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

Percentage Accurate: 99.9% → 99.9%

Time: 11.3s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.05 \cdot 10^{-140}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-26}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.05e-140)
   (* 0.5 (+ x y))
   (if (<= x 2.05e-26) (* (fabs (- y x)) 0.5) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.05e-140) {
		tmp = 0.5 * (x + y);
	} else if (x <= 2.05e-26) {
		tmp = fabs((y - x)) * 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 <= (-3.05d-140)) then
        tmp = 0.5d0 * (x + y)
    else if (x <= 2.05d-26) then
        tmp = abs((y - x)) * 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 <= -3.05e-140) {
		tmp = 0.5 * (x + y);
	} else if (x <= 2.05e-26) {
		tmp = Math.abs((y - x)) * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.05e-140:
		tmp = 0.5 * (x + y)
	elif x <= 2.05e-26:
		tmp = math.fabs((y - x)) * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.05e-140)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 2.05e-26)
		tmp = Float64(abs(Float64(y - x)) * 0.5);
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.05e-140)
		tmp = 0.5 * (x + y);
	elseif (x <= 2.05e-26)
		tmp = abs((y - x)) * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.05e-140], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.05e-26], N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.04999999999999982e-140

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. div-inv 99.9%

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

      3. fma-def 99.9%

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

      4. add-sqr-sqrt 86.1%

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

      5. fabs-sqr 86.1%

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

      6. add-sqr-sqrt 86.8%

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

      7. metadata-eval 86.8%

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

   3. Applied egg-rr 86.8%

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

   4. Taylor expanded in y around 0 86.8%

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

      1. +-commutative 86.8%

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

      2. associate-+r+ 86.8%

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

      3. distribute-lft1-in 86.8%

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

      4. metadata-eval 86.8%

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

      5. distribute-lft-out 86.8%

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

   6. Simplified 86.8%

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

   if -3.04999999999999982e-140 < x < 2.0499999999999999e-26

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 87.3%

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

   if 2.0499999999999999e-26 < x

   1. Initial program 99.6%

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

      1. flip-+ 37.3%

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

   3. Applied egg-rr 8.2%

      \[\leadsto \color{blue}{\frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{x - \left(y - x\right) \cdot 0.5}} \]
   4. Step-by-step derivation

      1. sub-neg 8.2%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\color{blue}{x + \left(-\left(y - x\right) \cdot 0.5\right)}} \]

      2. add-sqr-sqrt 8.2%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \left(-\left(y - x\right) \cdot 0.5\right)} \]

      3. sqrt-prod 8.2%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\color{blue}{\sqrt{x \cdot x}} + \left(-\left(y - x\right) \cdot 0.5\right)} \]

      4. add-cube-cbrt 8.2%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\sqrt{\color{blue}{\left(\sqrt[3]{x \cdot x} \cdot \sqrt[3]{x \cdot x}\right) \cdot \sqrt[3]{x \cdot x}}} + \left(-\left(y - x\right) \cdot 0.5\right)} \]

      5. sqrt-prod 8.2%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\color{blue}{\sqrt{\sqrt[3]{x \cdot x} \cdot \sqrt[3]{x \cdot x}} \cdot \sqrt{\sqrt[3]{x \cdot x}}} + \left(-\left(y - x\right) \cdot 0.5\right)} \]

      6. fma-def 8.2%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt[3]{x \cdot x} \cdot \sqrt[3]{x \cdot x}}, \sqrt{\sqrt[3]{x \cdot x}}, -\left(y - x\right) \cdot 0.5\right)}} \]

      7. cbrt-unprod 3.9%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\color{blue}{\sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}}, \sqrt{\sqrt[3]{x \cdot x}}, -\left(y - x\right) \cdot 0.5\right)} \]

      8. pow2 3.9%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{\color{blue}{{x}^{2}} \cdot \left(x \cdot x\right)}}, \sqrt{\sqrt[3]{x \cdot x}}, -\left(y - x\right) \cdot 0.5\right)} \]

      9. pow2 3.9%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{2} \cdot \color{blue}{{x}^{2}}}}, \sqrt{\sqrt[3]{x \cdot x}}, -\left(y - x\right) \cdot 0.5\right)} \]

      10. pow-prod-up 3.9%

          \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{\color{blue}{{x}^{\left(2 + 2\right)}}}}, \sqrt{\sqrt[3]{x \cdot x}}, -\left(y - x\right) \cdot 0.5\right)} \]

      11. metadata-eval 3.9%

          \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{\color{blue}{4}}}}, \sqrt{\sqrt[3]{x \cdot x}}, -\left(y - x\right) \cdot 0.5\right)} \]

      12. cbrt-prod 3.9%

          \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{4}}}, \sqrt{\color{blue}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}, -\left(y - x\right) \cdot 0.5\right)} \]

      13. pow2 3.9%

          \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{4}}}, \sqrt{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}}, -\left(y - x\right) \cdot 0.5\right)} \]

      14. distribute-rgt-neg-in 3.9%

          \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{4}}}, \sqrt{{\left(\sqrt[3]{x}\right)}^{2}}, \color{blue}{\left(y - x\right) \cdot \left(-0.5\right)}\right)} \]

      15. metadata-eval 3.9%

          \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{4}}}, \sqrt{{\left(\sqrt[3]{x}\right)}^{2}}, \left(y - x\right) \cdot \color{blue}{-0.5}\right)} \]

   5. Applied egg-rr 3.9%

      \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{4}}}, \sqrt{{\left(\sqrt[3]{x}\right)}^{2}}, \left(y - x\right) \cdot -0.5\right)}} \]
   6. Step-by-step derivation

      1. unpow2 3.9%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{4}}}, \sqrt{\color{blue}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}, \left(y - x\right) \cdot -0.5\right)} \]

      2. rem-sqrt-square 3.9%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{4}}}, \color{blue}{\left|\sqrt[3]{x}\right|}, \left(y - x\right) \cdot -0.5\right)} \]

      3. *-commutative 3.9%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{4}}}, \left|\sqrt[3]{x}\right|, \color{blue}{-0.5 \cdot \left(y - x\right)}\right)} \]

   7. Simplified 3.9%

      \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{4}}}, \left|\sqrt[3]{x}\right|, -0.5 \cdot \left(y - x\right)\right)}} \]

   8. Taylor expanded in x around inf 74.8%

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

      1. *-commutative 74.8%

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

   10. Simplified 74.8%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.05 \cdot 10^{-140}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-26}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]

Alternative 3: 57.5% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-22}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-59}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-124}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-97}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.1e-22)
   (* x 0.5)
   (if (<= x -3.1e-59)
     (* y 0.5)
     (if (<= x -5.8e-124) (* x 0.5) (if (<= x 6.5e-97) (* y 0.5) (* x 1.5))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.1e-22) {
		tmp = x * 0.5;
	} else if (x <= -3.1e-59) {
		tmp = y * 0.5;
	} else if (x <= -5.8e-124) {
		tmp = x * 0.5;
	} else if (x <= 6.5e-97) {
		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 <= (-4.1d-22)) then
        tmp = x * 0.5d0
    else if (x <= (-3.1d-59)) then
        tmp = y * 0.5d0
    else if (x <= (-5.8d-124)) then
        tmp = x * 0.5d0
    else if (x <= 6.5d-97) 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 <= -4.1e-22) {
		tmp = x * 0.5;
	} else if (x <= -3.1e-59) {
		tmp = y * 0.5;
	} else if (x <= -5.8e-124) {
		tmp = x * 0.5;
	} else if (x <= 6.5e-97) {
		tmp = y * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.1e-22:
		tmp = x * 0.5
	elif x <= -3.1e-59:
		tmp = y * 0.5
	elif x <= -5.8e-124:
		tmp = x * 0.5
	elif x <= 6.5e-97:
		tmp = y * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.1e-22)
		tmp = Float64(x * 0.5);
	elseif (x <= -3.1e-59)
		tmp = Float64(y * 0.5);
	elseif (x <= -5.8e-124)
		tmp = Float64(x * 0.5);
	elseif (x <= 6.5e-97)
		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 <= -4.1e-22)
		tmp = x * 0.5;
	elseif (x <= -3.1e-59)
		tmp = y * 0.5;
	elseif (x <= -5.8e-124)
		tmp = x * 0.5;
	elseif (x <= 6.5e-97)
		tmp = y * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.1e-22], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, -3.1e-59], N[(y * 0.5), $MachinePrecision], If[LessEqual[x, -5.8e-124], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 6.5e-97], N[(y * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.0999999999999999e-22 or -3.09999999999999999e-59 < x < -5.8000000000000004e-124

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. div-inv 99.9%

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

      3. fma-def 99.9%

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

      4. add-sqr-sqrt 84.8%

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

      5. fabs-sqr 84.8%

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

      6. add-sqr-sqrt 85.4%

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

      7. metadata-eval 85.4%

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

   3. Applied egg-rr 85.4%

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

   4. Taylor expanded in y around 0 71.1%

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

      1. distribute-lft1-in 71.1%

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

      2. metadata-eval 71.1%

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

      3. *-commutative 71.1%

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

   6. Simplified 71.1%

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

   if -4.0999999999999999e-22 < x < -3.09999999999999999e-59 or -5.8000000000000004e-124 < x < 6.5000000000000004e-97

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. div-inv 100.0%

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

      3. fma-def 100.0%

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

      4. add-sqr-sqrt 50.2%

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

      5. fabs-sqr 50.2%

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

      6. add-sqr-sqrt 52.6%

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

      7. metadata-eval 52.6%

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

   3. Applied egg-rr 52.6%

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

   4. Taylor expanded in y around inf 46.8%

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

   if 6.5000000000000004e-97 < x

   1. Initial program 99.7%

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

      1. flip-+ 42.5%

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

   3. Applied egg-rr 11.2%

      \[\leadsto \color{blue}{\frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{x - \left(y - x\right) \cdot 0.5}} \]
   4. Step-by-step derivation

      1. sub-neg 11.2%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\color{blue}{x + \left(-\left(y - x\right) \cdot 0.5\right)}} \]

      2. add-sqr-sqrt 11.2%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \left(-\left(y - x\right) \cdot 0.5\right)} \]

      3. sqrt-prod 11.2%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\color{blue}{\sqrt{x \cdot x}} + \left(-\left(y - x\right) \cdot 0.5\right)} \]

      4. add-cube-cbrt 11.2%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\sqrt{\color{blue}{\left(\sqrt[3]{x \cdot x} \cdot \sqrt[3]{x \cdot x}\right) \cdot \sqrt[3]{x \cdot x}}} + \left(-\left(y - x\right) \cdot 0.5\right)} \]

      5. sqrt-prod 11.2%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\color{blue}{\sqrt{\sqrt[3]{x \cdot x} \cdot \sqrt[3]{x \cdot x}} \cdot \sqrt{\sqrt[3]{x \cdot x}}} + \left(-\left(y - x\right) \cdot 0.5\right)} \]

      6. fma-def 11.2%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt[3]{x \cdot x} \cdot \sqrt[3]{x \cdot x}}, \sqrt{\sqrt[3]{x \cdot x}}, -\left(y - x\right) \cdot 0.5\right)}} \]

      7. cbrt-unprod 10.4%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\color{blue}{\sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}}, \sqrt{\sqrt[3]{x \cdot x}}, -\left(y - x\right) \cdot 0.5\right)} \]

      8. pow2 10.4%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{\color{blue}{{x}^{2}} \cdot \left(x \cdot x\right)}}, \sqrt{\sqrt[3]{x \cdot x}}, -\left(y - x\right) \cdot 0.5\right)} \]

      9. pow2 10.4%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{2} \cdot \color{blue}{{x}^{2}}}}, \sqrt{\sqrt[3]{x \cdot x}}, -\left(y - x\right) \cdot 0.5\right)} \]

      10. pow-prod-up 10.4%

          \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{\color{blue}{{x}^{\left(2 + 2\right)}}}}, \sqrt{\sqrt[3]{x \cdot x}}, -\left(y - x\right) \cdot 0.5\right)} \]

      11. metadata-eval 10.4%

          \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{\color{blue}{4}}}}, \sqrt{\sqrt[3]{x \cdot x}}, -\left(y - x\right) \cdot 0.5\right)} \]

      12. cbrt-prod 10.4%

          \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{4}}}, \sqrt{\color{blue}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}, -\left(y - x\right) \cdot 0.5\right)} \]

      13. pow2 10.4%

          \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{4}}}, \sqrt{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}}, -\left(y - x\right) \cdot 0.5\right)} \]

      14. distribute-rgt-neg-in 10.4%

          \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{4}}}, \sqrt{{\left(\sqrt[3]{x}\right)}^{2}}, \color{blue}{\left(y - x\right) \cdot \left(-0.5\right)}\right)} \]

      15. metadata-eval 10.4%

          \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{4}}}, \sqrt{{\left(\sqrt[3]{x}\right)}^{2}}, \left(y - x\right) \cdot \color{blue}{-0.5}\right)} \]

   5. Applied egg-rr 10.4%

      \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{4}}}, \sqrt{{\left(\sqrt[3]{x}\right)}^{2}}, \left(y - x\right) \cdot -0.5\right)}} \]
   6. Step-by-step derivation

      1. unpow2 10.4%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{4}}}, \sqrt{\color{blue}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}, \left(y - x\right) \cdot -0.5\right)} \]

      2. rem-sqrt-square 10.4%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{4}}}, \color{blue}{\left|\sqrt[3]{x}\right|}, \left(y - x\right) \cdot -0.5\right)} \]

      3. *-commutative 10.4%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{4}}}, \left|\sqrt[3]{x}\right|, \color{blue}{-0.5 \cdot \left(y - x\right)}\right)} \]

   7. Simplified 10.4%

      \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{4}}}, \left|\sqrt[3]{x}\right|, -0.5 \cdot \left(y - x\right)\right)}} \]

   8. Taylor expanded in x around inf 67.4%

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

      1. *-commutative 67.4%

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

   10. Simplified 67.4%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-22}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-59}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-124}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-97}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]

Alternative 4: 66.6% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.4 \cdot 10^{-33}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4.4e-33) (* 0.5 (+ x y)) (* x 1.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4.4e-33) {
		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 <= 4.4d-33) 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 <= 4.4e-33) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 4.4e-33:
		tmp = 0.5 * (x + y)
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4.4e-33)
		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 <= 4.4e-33)
		tmp = 0.5 * (x + y);
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4.4e-33], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.40000000000000011e-33

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. div-inv 99.9%

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

      3. fma-def 99.9%

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

      4. add-sqr-sqrt 65.3%

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

      5. fabs-sqr 65.3%

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

      6. add-sqr-sqrt 67.0%

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

      7. metadata-eval 67.0%

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

   3. Applied egg-rr 67.0%

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

   4. Taylor expanded in y around 0 67.0%

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

      1. +-commutative 67.0%

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

      2. associate-+r+ 67.0%

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

      3. distribute-lft1-in 67.0%

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

      4. metadata-eval 67.0%

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

      5. distribute-lft-out 67.0%

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

   6. Simplified 67.0%

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

   if 4.40000000000000011e-33 < x

   1. Initial program 99.7%

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

      1. flip-+ 38.1%

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

   3. Applied egg-rr 8.1%

      \[\leadsto \color{blue}{\frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{x - \left(y - x\right) \cdot 0.5}} \]
   4. Step-by-step derivation

      1. sub-neg 8.1%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\color{blue}{x + \left(-\left(y - x\right) \cdot 0.5\right)}} \]

      2. add-sqr-sqrt 8.1%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \left(-\left(y - x\right) \cdot 0.5\right)} \]

      3. sqrt-prod 8.1%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\color{blue}{\sqrt{x \cdot x}} + \left(-\left(y - x\right) \cdot 0.5\right)} \]

      4. add-cube-cbrt 8.1%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\sqrt{\color{blue}{\left(\sqrt[3]{x \cdot x} \cdot \sqrt[3]{x \cdot x}\right) \cdot \sqrt[3]{x \cdot x}}} + \left(-\left(y - x\right) \cdot 0.5\right)} \]

      5. sqrt-prod 8.1%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\color{blue}{\sqrt{\sqrt[3]{x \cdot x} \cdot \sqrt[3]{x \cdot x}} \cdot \sqrt{\sqrt[3]{x \cdot x}}} + \left(-\left(y - x\right) \cdot 0.5\right)} \]

      6. fma-def 8.1%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt[3]{x \cdot x} \cdot \sqrt[3]{x \cdot x}}, \sqrt{\sqrt[3]{x \cdot x}}, -\left(y - x\right) \cdot 0.5\right)}} \]

      7. cbrt-unprod 3.8%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\color{blue}{\sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}}, \sqrt{\sqrt[3]{x \cdot x}}, -\left(y - x\right) \cdot 0.5\right)} \]

      8. pow2 3.8%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{\color{blue}{{x}^{2}} \cdot \left(x \cdot x\right)}}, \sqrt{\sqrt[3]{x \cdot x}}, -\left(y - x\right) \cdot 0.5\right)} \]

      9. pow2 3.8%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{2} \cdot \color{blue}{{x}^{2}}}}, \sqrt{\sqrt[3]{x \cdot x}}, -\left(y - x\right) \cdot 0.5\right)} \]

      10. pow-prod-up 3.8%

          \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{\color{blue}{{x}^{\left(2 + 2\right)}}}}, \sqrt{\sqrt[3]{x \cdot x}}, -\left(y - x\right) \cdot 0.5\right)} \]

      11. metadata-eval 3.8%

          \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{\color{blue}{4}}}}, \sqrt{\sqrt[3]{x \cdot x}}, -\left(y - x\right) \cdot 0.5\right)} \]

      12. cbrt-prod 3.8%

          \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{4}}}, \sqrt{\color{blue}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}, -\left(y - x\right) \cdot 0.5\right)} \]

      13. pow2 3.8%

          \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{4}}}, \sqrt{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}}, -\left(y - x\right) \cdot 0.5\right)} \]

      14. distribute-rgt-neg-in 3.8%

          \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{4}}}, \sqrt{{\left(\sqrt[3]{x}\right)}^{2}}, \color{blue}{\left(y - x\right) \cdot \left(-0.5\right)}\right)} \]

      15. metadata-eval 3.8%

          \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{4}}}, \sqrt{{\left(\sqrt[3]{x}\right)}^{2}}, \left(y - x\right) \cdot \color{blue}{-0.5}\right)} \]

   5. Applied egg-rr 3.8%

      \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{4}}}, \sqrt{{\left(\sqrt[3]{x}\right)}^{2}}, \left(y - x\right) \cdot -0.5\right)}} \]
   6. Step-by-step derivation

      1. unpow2 3.8%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{4}}}, \sqrt{\color{blue}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}, \left(y - x\right) \cdot -0.5\right)} \]

      2. rem-sqrt-square 3.8%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{4}}}, \color{blue}{\left|\sqrt[3]{x}\right|}, \left(y - x\right) \cdot -0.5\right)} \]

      3. *-commutative 3.8%

         \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{4}}}, \left|\sqrt[3]{x}\right|, \color{blue}{-0.5 \cdot \left(y - x\right)}\right)} \]

   7. Simplified 3.8%

      \[\leadsto \frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt[3]{{x}^{4}}}, \left|\sqrt[3]{x}\right|, -0.5 \cdot \left(y - x\right)\right)}} \]

   8. Taylor expanded in x around inf 73.9%

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

      1. *-commutative 73.9%

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

   10. Simplified 73.9%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.4 \cdot 10^{-33}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]

Alternative 5: 30.7% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.95 \cdot 10^{-118}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 3.95e-118) x (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.95e-118) {
		tmp = x;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.95d-118) then
        tmp = x
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.95e-118) {
		tmp = x;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.95e-118:
		tmp = x
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.95e-118)
		tmp = x;
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.95e-118)
		tmp = x;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.95e-118], x, N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.9500000000000002e-118

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 12.8%

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

   if 3.9500000000000002e-118 < y

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. div-inv 99.9%

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

      3. fma-def 99.9%

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

      4. add-sqr-sqrt 82.2%

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

      5. fabs-sqr 82.2%

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

      6. add-sqr-sqrt 85.8%

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

      7. metadata-eval 85.8%

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

   3. Applied egg-rr 85.8%

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

   4. Taylor expanded in y around inf 59.5%

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

4. Final simplification 30.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.95 \cdot 10^{-118}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 6: 45.4% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-56}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 8.2e-56) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8.2e-56) {
		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 <= 8.2d-56) 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 <= 8.2e-56) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8.2e-56:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8.2e-56)
		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 <= 8.2e-56)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8.2e-56], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.2000000000000003e-56

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. div-inv 99.9%

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

      3. fma-def 99.9%

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

      4. add-sqr-sqrt 37.3%

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

      5. fabs-sqr 37.3%

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

      6. add-sqr-sqrt 42.8%

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

      7. metadata-eval 42.8%

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

   3. Applied egg-rr 42.8%

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

   4. Taylor expanded in y around 0 37.6%

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

      1. distribute-lft1-in 37.6%

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

      2. metadata-eval 37.6%

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

      3. *-commutative 37.6%

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

   6. Simplified 37.6%

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

   if 8.2000000000000003e-56 < y

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. div-inv 99.8%

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

      3. fma-def 99.8%

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

      4. add-sqr-sqrt 81.8%

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

      5. fabs-sqr 81.8%

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

      6. add-sqr-sqrt 85.5%

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

      7. metadata-eval 85.5%

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

   3. Applied egg-rr 85.5%

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

   4. Taylor expanded in y around inf 67.1%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-56}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 7: 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. Taylor expanded in x around inf 11.7%

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

3. Final simplification 11.7%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023274 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))