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

Percentage Accurate: 99.9% → 99.9%

Time: 3.7s

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. Add Preprocessing

3. Final simplification 99.9%

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

Alternative 2: 70.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-50}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.7e-50) (* (fabs (- y x)) 0.5) (* 0.5 (+ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.7e-50) {
		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 <= (-2.7d-50)) 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 <= -2.7e-50) {
		tmp = Math.abs((y - x)) * 0.5;
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.7e-50:
		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 <= -2.7e-50)
		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 <= -2.7e-50)
		tmp = abs((y - x)) * 0.5;
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.7e-50], 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 < -2.7e-50

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 71.1%

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

   if -2.7e-50 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 92.2%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. rem-square-sqrt 66.3%

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

      4. fabs-sqr 66.3%

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

      5. rem-square-sqrt 72.1%

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

      6. fma-define 72.1%

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

      7. distribute-lft-out-- 72.1%

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

      8. sub-neg 72.1%

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

      9. associate-+r+ 72.1%

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

      10. +-commutative 72.1%

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

      11. distribute-lft-neg-in 72.1%

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

      12. metadata-eval 72.1%

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

      13. distribute-rgt1-in 72.1%

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

      14. metadata-eval 72.1%

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

      15. distribute-lft-out 72.1%

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

   6. Simplified 72.1%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-50}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 45.2% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-29}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 5.2e-29) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.2e-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 <= 5.2d-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 <= 5.2e-29) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.2e-29:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.2e-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 <= 5.2e-29)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.2e-29], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.2000000000000004e-29

   1. Initial program 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. fma-define 99.9%

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

      3. add-sqr-sqrt 37.5%

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

      4. fabs-sqr 37.5%

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

      5. add-sqr-sqrt 43.7%

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

      6. frac-2neg 43.7%

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

      7. distribute-frac-neg 43.7%

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

      8. fmm-undef 43.7%

         \[\leadsto \color{blue}{1 \cdot x - \frac{y - x}{-2}} \]

      9. *-un-lft-identity 43.7%

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

      10. metadata-eval 43.7%

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

   4. Applied egg-rr 43.7%

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

   5. Taylor expanded in x around inf 35.8%

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

   if 5.2000000000000004e-29 < y

   1. Initial program 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. fma-define 99.9%

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

      3. add-sqr-sqrt 86.5%

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

      4. fabs-sqr 86.5%

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

      5. add-sqr-sqrt 89.5%

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

      6. frac-2neg 89.5%

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

      7. distribute-frac-neg 89.5%

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

      8. fmm-undef 89.5%

         \[\leadsto \color{blue}{1 \cdot x - \frac{y - x}{-2}} \]

      9. *-un-lft-identity 89.5%

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

      10. metadata-eval 89.5%

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

   4. Applied egg-rr 89.5%

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

   5. Taylor expanded in x around 0 77.2%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-29}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.8% 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. Add Preprocessing

3. Taylor expanded in x around inf 89.6%

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

4. Taylor expanded in x around 0 99.9%

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

   1. +-commutative 99.9%

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

   2. fma-define 99.9%

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

   3. rem-square-sqrt 51.1%

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

   4. fabs-sqr 51.1%

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

   5. rem-square-sqrt 56.4%

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

   6. fma-define 56.4%

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

   7. distribute-lft-out-- 56.4%

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

   8. sub-neg 56.4%

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

   9. associate-+r+ 56.4%

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

   10. +-commutative 56.4%

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

   11. distribute-lft-neg-in 56.4%

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

   12. metadata-eval 56.4%

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

   13. distribute-rgt1-in 56.4%

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

   14. metadata-eval 56.4%

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

   15. distribute-lft-out 56.4%

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

6. Simplified 56.4%

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

7. Final simplification 56.4%

   \[\leadsto 0.5 \cdot \left(x + y\right) \]
8. Add Preprocessing

Alternative 5: 30.7% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. fma-define 99.9%

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

   3. add-sqr-sqrt 51.1%

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

   4. fabs-sqr 51.1%

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

   5. add-sqr-sqrt 56.4%

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

   6. frac-2neg 56.4%

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

   7. distribute-frac-neg 56.4%

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

   8. fmm-undef 56.4%

      \[\leadsto \color{blue}{1 \cdot x - \frac{y - x}{-2}} \]

   9. *-un-lft-identity 56.4%

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

   10. metadata-eval 56.4%

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

4. Applied egg-rr 56.4%

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

5. Taylor expanded in x around inf 29.8%

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

6. Final simplification 29.8%

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

Alternative 6: 11.4% accurate, 107.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around inf 11.4%

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

4. Final simplification 11.4%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024115 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))