Graphics.Rendering.Chart.Plot.Vectors:renderPlotVectors from Chart-1.5.3

Percentage Accurate: 77.9% → 100.0%

Time: 3.9s

Alternatives: 7

Speedup: 1.3×

Specification

Math

\[\begin{array}{l} \\ x + \left(1 - x\right) \cdot \left(1 - y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (* (- 1.0 x) (- 1.0 y))))

C

double code(double x, double y) {
	return x + ((1.0 - x) * (1.0 - y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + ((1.0d0 - x) * (1.0d0 - y))
end function

Java

public static double code(double x, double y) {
	return x + ((1.0 - x) * (1.0 - y));
}

Python

def code(x, y):
	return x + ((1.0 - x) * (1.0 - y))

Julia

function code(x, y)
	return Float64(x + Float64(Float64(1.0 - x) * Float64(1.0 - y)))
end

MATLAB

function tmp = code(x, y)
	tmp = x + ((1.0 - x) * (1.0 - y));
end

Wolfram

code[x_, y_] := N[(x + N[(N[(1.0 - x), $MachinePrecision] * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 77.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(1 - x\right) \cdot \left(1 - y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (* (- 1.0 x) (- 1.0 y))))

C

double code(double x, double y) {
	return x + ((1.0 - x) * (1.0 - y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + ((1.0d0 - x) * (1.0d0 - y))
end function

Java

public static double code(double x, double y) {
	return x + ((1.0 - x) * (1.0 - y));
}

Python

def code(x, y):
	return x + ((1.0 - x) * (1.0 - y))

Julia

function code(x, y)
	return Float64(x + Float64(Float64(1.0 - x) * Float64(1.0 - y)))
end

MATLAB

function tmp = code(x, y)
	tmp = x + ((1.0 - x) * (1.0 - y));
end

Wolfram

code[x_, y_] := N[(x + N[(N[(1.0 - x), $MachinePrecision] * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, y, 1 - y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma x y (- 1.0 y)))

C

double code(double x, double y) {
	return fma(x, y, (1.0 - y));
}

Julia

function code(x, y)
	return fma(x, y, Float64(1.0 - y))
end

Wolfram

code[x_, y_] := N[(x * y + N[(1.0 - y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

   \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
2. Step-by-step derivation

   1. sub-neg 80.2%

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

   2. distribute-rgt-in 80.2%

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

   3. *-lft-identity 80.2%

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

   4. associate-+r+ 91.0%

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

   5. +-commutative 91.0%

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

   6. sub-neg 91.0%

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

   7. associate-+l+ 100.0%

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

   8. neg-mul-1 100.0%

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

   9. distribute-lft1-in 100.0%

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

   10. metadata-eval 100.0%

       \[\leadsto \left(1 + \color{blue}{0} \cdot x\right) + \left(-y\right) \cdot \left(1 - x\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \left(1 + \color{blue}{\left(0 \cdot -1\right)} \cdot x\right) + \left(-y\right) \cdot \left(1 - x\right) \]

   12. associate-*r* 100.0%

       \[\leadsto \left(1 + \color{blue}{0 \cdot \left(-1 \cdot x\right)}\right) + \left(-y\right) \cdot \left(1 - x\right) \]

   13. neg-mul-1 100.0%

       \[\leadsto \left(1 + 0 \cdot \color{blue}{\left(-x\right)}\right) + \left(-y\right) \cdot \left(1 - x\right) \]

   14. mul0-lft 100.0%

       \[\leadsto \left(1 + \color{blue}{0}\right) + \left(-y\right) \cdot \left(1 - x\right) \]

   15. metadata-eval 100.0%

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

   16. sub-neg 100.0%

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

   17. distribute-rgt-in 100.0%

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

   18. *-lft-identity 100.0%

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

   19. associate-+r+ 100.0%

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

   20. sub-neg 100.0%

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

   21. +-commutative 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 62.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5200000000:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-23}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-72}:\\ \;\;\;\;-y\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-296}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-245}:\\ \;\;\;\;-y\\ \mathbf{elif}\;x \leq 280000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5200000000.0)
   (* x y)
   (if (<= x -5e-23)
     1.0
     (if (<= x -2.3e-72)
       (- y)
       (if (<= x 4.5e-296)
         1.0
         (if (<= x 2.3e-245) (- y) (if (<= x 280000000.0) 1.0 (* x y))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5200000000.0) {
		tmp = x * y;
	} else if (x <= -5e-23) {
		tmp = 1.0;
	} else if (x <= -2.3e-72) {
		tmp = -y;
	} else if (x <= 4.5e-296) {
		tmp = 1.0;
	} else if (x <= 2.3e-245) {
		tmp = -y;
	} else if (x <= 280000000.0) {
		tmp = 1.0;
	} else {
		tmp = 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 (x <= (-5200000000.0d0)) then
        tmp = x * y
    else if (x <= (-5d-23)) then
        tmp = 1.0d0
    else if (x <= (-2.3d-72)) then
        tmp = -y
    else if (x <= 4.5d-296) then
        tmp = 1.0d0
    else if (x <= 2.3d-245) then
        tmp = -y
    else if (x <= 280000000.0d0) then
        tmp = 1.0d0
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5200000000.0) {
		tmp = x * y;
	} else if (x <= -5e-23) {
		tmp = 1.0;
	} else if (x <= -2.3e-72) {
		tmp = -y;
	} else if (x <= 4.5e-296) {
		tmp = 1.0;
	} else if (x <= 2.3e-245) {
		tmp = -y;
	} else if (x <= 280000000.0) {
		tmp = 1.0;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5200000000.0:
		tmp = x * y
	elif x <= -5e-23:
		tmp = 1.0
	elif x <= -2.3e-72:
		tmp = -y
	elif x <= 4.5e-296:
		tmp = 1.0
	elif x <= 2.3e-245:
		tmp = -y
	elif x <= 280000000.0:
		tmp = 1.0
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5200000000.0)
		tmp = Float64(x * y);
	elseif (x <= -5e-23)
		tmp = 1.0;
	elseif (x <= -2.3e-72)
		tmp = Float64(-y);
	elseif (x <= 4.5e-296)
		tmp = 1.0;
	elseif (x <= 2.3e-245)
		tmp = Float64(-y);
	elseif (x <= 280000000.0)
		tmp = 1.0;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5200000000.0)
		tmp = x * y;
	elseif (x <= -5e-23)
		tmp = 1.0;
	elseif (x <= -2.3e-72)
		tmp = -y;
	elseif (x <= 4.5e-296)
		tmp = 1.0;
	elseif (x <= 2.3e-245)
		tmp = -y;
	elseif (x <= 280000000.0)
		tmp = 1.0;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5200000000.0], N[(x * y), $MachinePrecision], If[LessEqual[x, -5e-23], 1.0, If[LessEqual[x, -2.3e-72], (-y), If[LessEqual[x, 4.5e-296], 1.0, If[LessEqual[x, 2.3e-245], (-y), If[LessEqual[x, 280000000.0], 1.0, N[(x * y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.2e9 or 2.8e8 < x

   1. Initial program 58.4%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]

   2. Taylor expanded in x around inf 58.0%

      \[\leadsto \color{blue}{\left(1 + -1 \cdot \left(1 - y\right)\right) \cdot x} \]
   3. Step-by-step derivation

      1. mul-1-neg 58.0%

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

      2. unsub-neg 58.0%

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

      3. sub-neg 58.0%

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

      4. associate--r+ 80.8%

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

      5. metadata-eval 80.8%

         \[\leadsto \left(\color{blue}{0} - \left(-y\right)\right) \cdot x \]

      6. neg-sub0 80.8%

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

      7. remove-double-neg 80.8%

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

   4. Simplified 80.8%

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

   if -5.2e9 < x < -5.0000000000000002e-23 or -2.29999999999999995e-72 < x < 4.5000000000000002e-296 or 2.3000000000000002e-245 < x < 2.8e8

   1. Initial program 100.0%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]

   2. Taylor expanded in y around 0 61.8%

      \[\leadsto \color{blue}{1} \]

   if -5.0000000000000002e-23 < x < -2.29999999999999995e-72 or 4.5000000000000002e-296 < x < 2.3000000000000002e-245

   1. Initial program 100.0%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]

   2. Taylor expanded in y around inf 85.3%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(1 - x\right)\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 85.3%

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

      2. distribute-rgt-neg-in 85.3%

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

      3. sub-neg 85.3%

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

      4. distribute-neg-in 85.3%

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

      5. metadata-eval 85.3%

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

      6. remove-double-neg 85.3%

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

   4. Simplified 85.3%

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

   5. Taylor expanded in x around 0 85.3%

      \[\leadsto \color{blue}{-1 \cdot y} \]
   6. Step-by-step derivation

      1. neg-mul-1 85.3%

         \[\leadsto \color{blue}{-y} \]

   7. Simplified 85.3%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5200000000:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-23}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-72}:\\ \;\;\;\;-y\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-296}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-245}:\\ \;\;\;\;-y\\ \mathbf{elif}\;x \leq 280000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1800000000 \lor \neg \left(x \leq 1.05 \cdot 10^{-15}\right):\\ \;\;\;\;y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1800000000.0) (not (<= x 1.05e-15)))
   (* y (+ x -1.0))
   (- 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1800000000.0) || !(x <= 1.05e-15)) {
		tmp = y * (x + -1.0);
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1800000000.0d0)) .or. (.not. (x <= 1.05d-15))) then
        tmp = y * (x + (-1.0d0))
    else
        tmp = 1.0d0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1800000000.0) || !(x <= 1.05e-15)) {
		tmp = y * (x + -1.0);
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1800000000.0) or not (x <= 1.05e-15):
		tmp = y * (x + -1.0)
	else:
		tmp = 1.0 - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1800000000.0) || !(x <= 1.05e-15))
		tmp = Float64(y * Float64(x + -1.0));
	else
		tmp = Float64(1.0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1800000000.0) || ~((x <= 1.05e-15)))
		tmp = y * (x + -1.0);
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1800000000.0], N[Not[LessEqual[x, 1.05e-15]], $MachinePrecision]], N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8e9 or 1.0499999999999999e-15 < x

   1. Initial program 58.7%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]

   2. Taylor expanded in y around inf 81.3%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(1 - x\right)\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 81.3%

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

      2. distribute-rgt-neg-in 81.3%

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

      3. sub-neg 81.3%

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

      4. distribute-neg-in 81.3%

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

      5. metadata-eval 81.3%

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

      6. remove-double-neg 81.3%

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

   4. Simplified 81.3%

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

   if -1.8e9 < x < 1.0499999999999999e-15

   1. Initial program 100.0%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]

   2. Taylor expanded in x around 0 99.8%

      \[\leadsto \color{blue}{1 - y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1800000000 \lor \neg \left(x \leq 1.05 \cdot 10^{-15}\right):\\ \;\;\;\;y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]

Alternative 4: 86.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7400000000:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 14500000:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7400000000.0) (* x y) (if (<= x 14500000.0) (- 1.0 y) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7400000000.0) {
		tmp = x * y;
	} else if (x <= 14500000.0) {
		tmp = 1.0 - y;
	} else {
		tmp = 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 (x <= (-7400000000.0d0)) then
        tmp = x * y
    else if (x <= 14500000.0d0) then
        tmp = 1.0d0 - y
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -7400000000.0) {
		tmp = x * y;
	} else if (x <= 14500000.0) {
		tmp = 1.0 - y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7400000000.0:
		tmp = x * y
	elif x <= 14500000.0:
		tmp = 1.0 - y
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7400000000.0)
		tmp = Float64(x * y);
	elseif (x <= 14500000.0)
		tmp = Float64(1.0 - y);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7400000000.0)
		tmp = x * y;
	elseif (x <= 14500000.0)
		tmp = 1.0 - y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7400000000.0], N[(x * y), $MachinePrecision], If[LessEqual[x, 14500000.0], N[(1.0 - y), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.4e9 or 1.45e7 < x

   1. Initial program 58.4%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]

   2. Taylor expanded in x around inf 58.0%

      \[\leadsto \color{blue}{\left(1 + -1 \cdot \left(1 - y\right)\right) \cdot x} \]
   3. Step-by-step derivation

      1. mul-1-neg 58.0%

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

      2. unsub-neg 58.0%

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

      3. sub-neg 58.0%

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

      4. associate--r+ 80.8%

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

      5. metadata-eval 80.8%

         \[\leadsto \left(\color{blue}{0} - \left(-y\right)\right) \cdot x \]

      6. neg-sub0 80.8%

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

      7. remove-double-neg 80.8%

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

   4. Simplified 80.8%

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

   if -7.4e9 < x < 1.45e7

   1. Initial program 100.0%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]

   2. Taylor expanded in x around 0 99.8%

      \[\leadsto \color{blue}{1 - y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7400000000:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 14500000:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 5: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(1 + x \cdot y\right) - y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (+ 1.0 (* x y)) y))

C

double code(double x, double y) {
	return (1.0 + (x * y)) - y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 + (x * y)) - y
end function

Java

public static double code(double x, double y) {
	return (1.0 + (x * y)) - y;
}

Python

def code(x, y):
	return (1.0 + (x * y)) - y

Julia

function code(x, y)
	return Float64(Float64(1.0 + Float64(x * y)) - y)
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 + (x * y)) - y;
end

Wolfram

code[x_, y_] := N[(N[(1.0 + N[(x * y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]

Derivation

1. Initial program 80.2%

   \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]

2. Taylor expanded in x around -inf 100.0%

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

3. Final simplification 100.0%

   \[\leadsto \left(1 + x \cdot y\right) - y \]

Alternative 6: 62.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 0.014:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) (- y) (if (<= y 0.014) 1.0 (- y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = -y;
	} else if (y <= 0.014) {
		tmp = 1.0;
	} else {
		tmp = -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 <= (-1.0d0)) then
        tmp = -y
    else if (y <= 0.014d0) then
        tmp = 1.0d0
    else
        tmp = -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = -y;
	} else if (y <= 0.014) {
		tmp = 1.0;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = -y
	elif y <= 0.014:
		tmp = 1.0
	else:
		tmp = -y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(-y);
	elseif (y <= 0.014)
		tmp = 1.0;
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = -y;
	elseif (y <= 0.014)
		tmp = 1.0;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], (-y), If[LessEqual[y, 0.014], 1.0, (-y)]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.0140000000000000003 < y

   1. Initial program 99.9%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]

   2. Taylor expanded in y around inf 99.2%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(1 - x\right)\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 99.2%

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

      2. distribute-rgt-neg-in 99.2%

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

      3. sub-neg 99.2%

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

      4. distribute-neg-in 99.2%

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

      5. metadata-eval 99.2%

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

      6. remove-double-neg 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in x around 0 48.7%

      \[\leadsto \color{blue}{-1 \cdot y} \]
   6. Step-by-step derivation

      1. neg-mul-1 48.7%

         \[\leadsto \color{blue}{-y} \]

   7. Simplified 48.7%

      \[\leadsto \color{blue}{-y} \]

   if -1 < y < 0.0140000000000000003

   1. Initial program 59.4%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]

   2. Taylor expanded in y around 0 73.1%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 0.014:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 7: 39.2% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 80.2%

   \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]

2. Taylor expanded in y around 0 37.2%

   \[\leadsto \color{blue}{1} \]

3. Final simplification 37.2%

   \[\leadsto 1 \]

Developer target: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ y \cdot x - \left(y - 1\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (* y x) (- y 1.0)))

C

double code(double x, double y) {
	return (y * x) - (y - 1.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (y * x) - (y - 1.0d0)
end function

Java

public static double code(double x, double y) {
	return (y * x) - (y - 1.0);
}

Python

def code(x, y):
	return (y * x) - (y - 1.0)

Julia

function code(x, y)
	return Float64(Float64(y * x) - Float64(y - 1.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (y * x) - (y - 1.0);
end

Wolfram

code[x_, y_] := N[(N[(y * x), $MachinePrecision] - N[(y - 1.0), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023200 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.Vectors:renderPlotVectors from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (- (* y x) (- y 1.0))

  (+ x (* (- 1.0 x) (- 1.0 y))))