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

Percentage Accurate: 77.2% → 100.0%

Time: 3.1s

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: 77.2% 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 82.1%

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

   1. sub-neg 82.1%

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

   2. distribute-rgt-in 82.1%

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

   3. *-lft-identity 82.1%

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

   4. associate-+r+ 90.2%

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

   5. +-commutative 90.2%

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

   6. sub-neg 90.2%

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

   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: 62.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+45}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq -5.4 \cdot 10^{-33}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+221}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.4e+45)
   (- y)
   (if (<= y -5.4e-33)
     (* x y)
     (if (<= y 8.8e-10) 1.0 (if (<= y 3.7e+221) (* x y) (- y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.4e+45) {
		tmp = -y;
	} else if (y <= -5.4e-33) {
		tmp = x * y;
	} else if (y <= 8.8e-10) {
		tmp = 1.0;
	} else if (y <= 3.7e+221) {
		tmp = x * y;
	} 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 <= (-4.4d+45)) then
        tmp = -y
    else if (y <= (-5.4d-33)) then
        tmp = x * y
    else if (y <= 8.8d-10) then
        tmp = 1.0d0
    else if (y <= 3.7d+221) then
        tmp = x * y
    else
        tmp = -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.4e+45) {
		tmp = -y;
	} else if (y <= -5.4e-33) {
		tmp = x * y;
	} else if (y <= 8.8e-10) {
		tmp = 1.0;
	} else if (y <= 3.7e+221) {
		tmp = x * y;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.4e+45:
		tmp = -y
	elif y <= -5.4e-33:
		tmp = x * y
	elif y <= 8.8e-10:
		tmp = 1.0
	elif y <= 3.7e+221:
		tmp = x * y
	else:
		tmp = -y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.4e+45)
		tmp = Float64(-y);
	elseif (y <= -5.4e-33)
		tmp = Float64(x * y);
	elseif (y <= 8.8e-10)
		tmp = 1.0;
	elseif (y <= 3.7e+221)
		tmp = Float64(x * y);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.4e+45)
		tmp = -y;
	elseif (y <= -5.4e-33)
		tmp = x * y;
	elseif (y <= 8.8e-10)
		tmp = 1.0;
	elseif (y <= 3.7e+221)
		tmp = x * y;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.4e+45], (-y), If[LessEqual[y, -5.4e-33], N[(x * y), $MachinePrecision], If[LessEqual[y, 8.8e-10], 1.0, If[LessEqual[y, 3.7e+221], N[(x * y), $MachinePrecision], (-y)]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.4000000000000001e45 or 3.70000000000000001e221 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. metadata-eval 100.0%

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

      6. remove-double-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 62.3%

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

      1. neg-mul-1 62.3%

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

   7. Simplified 62.3%

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

   if -4.4000000000000001e45 < y < -5.4000000000000002e-33 or 8.7999999999999996e-10 < y < 3.70000000000000001e221

   1. Initial program 93.4%

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

   2. Taylor expanded in x around inf 60.9%

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

      1. mul-1-neg 60.9%

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

      2. unsub-neg 60.9%

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

      3. sub-neg 60.9%

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

      4. associate--r+ 67.5%

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

      5. metadata-eval 67.5%

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

      6. neg-sub0 67.5%

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

      7. remove-double-neg 67.5%

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

   4. Simplified 67.5%

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

   if -5.4000000000000002e-33 < y < 8.7999999999999996e-10

   1. Initial program 62.6%

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

   2. Taylor expanded in y around 0 83.3%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+45}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq -5.4 \cdot 10^{-33}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+221}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 3: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-59} \lor \neg \left(y \leq 2.45\right):\\ \;\;\;\;y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.65e-59) (not (<= y 2.45))) (* y (+ x -1.0)) (- 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.65e-59) || !(y <= 2.45)) {
		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 ((y <= (-1.65d-59)) .or. (.not. (y <= 2.45d0))) 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 ((y <= -1.65e-59) || !(y <= 2.45)) {
		tmp = y * (x + -1.0);
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.65e-59) or not (y <= 2.45):
		tmp = y * (x + -1.0)
	else:
		tmp = 1.0 - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.65e-59) || !(y <= 2.45))
		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 ((y <= -1.65e-59) || ~((y <= 2.45)))
		tmp = y * (x + -1.0);
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.65e-59], N[Not[LessEqual[y, 2.45]], $MachinePrecision]], N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.64999999999999991e-59 or 2.4500000000000002 < y

   1. Initial program 95.0%

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

   2. Taylor expanded in y around inf 97.2%

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

      1. mul-1-neg 97.2%

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

      2. distribute-rgt-neg-in 97.2%

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

      3. sub-neg 97.2%

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

      4. distribute-neg-in 97.2%

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

      5. metadata-eval 97.2%

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

      6. remove-double-neg 97.2%

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

   4. Simplified 97.2%

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

   if -1.64999999999999991e-59 < y < 2.4500000000000002

   1. Initial program 63.6%

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

   2. Taylor expanded in x around 0 84.8%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-59} \lor \neg \left(y \leq 2.45\right):\\ \;\;\;\;y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]

Alternative 4: 86.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -12800000:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 680000:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -12800000.0) (* x y) (if (<= x 680000.0) (- 1.0 y) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -12800000.0) {
		tmp = x * y;
	} else if (x <= 680000.0) {
		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 <= (-12800000.0d0)) then
        tmp = x * y
    else if (x <= 680000.0d0) 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 <= -12800000.0) {
		tmp = x * y;
	} else if (x <= 680000.0) {
		tmp = 1.0 - y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -12800000.0:
		tmp = x * y
	elif x <= 680000.0:
		tmp = 1.0 - y
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -12800000.0)
		tmp = Float64(x * y);
	elseif (x <= 680000.0)
		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 <= -12800000.0)
		tmp = x * y;
	elseif (x <= 680000.0)
		tmp = 1.0 - y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -12800000.0], N[(x * y), $MachinePrecision], If[LessEqual[x, 680000.0], N[(1.0 - y), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.28e7 or 6.8e5 < x

   1. Initial program 61.2%

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

   2. Taylor expanded in x around inf 60.2%

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

      1. mul-1-neg 60.2%

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

      2. unsub-neg 60.2%

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

      3. sub-neg 60.2%

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

      4. associate--r+ 77.8%

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

      5. metadata-eval 77.8%

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

      6. neg-sub0 77.8%

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

      7. remove-double-neg 77.8%

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

   4. Simplified 77.8%

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

   if -1.28e7 < x < 6.8e5

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 97.2%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -12800000:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 680000:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

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 82.1%

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

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: 61.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-14}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 37000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.65e-14) (- y) (if (<= y 37000000.0) 1.0 (- y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.65e-14) {
		tmp = -y;
	} else if (y <= 37000000.0) {
		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 <= (-1.65d-14)) then
        tmp = -y
    else if (y <= 37000000.0d0) 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 <= -1.65e-14) {
		tmp = -y;
	} else if (y <= 37000000.0) {
		tmp = 1.0;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.65e-14:
		tmp = -y
	elif y <= 37000000.0:
		tmp = 1.0
	else:
		tmp = -y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.65e-14)
		tmp = Float64(-y);
	elseif (y <= 37000000.0)
		tmp = 1.0;
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.65e-14)
		tmp = -y;
	elseif (y <= 37000000.0)
		tmp = 1.0;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.65e-14], (-y), If[LessEqual[y, 37000000.0], 1.0, (-y)]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6499999999999999e-14 or 3.7e7 < y

   1. Initial program 99.5%

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

   2. Taylor expanded in y around inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. distribute-rgt-neg-in 99.6%

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

      3. sub-neg 99.6%

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

      4. distribute-neg-in 99.6%

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

      5. metadata-eval 99.6%

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

      6. remove-double-neg 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in x around 0 48.3%

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

      1. neg-mul-1 48.3%

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

   7. Simplified 48.3%

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

   if -1.6499999999999999e-14 < y < 3.7e7

   1. Initial program 61.5%

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

   2. Taylor expanded in y around 0 79.1%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-14}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 37000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 7: 38.4% 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 82.1%

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

2. Taylor expanded in y around 0 37.7%

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

3. Final simplification 37.7%

   \[\leadsto 1 \]

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 2023257 
(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))))