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

Percentage Accurate: 99.9% → 99.9%

Time: 6.1s

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: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-94}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 3.25 \cdot 10^{+15}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.7e-94)
   (* 0.5 (+ x y))
   (if (<= x 3.25e+15) (* (fabs (- y x)) 0.5) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.7e-94) {
		tmp = 0.5 * (x + y);
	} else if (x <= 3.25e+15) {
		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 <= (-2.7d-94)) then
        tmp = 0.5d0 * (x + y)
    else if (x <= 3.25d+15) 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 <= -2.7e-94) {
		tmp = 0.5 * (x + y);
	} else if (x <= 3.25e+15) {
		tmp = Math.abs((y - x)) * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.7e-94:
		tmp = 0.5 * (x + y)
	elif x <= 3.25e+15:
		tmp = math.fabs((y - x)) * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.7e-94)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 3.25e+15)
		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 <= -2.7e-94)
		tmp = 0.5 * (x + y);
	elseif (x <= 3.25e+15)
		tmp = abs((y - x)) * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.7e-94], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.25e+15], 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 < -2.7000000000000001e-94

   1. Initial program 100.0%

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

      1. flip-+ 45.4%

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

   3. Applied egg-rr 38.4%

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

   4. Taylor expanded in x around 0 85.2%

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

      1. distribute-lft-out 85.2%

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

      2. +-commutative 85.2%

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

   6. Simplified 85.2%

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

   if -2.7000000000000001e-94 < x < 3.25e15

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 82.3%

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

   if 3.25e15 < x

   1. Initial program 99.8%

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

      1. flip-+ 35.8%

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

   3. Applied egg-rr 7.9%

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

      1. add-sqr-sqrt 7.9%

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

      2. sqrt-prod 7.9%

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

      3. add-cube-cbrt 7.9%

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

      4. sqrt-prod 7.9%

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

      5. fma-neg 7.9%

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

      6. cbrt-unprod 5.3%

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

      7. pow2 5.3%

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

      8. pow2 5.3%

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

      9. pow-prod-up 5.3%

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

      10. metadata-eval 5.3%

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

      11. cbrt-prod 5.3%

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

      12. pow2 5.3%

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

      13. distribute-rgt-neg-in 5.3%

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

      14. metadata-eval 5.3%

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

   5. Applied egg-rr 5.3%

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

      1. unpow2 5.3%

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

      2. rem-sqrt-square 5.3%

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

      3. *-commutative 5.3%

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

   7. Simplified 5.3%

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

   8. Taylor expanded in x around inf 78.6%

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

      1. *-commutative 78.6%

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

   10. Simplified 78.6%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-94}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 3.25 \cdot 10^{+15}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]

Alternative 3: 59.8% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-95}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-73}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9e-95) (* x 0.5) (if (<= x 2.6e-73) (* y 0.5) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9e-95) {
		tmp = x * 0.5;
	} else if (x <= 2.6e-73) {
		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 <= (-9d-95)) then
        tmp = x * 0.5d0
    else if (x <= 2.6d-73) 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 <= -9e-95) {
		tmp = x * 0.5;
	} else if (x <= 2.6e-73) {
		tmp = y * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9e-95:
		tmp = x * 0.5
	elif x <= 2.6e-73:
		tmp = y * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9e-95)
		tmp = Float64(x * 0.5);
	elseif (x <= 2.6e-73)
		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 <= -9e-95)
		tmp = x * 0.5;
	elseif (x <= 2.6e-73)
		tmp = y * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9e-95], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 2.6e-73], N[(y * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9e-95

   1. Initial program 100.0%

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

      1. flip-+ 44.9%

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

   3. Applied egg-rr 37.9%

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

   4. Taylor expanded in x around inf 71.0%

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

   if -9e-95 < x < 2.6000000000000001e-73

   1. Initial program 100.0%

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

      1. flip-+ 61.4%

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

   3. Applied egg-rr 35.7%

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

   4. Taylor expanded in x around 0 42.4%

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

   if 2.6000000000000001e-73 < x

   1. Initial program 99.9%

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

      1. flip-+ 40.0%

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

   3. Applied egg-rr 10.1%

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

      1. add-sqr-sqrt 10.1%

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

      2. sqrt-prod 10.1%

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

      3. add-cube-cbrt 10.1%

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

      4. sqrt-prod 10.1%

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

      5. fma-neg 10.1%

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

      6. cbrt-unprod 8.3%

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

      7. pow2 8.3%

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

      8. pow2 8.3%

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

      9. pow-prod-up 8.3%

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

      10. metadata-eval 8.3%

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

      11. cbrt-prod 8.3%

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

      12. pow2 8.3%

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

      13. distribute-rgt-neg-in 8.3%

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

      14. metadata-eval 8.3%

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

   5. Applied egg-rr 8.3%

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

      1. unpow2 8.3%

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

      2. rem-sqrt-square 8.3%

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

      3. *-commutative 8.3%

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

   7. Simplified 8.3%

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

   8. Taylor expanded in x around inf 65.4%

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

      1. *-commutative 65.4%

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

   10. Simplified 65.4%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-95}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-73}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]

Alternative 4: 67.7% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.1e+14) (* 0.5 (+ x y)) (* x 1.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.1e+14) {
		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 <= 3.1d+14) 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 <= 3.1e+14) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 3.1e+14:
		tmp = 0.5 * (x + y)
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.1e+14)
		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 <= 3.1e+14)
		tmp = 0.5 * (x + y);
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.1e+14], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.1e14

   1. Initial program 100.0%

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

      1. flip-+ 54.1%

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

   3. Applied egg-rr 34.4%

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

   4. Taylor expanded in x around 0 60.5%

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

      1. distribute-lft-out 60.5%

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

      2. +-commutative 60.5%

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

   6. Simplified 60.5%

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

   if 3.1e14 < x

   1. Initial program 99.8%

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

      1. flip-+ 35.8%

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

   3. Applied egg-rr 7.9%

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

      1. add-sqr-sqrt 7.9%

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

      2. sqrt-prod 7.9%

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

      3. add-cube-cbrt 7.9%

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

      4. sqrt-prod 7.9%

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

      5. fma-neg 7.9%

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

      6. cbrt-unprod 5.3%

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

      7. pow2 5.3%

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

      8. pow2 5.3%

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

      9. pow-prod-up 5.3%

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

      10. metadata-eval 5.3%

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

      11. cbrt-prod 5.3%

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

      12. pow2 5.3%

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

      13. distribute-rgt-neg-in 5.3%

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

      14. metadata-eval 5.3%

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

   5. Applied egg-rr 5.3%

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

      1. unpow2 5.3%

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

      2. rem-sqrt-square 5.3%

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

      3. *-commutative 5.3%

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

   7. Simplified 5.3%

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

   8. Taylor expanded in x around inf 78.6%

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

      1. *-commutative 78.6%

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

   10. Simplified 78.6%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]

Alternative 5: 46.1% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-64}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 8e-64) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8e-64) {
		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 <= 8d-64) 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 <= 8e-64) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8e-64:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8e-64)
		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 <= 8e-64)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8e-64], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.99999999999999972e-64

   1. Initial program 99.9%

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

      1. flip-+ 50.4%

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

   3. Applied egg-rr 21.5%

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

   4. Taylor expanded in x around inf 32.9%

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

   if 7.99999999999999972e-64 < y

   1. Initial program 99.9%

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

      1. flip-+ 50.0%

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

   3. Applied egg-rr 44.0%

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

   4. Taylor expanded in x around 0 71.9%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-64}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 6: 31.2% 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. flip-+ 50.3%

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

3. Applied egg-rr 28.8%

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

4. Taylor expanded in x around inf 28.5%

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

5. Final simplification 28.5%

   \[\leadsto x \cdot 0.5 \]

Alternative 7: 11.5% 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 10.9%

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

3. Final simplification 10.9%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023293 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))