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

Percentage Accurate: 99.9% → 99.9%

Time: 2.2s

Alternatives: 6

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: 70.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{-99}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.2e-99) (* (fabs (- y x)) 0.5) (* 0.5 (+ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.2e-99) {
		tmp = fabs((y - x)) * 0.5;
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-8.2d-99)) then
        tmp = abs((y - x)) * 0.5d0
    else
        tmp = 0.5d0 * (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8.2e-99) {
		tmp = Math.abs((y - x)) * 0.5;
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.2e-99:
		tmp = math.fabs((y - x)) * 0.5
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.2e-99)
		tmp = Float64(abs(Float64(y - x)) * 0.5);
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8.2e-99)
		tmp = abs((y - x)) * 0.5;
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.2e-99], N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.20000000000000057e-99

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 70.9%

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

   if -8.20000000000000057e-99 < 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. add-sqr-sqrt 99.4%

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

      3. fma-def 99.4%

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

      4. div-inv 99.4%

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

      5. add-sqr-sqrt 69.1%

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

      6. fabs-sqr 69.1%

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

      7. add-sqr-sqrt 69.2%

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

      8. metadata-eval 69.2%

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

      9. div-inv 69.2%

         \[\leadsto \mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot 0.5}, \sqrt{\color{blue}{\left|y - x\right| \cdot \frac{1}{2}}}, x\right) \]

      10. add-sqr-sqrt 69.1%

          \[\leadsto \mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot 0.5}, \sqrt{\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right| \cdot \frac{1}{2}}, x\right) \]

      11. fabs-sqr 69.1%

          \[\leadsto \mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot 0.5}, \sqrt{\color{blue}{\left(\sqrt{y - x} \cdot \sqrt{y - x}\right)} \cdot \frac{1}{2}}, x\right) \]

      12. add-sqr-sqrt 69.2%

          \[\leadsto \mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot 0.5}, \sqrt{\color{blue}{\left(y - x\right)} \cdot \frac{1}{2}}, x\right) \]

      13. metadata-eval 69.2%

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

   3. Applied egg-rr 69.2%

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

   4. Taylor expanded in y around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 0.0%

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

      5. *-rgt-identity 0.0%

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

      6. *-commutative 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 0.0%

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

      9. *-commutative 0.0%

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

      10. associate-*l* 0.0%

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

      11. unpow2 0.0%

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

      12. rem-square-sqrt 75.0%

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

      13. metadata-eval 75.0%

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

      14. distribute-lft-out 75.0%

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

      15. metadata-eval 75.0%

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

      16. distribute-rgt-out 75.0%

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

      17. +-commutative 75.0%

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

   6. Simplified 75.0%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{-99}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]

Alternative 3: 46.0% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-29}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.3e-29) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.3e-29) {
		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 <= 1.3d-29) 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 <= 1.3e-29) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.3e-29:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.3e-29)
		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 <= 1.3e-29)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.3e-29], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3000000000000001e-29

   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. add-sqr-sqrt 99.3%

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

      3. fma-def 99.3%

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

      4. div-inv 99.3%

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

      5. add-sqr-sqrt 32.7%

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

      6. fabs-sqr 32.7%

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

      7. add-sqr-sqrt 32.7%

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

      8. metadata-eval 32.7%

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

      9. div-inv 32.7%

         \[\leadsto \mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot 0.5}, \sqrt{\color{blue}{\left|y - x\right| \cdot \frac{1}{2}}}, x\right) \]

      10. add-sqr-sqrt 32.7%

          \[\leadsto \mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot 0.5}, \sqrt{\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right| \cdot \frac{1}{2}}, x\right) \]

      11. fabs-sqr 32.7%

          \[\leadsto \mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot 0.5}, \sqrt{\color{blue}{\left(\sqrt{y - x} \cdot \sqrt{y - x}\right)} \cdot \frac{1}{2}}, x\right) \]

      12. add-sqr-sqrt 32.7%

          \[\leadsto \mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot 0.5}, \sqrt{\color{blue}{\left(y - x\right)} \cdot \frac{1}{2}}, x\right) \]

      13. metadata-eval 32.7%

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

   3. Applied egg-rr 32.7%

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

   4. Taylor expanded in y around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 0.0%

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

      4. *-commutative 0.0%

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

      5. associate-*l* 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 35.4%

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

      8. metadata-eval 35.4%

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

      9. *-commutative 35.4%

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

      10. distribute-lft1-in 35.4%

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

      11. metadata-eval 35.4%

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

   6. Simplified 35.4%

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

   if 1.3000000000000001e-29 < y

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. add-sqr-sqrt 99.3%

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

      3. fma-def 99.3%

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

      4. div-inv 99.3%

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

      5. add-sqr-sqrt 91.5%

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

      6. fabs-sqr 91.5%

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

      7. add-sqr-sqrt 91.6%

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

      8. metadata-eval 91.6%

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

      9. div-inv 91.6%

         \[\leadsto \mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot 0.5}, \sqrt{\color{blue}{\left|y - x\right| \cdot \frac{1}{2}}}, x\right) \]

      10. add-sqr-sqrt 91.5%

          \[\leadsto \mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot 0.5}, \sqrt{\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right| \cdot \frac{1}{2}}, x\right) \]

      11. fabs-sqr 91.5%

          \[\leadsto \mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot 0.5}, \sqrt{\color{blue}{\left(\sqrt{y - x} \cdot \sqrt{y - x}\right)} \cdot \frac{1}{2}}, x\right) \]

      12. add-sqr-sqrt 91.6%

          \[\leadsto \mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot 0.5}, \sqrt{\color{blue}{\left(y - x\right)} \cdot \frac{1}{2}}, x\right) \]

      13. metadata-eval 91.6%

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

   3. Applied egg-rr 91.6%

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

   4. Taylor expanded in x around 0 78.0%

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

      1. unpow2 78.0%

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

      2. rem-square-sqrt 79.6%

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

   6. Simplified 79.6%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-29}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 4: 55.1% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(x + y\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. add-sqr-sqrt 99.3%

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

   3. fma-def 99.3%

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

   4. div-inv 99.3%

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

   5. add-sqr-sqrt 50.4%

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

   6. fabs-sqr 50.4%

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

   7. add-sqr-sqrt 50.4%

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

   8. metadata-eval 50.4%

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

   9. div-inv 50.4%

      \[\leadsto \mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot 0.5}, \sqrt{\color{blue}{\left|y - x\right| \cdot \frac{1}{2}}}, x\right) \]

   10. add-sqr-sqrt 50.4%

       \[\leadsto \mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot 0.5}, \sqrt{\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right| \cdot \frac{1}{2}}, x\right) \]

   11. fabs-sqr 50.4%

       \[\leadsto \mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot 0.5}, \sqrt{\color{blue}{\left(\sqrt{y - x} \cdot \sqrt{y - x}\right)} \cdot \frac{1}{2}}, x\right) \]

   12. add-sqr-sqrt 50.4%

       \[\leadsto \mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot 0.5}, \sqrt{\color{blue}{\left(y - x\right)} \cdot \frac{1}{2}}, x\right) \]

   13. metadata-eval 50.4%

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

3. Applied egg-rr 50.4%

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

4. Taylor expanded in y around 0 0.0%

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

   1. +-commutative 0.0%

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

   2. *-commutative 0.0%

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

   3. unpow2 0.0%

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

   4. rem-square-sqrt 0.0%

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

   5. *-rgt-identity 0.0%

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

   6. *-commutative 0.0%

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

   7. unpow2 0.0%

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

   8. rem-square-sqrt 0.0%

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

   9. *-commutative 0.0%

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

   10. associate-*l* 0.0%

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

   11. unpow2 0.0%

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

   12. rem-square-sqrt 55.8%

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

   13. metadata-eval 55.8%

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

   14. distribute-lft-out 55.8%

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

   15. metadata-eval 55.8%

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

   16. distribute-rgt-out 55.8%

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

   17. +-commutative 55.8%

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

6. Simplified 55.8%

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

7. Final simplification 55.8%

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

Alternative 5: 30.4% 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. Step-by-step derivation

   1. +-commutative 99.9%

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

   2. add-sqr-sqrt 99.3%

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

   3. fma-def 99.3%

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

   4. div-inv 99.3%

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

   5. add-sqr-sqrt 50.4%

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

   6. fabs-sqr 50.4%

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

   7. add-sqr-sqrt 50.4%

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

   8. metadata-eval 50.4%

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

   9. div-inv 50.4%

      \[\leadsto \mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot 0.5}, \sqrt{\color{blue}{\left|y - x\right| \cdot \frac{1}{2}}}, x\right) \]

   10. add-sqr-sqrt 50.4%

       \[\leadsto \mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot 0.5}, \sqrt{\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right| \cdot \frac{1}{2}}, x\right) \]

   11. fabs-sqr 50.4%

       \[\leadsto \mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot 0.5}, \sqrt{\color{blue}{\left(\sqrt{y - x} \cdot \sqrt{y - x}\right)} \cdot \frac{1}{2}}, x\right) \]

   12. add-sqr-sqrt 50.4%

       \[\leadsto \mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot 0.5}, \sqrt{\color{blue}{\left(y - x\right)} \cdot \frac{1}{2}}, x\right) \]

   13. metadata-eval 50.4%

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

3. Applied egg-rr 50.4%

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

4. Taylor expanded in y around 0 0.0%

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

   1. *-commutative 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 0.0%

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

   4. *-commutative 0.0%

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

   5. associate-*l* 0.0%

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

   6. unpow2 0.0%

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

   7. rem-square-sqrt 29.4%

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

   8. metadata-eval 29.4%

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

   9. *-commutative 29.4%

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

   10. distribute-lft1-in 29.4%

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

   11. metadata-eval 29.4%

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

6. Simplified 29.4%

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

7. Final simplification 29.4%

   \[\leadsto x \cdot 0.5 \]

Alternative 6: 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 11.5%

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

3. Final simplification 11.5%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023196 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))