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

Percentage Accurate: 99.9% → 99.9%

Time: 7.4s

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 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-69}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.4e-69) (* 0.5 (+ x y)) (* (fabs (- y x)) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.4e-69) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = fabs((y - x)) * 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 (x <= (-3.4d-69)) then
        tmp = 0.5d0 * (x + y)
    else
        tmp = abs((y - x)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.4e-69) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = Math.abs((y - x)) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.4e-69:
		tmp = 0.5 * (x + y)
	else:
		tmp = math.fabs((y - x)) * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.4e-69)
		tmp = Float64(0.5 * Float64(x + y));
	else
		tmp = Float64(abs(Float64(y - x)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.4e-69)
		tmp = 0.5 * (x + y);
	else
		tmp = abs((y - x)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.4e-69], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.40000000000000008e-69

   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 83.4%

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

      5. fabs-sqr 83.4%

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

      6. add-sqr-sqrt 84.3%

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

      7. metadata-eval 84.3%

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

   3. Applied egg-rr 84.3%

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

   4. Taylor expanded in y around 0 84.3%

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

      1. associate-+r+ 84.3%

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

      2. distribute-rgt1-in 84.3%

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

      3. metadata-eval 84.3%

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

      4. distribute-lft-out 84.3%

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

   6. Simplified 84.3%

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

   if -3.40000000000000008e-69 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 66.0%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-69}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \end{array} \]

Alternative 3: 42.1% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-54}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 1.42 \cdot 10^{-9}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.7e-54) (* x 0.5) (if (<= x 1.42e-9) (* y 0.5) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.7e-54) {
		tmp = x * 0.5;
	} else if (x <= 1.42e-9) {
		tmp = y * 0.5;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.7d-54)) then
        tmp = x * 0.5d0
    else if (x <= 1.42d-9) then
        tmp = y * 0.5d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.7e-54) {
		tmp = x * 0.5;
	} else if (x <= 1.42e-9) {
		tmp = y * 0.5;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.7e-54:
		tmp = x * 0.5
	elif x <= 1.42e-9:
		tmp = y * 0.5
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.7e-54)
		tmp = Float64(x * 0.5);
	elseif (x <= 1.42e-9)
		tmp = Float64(y * 0.5);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.7e-54)
		tmp = x * 0.5;
	elseif (x <= 1.42e-9)
		tmp = y * 0.5;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.7e-54], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 1.42e-9], N[(y * 0.5), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if x < -3.7000000000000003e-54

   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 82.6%

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

      5. fabs-sqr 82.6%

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

      6. add-sqr-sqrt 83.5%

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

      7. metadata-eval 83.5%

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

   3. Applied egg-rr 83.5%

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

   4. Taylor expanded in y around 0 69.7%

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

      1. distribute-rgt1-in 69.7%

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

      2. metadata-eval 69.7%

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

      3. *-commutative 69.7%

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

   6. Simplified 69.7%

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

   if -3.7000000000000003e-54 < x < 1.4200000000000001e-9

   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 42.5%

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

      5. fabs-sqr 42.5%

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

      6. add-sqr-sqrt 45.7%

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

      7. metadata-eval 45.7%

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

   3. Applied egg-rr 45.7%

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

   4. Taylor expanded in y around inf 41.0%

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

   if 1.4200000000000001e-9 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 17.9%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-54}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 1.42 \cdot 10^{-9}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 67.5% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-243}:\\ \;\;\;\;\frac{y}{0.5} \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.45e-243) (* (/ y 0.5) -0.25) (* 0.5 (+ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.45e-243) {
		tmp = (y / 0.5) * -0.25;
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.45d-243)) then
        tmp = (y / 0.5d0) * (-0.25d0)
    else
        tmp = 0.5d0 * (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.45e-243) {
		tmp = (y / 0.5) * -0.25;
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.45e-243:
		tmp = (y / 0.5) * -0.25
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.45e-243)
		tmp = Float64(Float64(y / 0.5) * -0.25);
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.45e-243)
		tmp = (y / 0.5) * -0.25;
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.45e-243], N[(N[(y / 0.5), $MachinePrecision] * -0.25), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.44999999999999988e-243

   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 12.4%

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

      5. fabs-sqr 12.4%

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

      6. add-sqr-sqrt 18.0%

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

      7. metadata-eval 18.0%

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

   3. Applied egg-rr 18.0%

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

      1. fma-udef 18.0%

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

      2. flip-+ 8.5%

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

   5. Applied egg-rr 8.5%

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

      1. add-sqr-sqrt 5.0%

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

      2. sqrt-prod 55.1%

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

      3. associate-*r* 55.1%

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

      4. sqrt-prod 54.8%

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

      5. fma-neg 54.8%

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

      6. *-commutative 54.8%

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

      7. associate-*l* 54.8%

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

      8. pow2 54.8%

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

   7. Applied egg-rr 54.8%

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

   8. Taylor expanded in y around -inf 61.0%

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

      1. *-commutative 61.0%

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

      2. unpow2 61.0%

         \[\leadsto \frac{y}{\color{blue}{\sqrt{0.5} \cdot \sqrt{0.5}}} \cdot -0.25 \]

      3. rem-square-sqrt 62.2%

         \[\leadsto \frac{y}{\color{blue}{0.5}} \cdot -0.25 \]

   10. Simplified 62.2%

       \[\leadsto \color{blue}{\frac{y}{0.5} \cdot -0.25} \]

   if -1.44999999999999988e-243 < 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 67.0%

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

      5. fabs-sqr 67.0%

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

      6. add-sqr-sqrt 73.4%

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

      7. metadata-eval 73.4%

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

   3. Applied egg-rr 73.4%

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

   4. Taylor expanded in y around 0 73.4%

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

      1. associate-+r+ 73.4%

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

      2. distribute-rgt1-in 73.4%

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

      3. metadata-eval 73.4%

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

      4. distribute-lft-out 73.4%

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

   6. Simplified 73.4%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-243}:\\ \;\;\;\;\frac{y}{0.5} \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]

Alternative 5: 31.1% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-247}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 3.8e-247) x (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.8e-247) {
		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 <= 3.8d-247) 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 <= 3.8e-247) {
		tmp = x;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.8e-247:
		tmp = x
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.8e-247)
		tmp = x;
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.8e-247)
		tmp = x;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.8e-247], x, N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.79999999999999988e-247

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 11.1%

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

   if 3.79999999999999988e-247 < 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 73.2%

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

      5. fabs-sqr 73.2%

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

      6. add-sqr-sqrt 78.5%

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

      7. metadata-eval 78.5%

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

   3. Applied egg-rr 78.5%

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

   4. Taylor expanded in y around inf 53.7%

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

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-247}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 6: 54.6% 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 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 41.4%

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

   5. fabs-sqr 41.4%

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

   6. add-sqr-sqrt 47.4%

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

   7. metadata-eval 47.4%

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

3. Applied egg-rr 47.4%

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

4. Taylor expanded in y around 0 47.4%

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

   1. associate-+r+ 47.4%

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

   2. distribute-rgt1-in 47.4%

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

   3. metadata-eval 47.4%

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

   4. distribute-lft-out 47.4%

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

6. Simplified 47.4%

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

7. Final simplification 47.4%

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

Alternative 7: 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 100.0%

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

2. Taylor expanded in x around inf 11.0%

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

3. Final simplification 11.0%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023275 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))