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

Percentage Accurate: 99.9% → 99.9%

Time: 8.9s

Alternatives: 8

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: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-78}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+40}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.2e-78)
   (* 0.5 (+ x y))
   (if (<= x 2.2e+40) (+ x (/ (fabs y) 2.0)) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.2e-78) {
		tmp = 0.5 * (x + y);
	} else if (x <= 2.2e+40) {
		tmp = x + (fabs(y) / 2.0);
	} 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 <= (-4.2d-78)) then
        tmp = 0.5d0 * (x + y)
    else if (x <= 2.2d+40) then
        tmp = x + (abs(y) / 2.0d0)
    else
        tmp = x * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.2e-78) {
		tmp = 0.5 * (x + y);
	} else if (x <= 2.2e+40) {
		tmp = x + (Math.abs(y) / 2.0);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.2e-78:
		tmp = 0.5 * (x + y)
	elif x <= 2.2e+40:
		tmp = x + (math.fabs(y) / 2.0)
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.2e-78)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 2.2e+40)
		tmp = Float64(x + Float64(abs(y) / 2.0));
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.2e-78)
		tmp = 0.5 * (x + y);
	elseif (x <= 2.2e+40)
		tmp = x + (abs(y) / 2.0);
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.2e-78], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.2e+40], N[(x + N[(N[Abs[y], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.2000000000000001e-78

   1. Initial program 99.9%

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

   3. Taylor expanded in y around -inf 99.9%

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

      1. fabs-neg 99.9%

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

      2. mul-1-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. fabs-sub 99.9%

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

      5. rem-square-sqrt 84.7%

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

      6. fabs-sqr 84.7%

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

      7. rem-square-sqrt 85.4%

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

   5. Simplified 85.4%

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

   6. Taylor expanded in x around 0 85.5%

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

      1. +-commutative 85.5%

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

      2. distribute-lft-in 85.5%

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

   8. Simplified 85.5%

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

   if -4.2000000000000001e-78 < x < 2.1999999999999999e40

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 85.3%

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

   if 2.1999999999999999e40 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 86.0%

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

      1. neg-mul-1 86.0%

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

   5. Simplified 86.0%

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

   6. Taylor expanded in x around 0 86.0%

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

      1. +-commutative 86.0%

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

      2. fma-define 86.0%

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

      3. fabs-neg 86.0%

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

      4. rem-square-sqrt 85.9%

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

      5. fabs-sqr 85.9%

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

      6. rem-square-sqrt 86.0%

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

      7. fma-define 86.0%

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

      8. *-lft-identity 86.0%

         \[\leadsto 0.5 \cdot x + \color{blue}{1 \cdot x} \]

      9. distribute-rgt-out 86.0%

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

      10. metadata-eval 86.0%

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

   8. Simplified 86.0%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-78}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+40}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-60}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+40}:\\ \;\;\;\;\left|y \cdot 0.5\right|\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.6e-60)
   (* 0.5 (+ x y))
   (if (<= x 7e+40) (fabs (* y 0.5)) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.6e-60) {
		tmp = 0.5 * (x + y);
	} else if (x <= 7e+40) {
		tmp = fabs((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 <= (-3.6d-60)) then
        tmp = 0.5d0 * (x + y)
    else if (x <= 7d+40) then
        tmp = abs((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 <= -3.6e-60) {
		tmp = 0.5 * (x + y);
	} else if (x <= 7e+40) {
		tmp = Math.abs((y * 0.5));
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.6e-60:
		tmp = 0.5 * (x + y)
	elif x <= 7e+40:
		tmp = math.fabs((y * 0.5))
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.6e-60)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 7e+40)
		tmp = abs(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 <= -3.6e-60)
		tmp = 0.5 * (x + y);
	elseif (x <= 7e+40)
		tmp = abs((y * 0.5));
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.6e-60], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7e+40], N[Abs[N[(y * 0.5), $MachinePrecision]], $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.6e-60

   1. Initial program 99.9%

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

   3. Taylor expanded in y around -inf 99.9%

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

      1. fabs-neg 99.9%

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

      2. mul-1-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. fabs-sub 99.9%

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

      5. rem-square-sqrt 84.7%

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

      6. fabs-sqr 84.7%

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

      7. rem-square-sqrt 85.4%

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

   5. Simplified 85.4%

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

   6. Taylor expanded in x around 0 85.5%

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

      1. +-commutative 85.5%

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

      2. distribute-lft-in 85.5%

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

   8. Simplified 85.5%

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

   if -3.6e-60 < x < 6.9999999999999998e40

   1. Initial program 100.0%

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

   3. Taylor expanded in y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. rem-square-sqrt 37.1%

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

      6. fabs-sqr 37.1%

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

      7. rem-square-sqrt 40.0%

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

   5. Simplified 40.0%

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

   6. Taylor expanded in x around 0 32.0%

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

      1. add-sqr-sqrt 30.7%

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

      2. sqrt-unprod 51.6%

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

      3. *-commutative 51.6%

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

      4. *-commutative 51.6%

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

      5. swap-sqr 51.6%

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

      6. pow2 51.6%

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

      7. metadata-eval 51.6%

         \[\leadsto \sqrt{{y}^{2} \cdot \color{blue}{0.25}} \]

   8. Applied egg-rr 51.6%

      \[\leadsto \color{blue}{\sqrt{{y}^{2} \cdot 0.25}} \]
   9. Step-by-step derivation

      1. unpow2 51.6%

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

      2. metadata-eval 51.6%

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

      3. swap-sqr 51.6%

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

      4. rem-sqrt-square 82.9%

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

      5. *-commutative 82.9%

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

   10. Simplified 82.9%

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

   if 6.9999999999999998e40 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 86.0%

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

      1. neg-mul-1 86.0%

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

   5. Simplified 86.0%

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

   6. Taylor expanded in x around 0 86.0%

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

      1. +-commutative 86.0%

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

      2. fma-define 86.0%

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

      3. fabs-neg 86.0%

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

      4. rem-square-sqrt 85.9%

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

      5. fabs-sqr 85.9%

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

      6. rem-square-sqrt 86.0%

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

      7. fma-define 86.0%

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

      8. *-lft-identity 86.0%

         \[\leadsto 0.5 \cdot x + \color{blue}{1 \cdot x} \]

      9. distribute-rgt-out 86.0%

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

      10. metadata-eval 86.0%

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

   8. Simplified 86.0%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-60}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+40}:\\ \;\;\;\;\left|y \cdot 0.5\right|\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.5% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-134}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-55}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.1e-134) (* x 0.5) (if (<= x 2e-55) (* y 0.5) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.1e-134) {
		tmp = x * 0.5;
	} else if (x <= 2e-55) {
		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 <= (-1.1d-134)) then
        tmp = x * 0.5d0
    else if (x <= 2d-55) 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 <= -1.1e-134) {
		tmp = x * 0.5;
	} else if (x <= 2e-55) {
		tmp = y * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.1e-134:
		tmp = x * 0.5
	elif x <= 2e-55:
		tmp = y * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.1e-134)
		tmp = Float64(x * 0.5);
	elseif (x <= 2e-55)
		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 <= -1.1e-134)
		tmp = x * 0.5;
	elseif (x <= 2e-55)
		tmp = y * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.1e-134], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 2e-55], N[(y * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1e-134

   1. Initial program 100.0%

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

   3. Taylor expanded in y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. rem-square-sqrt 79.1%

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

      6. fabs-sqr 79.1%

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

      7. rem-square-sqrt 79.9%

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

   5. Simplified 79.9%

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

   6. Taylor expanded in x around inf 60.1%

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

   if -1.1e-134 < x < 1.99999999999999999e-55

   1. Initial program 100.0%

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

   3. Taylor expanded in y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. rem-square-sqrt 39.0%

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

      6. fabs-sqr 39.0%

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

      7. rem-square-sqrt 41.1%

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

   5. Simplified 41.1%

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

   6. Taylor expanded in x around 0 37.0%

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

   if 1.99999999999999999e-55 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 76.9%

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

      1. neg-mul-1 76.9%

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

   5. Simplified 76.9%

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

   6. Taylor expanded in x around 0 76.9%

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

      1. +-commutative 76.9%

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

      2. fma-define 76.9%

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

      3. fabs-neg 76.9%

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

      4. rem-square-sqrt 76.8%

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

      5. fabs-sqr 76.8%

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

      6. rem-square-sqrt 76.9%

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

      7. fma-define 76.9%

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

      8. *-lft-identity 76.9%

         \[\leadsto 0.5 \cdot x + \color{blue}{1 \cdot x} \]

      9. distribute-rgt-out 76.9%

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

      10. metadata-eval 76.9%

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

   8. Simplified 76.9%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-134}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-55}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-56}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 4e-56) (* 0.5 (+ x y)) (* x 1.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4e-56) {
		tmp = 0.5 * (x + y);
	} 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 <= 4d-56) then
        tmp = 0.5d0 * (x + y)
    else
        tmp = x * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 4e-56) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 4e-56:
		tmp = 0.5 * (x + y)
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4e-56)
		tmp = Float64(0.5 * Float64(x + y));
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 4e-56)
		tmp = 0.5 * (x + y);
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4e-56], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.0000000000000002e-56

   1. Initial program 100.0%

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

   3. Taylor expanded in y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. rem-square-sqrt 61.2%

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

      6. fabs-sqr 61.2%

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

      7. rem-square-sqrt 62.6%

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

   5. Simplified 62.6%

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

   6. Taylor expanded in x around 0 62.6%

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

      1. +-commutative 62.6%

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

      2. distribute-lft-in 62.6%

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

   8. Simplified 62.6%

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

   if 4.0000000000000002e-56 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 76.9%

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

      1. neg-mul-1 76.9%

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

   5. Simplified 76.9%

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

   6. Taylor expanded in x around 0 76.9%

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

      1. +-commutative 76.9%

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

      2. fma-define 76.9%

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

      3. fabs-neg 76.9%

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

      4. rem-square-sqrt 76.8%

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

      5. fabs-sqr 76.8%

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

      6. rem-square-sqrt 76.9%

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

      7. fma-define 76.9%

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

      8. *-lft-identity 76.9%

         \[\leadsto 0.5 \cdot x + \color{blue}{1 \cdot x} \]

      9. distribute-rgt-out 76.9%

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

      10. metadata-eval 76.9%

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

   8. Simplified 76.9%

      \[\leadsto \color{blue}{x \cdot 1.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-56}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 45.7% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.05 \cdot 10^{-73}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.05e-73) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.05e-73) {
		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 <= 2.05d-73) 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 <= 2.05e-73) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.05e-73:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.05e-73)
		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 <= 2.05e-73)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.05e-73], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.05000000000000008e-73

   1. Initial program 99.9%

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

   3. Taylor expanded in y around -inf 99.9%

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

      1. fabs-neg 99.9%

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

      2. mul-1-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. fabs-sub 99.9%

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

      5. rem-square-sqrt 31.0%

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

      6. fabs-sqr 31.0%

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

      7. rem-square-sqrt 36.4%

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

   5. Simplified 36.4%

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

   6. Taylor expanded in x around inf 36.0%

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

   if 2.05000000000000008e-73 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around -inf 99.9%

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

      1. fabs-neg 99.9%

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

      2. mul-1-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. fabs-sub 99.9%

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

      5. rem-square-sqrt 85.1%

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

      6. fabs-sqr 85.1%

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

      7. rem-square-sqrt 88.2%

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

   5. Simplified 88.2%

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

   6. Taylor expanded in x around 0 67.7%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.05 \cdot 10^{-73}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 30.8% 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. Taylor expanded in y around -inf 99.9%

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

   1. fabs-neg 99.9%

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

   2. mul-1-neg 99.9%

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

   3. sub-neg 99.9%

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

   4. fabs-sub 99.9%

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

   5. rem-square-sqrt 47.3%

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

   6. fabs-sqr 47.3%

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

   7. rem-square-sqrt 52.0%

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

5. Simplified 52.0%

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

6. Taylor expanded in x around inf 30.9%

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

7. Final simplification 30.9%

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

Alternative 8: 11.4% accurate, 107.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around inf 10.8%

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

Reproduce

herbie shell --seed 2024139 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))