <= 4.8e-246)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x + (abs((y - x)) / 2.0)) <= 4.8e-246)
		tmp = 0.5 * x;
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x + N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 4.8e-246], N[(0.5 * x), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (fabs.f64 (-.f64 y x)) #s(literal 2 binary64))) < 4.7999999999999996e-246

   1. Initial program 100.0%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 93.5

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

   4. Applied rewrites 93.5%

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

   5. Taylor expanded in y around 0

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 94.4

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

   7. Applied rewrites 94.4%

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

   if 4.7999999999999996e-246 < (+.f64 x (/.f64 (fabs.f64 (-.f64 y x)) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 36.4

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

   4. Applied rewrites 36.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot y} \]
   6. Step-by-step derivation

      1. lower-*.f64 36.9

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

   7. Applied rewrites 36.9%

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

Alternative 3: 61.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{-81}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq -1 \cdot 10^{-213}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+14}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.7e-81)
   (* -0.5 y)
   (if (<= y -1e-213) (* 0.5 x) (if (<= y 1.95e+14) (* 1.5 x) (* 0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.7e-81) {
		tmp = -0.5 * y;
	} else if (y <= -1e-213) {
		tmp = 0.5 * x;
	} else if (y <= 1.95e+14) {
		tmp = 1.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.7d-81)) then
        tmp = (-0.5d0) * y
    else if (y <= (-1d-213)) then
        tmp = 0.5d0 * x
    else if (y <= 1.95d+14) then
        tmp = 1.5d0 * x
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.7e-81) {
		tmp = -0.5 * y;
	} else if (y <= -1e-213) {
		tmp = 0.5 * x;
	} else if (y <= 1.95e+14) {
		tmp = 1.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.7e-81:
		tmp = -0.5 * y
	elif y <= -1e-213:
		tmp = 0.5 * x
	elif y <= 1.95e+14:
		tmp = 1.5 * x
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.7e-81)
		tmp = Float64(-0.5 * y);
	elseif (y <= -1e-213)
		tmp = Float64(0.5 * x);
	elseif (y <= 1.95e+14)
		tmp = Float64(1.5 * x);
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.7e-81)
		tmp = -0.5 * y;
	elseif (y <= -1e-213)
		tmp = 0.5 * x;
	elseif (y <= 1.95e+14)
		tmp = 1.5 * x;
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.7e-81], N[(-0.5 * y), $MachinePrecision], If[LessEqual[y, -1e-213], N[(0.5 * x), $MachinePrecision], If[LessEqual[y, 1.95e+14], N[(1.5 * x), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.69999999999999986e-81

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. pow2 N/A

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

      6. sqrt-prod N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 53.1

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

      10. lift-/.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. sqrt-prod N/A

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

      14. rem-square-sqrt N/A

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

      15. frac-2neg N/A

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

   4. Applied rewrites 53.1%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot y} \]
   6. Step-by-step derivation

      1. lower-*.f64 67.4

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

   7. Applied rewrites 67.4%

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

   if -3.69999999999999986e-81 < y < -9.9999999999999995e-214

   1. Initial program 99.9%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 55.2

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

   4. Applied rewrites 55.2%

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

   5. Taylor expanded in y around 0

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 60.4

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

   7. Applied rewrites 60.4%

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

   if -9.9999999999999995e-214 < y < 1.95e14

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. pow2 N/A

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

      6. sqrt-prod N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 58.5

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

      10. lift-/.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. sqrt-prod N/A

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

      14. rem-square-sqrt N/A

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

      15. frac-2neg N/A

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

   4. Applied rewrites 50.6%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{3}{2} \cdot x} \]
   6. Step-by-step derivation

      1. lower-*.f64 53.0

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

   7. Applied rewrites 53.0%

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

   if 1.95e14 < y

   1. Initial program 100.0%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 94.7

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

   4. Applied rewrites 94.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot y} \]
   6. Step-by-step derivation

      1. lower-*.f64 84.4

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

   7. Applied rewrites 84.4%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 78.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-213}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-257}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1e-213)
   (* (- x y) 0.5)
   (if (<= y 7.5e-257) (fma (- x y) 0.5 x) (* 0.5 (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1e-213) {
		tmp = (x - y) * 0.5;
	} else if (y <= 7.5e-257) {
		tmp = fma((x - y), 0.5, x);
	} else {
		tmp = 0.5 * (y + x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1e-213)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (y <= 7.5e-257)
		tmp = fma(Float64(x - y), 0.5, x);
	else
		tmp = Float64(0.5 * Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1e-213], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[y, 7.5e-257], N[(N[(x - y), $MachinePrecision] * 0.5 + x), $MachinePrecision], N[(0.5 * N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.9999999999999995e-214

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fabs-sub N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. lower-fabs.f64 N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 58.0

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

   5. Applied rewrites 58.0%

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

   7. Applied rewrites 85.7%

      \[\leadsto \left(x - y\right) \cdot 0.5 \]

   if -9.9999999999999995e-214 < y < 7.4999999999999995e-257

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. fabs-sub N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. lower-fabs.f64 N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 77.6%

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

   if 7.4999999999999995e-257 < y

   1. Initial program 99.9%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 76.8

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

   4. Applied rewrites 76.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + \frac{1}{2} \cdot y} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 81.4

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

   7. Applied rewrites 81.4%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-213}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-257}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-246}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-257}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.2e-246)
   (* (- x y) 0.5)
   (if (<= y 7.5e-257) (* 1.5 x) (* 0.5 (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.2e-246) {
		tmp = (x - y) * 0.5;
	} else if (y <= 7.5e-257) {
		tmp = 1.5 * x;
	} else {
		tmp = 0.5 * (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.2d-246)) then
        tmp = (x - y) * 0.5d0
    else if (y <= 7.5d-257) then
        tmp = 1.5d0 * x
    else
        tmp = 0.5d0 * (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.2e-246) {
		tmp = (x - y) * 0.5;
	} else if (y <= 7.5e-257) {
		tmp = 1.5 * x;
	} else {
		tmp = 0.5 * (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.2e-246:
		tmp = (x - y) * 0.5
	elif y <= 7.5e-257:
		tmp = 1.5 * x
	else:
		tmp = 0.5 * (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.2e-246)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (y <= 7.5e-257)
		tmp = Float64(1.5 * x);
	else
		tmp = Float64(0.5 * Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.2e-246)
		tmp = (x - y) * 0.5;
	elseif (y <= 7.5e-257)
		tmp = 1.5 * x;
	else
		tmp = 0.5 * (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.2e-246], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[y, 7.5e-257], N[(1.5 * x), $MachinePrecision], N[(0.5 * N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.19999999999999998e-246

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fabs-sub N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. lower-fabs.f64 N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 56.4

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

   5. Applied rewrites 56.4%

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

   7. Applied rewrites 83.8%

      \[\leadsto \left(x - y\right) \cdot 0.5 \]

   if -2.19999999999999998e-246 < y < 7.4999999999999995e-257

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. pow2 N/A

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

      6. sqrt-prod N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 71.8

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

      10. lift-/.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. sqrt-prod N/A

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

      14. rem-square-sqrt N/A

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

      15. frac-2neg N/A

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

   4. Applied rewrites 71.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{3}{2} \cdot x} \]
   6. Step-by-step derivation

      1. lower-*.f64 69.4

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

   7. Applied rewrites 69.4%

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

   if 7.4999999999999995e-257 < y

   1. Initial program 99.9%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 76.8

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

   4. Applied rewrites 76.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + \frac{1}{2} \cdot y} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 81.4

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

   7. Applied rewrites 81.4%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-246}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-257}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-246}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+14}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.2e-246)
   (* (- x y) 0.5)
   (if (<= y 1.95e+14) (* 1.5 x) (* 0.5 y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.2e-246) {
		tmp = (x - y) * 0.5;
	} else if (y <= 1.95e+14) {
		tmp = 1.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.2d-246)) then
        tmp = (x - y) * 0.5d0
    else if (y <= 1.95d+14) then
        tmp = 1.5d0 * x
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.2e-246) {
		tmp = (x - y) * 0.5;
	} else if (y <= 1.95e+14) {
		tmp = 1.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.2e-246:
		tmp = (x - y) * 0.5
	elif y <= 1.95e+14:
		tmp = 1.5 * x
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.2e-246)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (y <= 1.95e+14)
		tmp = Float64(1.5 * x);
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.2e-246)
		tmp = (x - y) * 0.5;
	elseif (y <= 1.95e+14)
		tmp = 1.5 * x;
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.2e-246], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[y, 1.95e+14], N[(1.5 * x), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.19999999999999998e-246

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fabs-sub N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. lower-fabs.f64 N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 56.4

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

   5. Applied rewrites 56.4%

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

   7. Applied rewrites 83.8%

      \[\leadsto \left(x - y\right) \cdot 0.5 \]

   if -2.19999999999999998e-246 < y < 1.95e14

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. pow2 N/A

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

      6. sqrt-prod N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 58.2

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

      10. lift-/.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. sqrt-prod N/A

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

      14. rem-square-sqrt N/A

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

      15. frac-2neg N/A

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

   4. Applied rewrites 49.4%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{3}{2} \cdot x} \]
   6. Step-by-step derivation

      1. lower-*.f64 52.7

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

   7. Applied rewrites 52.7%

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

   if 1.95e14 < y

   1. Initial program 100.0%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 94.7

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

   4. Applied rewrites 94.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot y} \]
   6. Step-by-step derivation

      1. lower-*.f64 84.4

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

   7. Applied rewrites 84.4%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-246}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+14}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{-81}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq 0.085:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.7e-81) (* -0.5 y) (if (<= y 0.085) (* 0.5 x) (* 0.5 y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.7e-81) {
		tmp = -0.5 * y;
	} else if (y <= 0.085) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.7d-81)) then
        tmp = (-0.5d0) * y
    else if (y <= 0.085d0) then
        tmp = 0.5d0 * x
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.7e-81) {
		tmp = -0.5 * y;
	} else if (y <= 0.085) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.7e-81:
		tmp = -0.5 * y
	elif y <= 0.085:
		tmp = 0.5 * x
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.7e-81)
		tmp = Float64(-0.5 * y);
	elseif (y <= 0.085)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.7e-81)
		tmp = -0.5 * y;
	elseif (y <= 0.085)
		tmp = 0.5 * x;
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.7e-81], N[(-0.5 * y), $MachinePrecision], If[LessEqual[y, 0.085], N[(0.5 * x), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.69999999999999986e-81

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. pow2 N/A

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

      6. sqrt-prod N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 53.1

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

      10. lift-/.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. sqrt-prod N/A

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

      14. rem-square-sqrt N/A

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

      15. frac-2neg N/A

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

   4. Applied rewrites 53.1%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot y} \]
   6. Step-by-step derivation

      1. lower-*.f64 67.4

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

   7. Applied rewrites 67.4%

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

   if -3.69999999999999986e-81 < y < 0.0850000000000000061

   1. Initial program 99.9%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 50.0

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

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in y around 0

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 47.7

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

   7. Applied rewrites 47.7%

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

   if 0.0850000000000000061 < y

   1. Initial program 100.0%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 92.0

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

   4. Applied rewrites 92.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot y} \]
   6. Step-by-step derivation

      1. lower-*.f64 82.2

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

   7. Applied rewrites 82.2%

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

Alternative 8: 30.5% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-fabs.f64 N/A

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

   2. rem-sqrt-square-rev N/A

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

   3. sqrt-prod N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 51.6

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

4. Applied rewrites 51.6%

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

5. Taylor expanded in y around 0

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

   1. distribute-rgt1-in N/A

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

   2. metadata-eval N/A

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

   3. lower-*.f64 30.9

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

7. Applied rewrites 30.9%

   \[\leadsto \color{blue}{0.5 \cdot x} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024337 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))