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

Percentage Accurate: 78.2% → 100.0%

Time: 6.4s

Alternatives: 7

Speedup: 1.5×

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: 78.2% 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, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.2%

   \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. mul-1-neg N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. distribute-neg-in N/A

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

   9. remove-double-neg N/A

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

   10. sub-neg N/A

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

   11. lower--.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 2: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+15}:\\ \;\;\;\;\left(x - 1\right) \cdot y\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-46}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2e+15)
   (* (- x 1.0) y)
   (if (<= y 1.45e-46) (- 1.0 y) (fma y x (- y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2e+15) {
		tmp = (x - 1.0) * y;
	} else if (y <= 1.45e-46) {
		tmp = 1.0 - y;
	} else {
		tmp = fma(y, x, -y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2e+15)
		tmp = Float64(Float64(x - 1.0) * y);
	elseif (y <= 1.45e-46)
		tmp = Float64(1.0 - y);
	else
		tmp = fma(y, x, Float64(-y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2e+15], N[(N[(x - 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1.45e-46], N[(1.0 - y), $MachinePrecision], N[(y * x + (-y)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2e15

   1. Initial program 100.0%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -2e15 < y < 1.45000000000000002e-46

   1. Initial program 56.2%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower--.f64 77.5

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

   5. Applied rewrites 77.5%

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

   if 1.45000000000000002e-46 < y

   1. Initial program 92.9%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 95.2

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

   5. Applied rewrites 95.2%

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

   7. Applied rewrites 95.2%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{x}, -y\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - 1\right) \cdot y\\ \mathbf{if}\;y \leq -2 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-46}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- x 1.0) y)))
   (if (<= y -2e+15) t_0 (if (<= y 1.45e-46) (- 1.0 y) t_0))))

C

double code(double x, double y) {
	double t_0 = (x - 1.0) * y;
	double tmp;
	if (y <= -2e+15) {
		tmp = t_0;
	} else if (y <= 1.45e-46) {
		tmp = 1.0 - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - 1.0d0) * y
    if (y <= (-2d+15)) then
        tmp = t_0
    else if (y <= 1.45d-46) then
        tmp = 1.0d0 - y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - 1.0) * y;
	double tmp;
	if (y <= -2e+15) {
		tmp = t_0;
	} else if (y <= 1.45e-46) {
		tmp = 1.0 - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - 1.0) * y
	tmp = 0
	if y <= -2e+15:
		tmp = t_0
	elif y <= 1.45e-46:
		tmp = 1.0 - y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - 1.0) * y)
	tmp = 0.0
	if (y <= -2e+15)
		tmp = t_0;
	elseif (y <= 1.45e-46)
		tmp = Float64(1.0 - y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - 1.0) * y;
	tmp = 0.0;
	if (y <= -2e+15)
		tmp = t_0;
	elseif (y <= 1.45e-46)
		tmp = 1.0 - y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - 1.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -2e+15], t$95$0, If[LessEqual[y, 1.45e-46], N[(1.0 - y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2e15 or 1.45000000000000002e-46 < y

   1. Initial program 96.2%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 97.4

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

   5. Applied rewrites 97.4%

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

   if -2e15 < y < 1.45000000000000002e-46

   1. Initial program 56.2%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower--.f64 77.5

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

   5. Applied rewrites 77.5%

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

Alternative 4: 62.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - y \leq -20000000:\\ \;\;\;\;-y\\ \mathbf{elif}\;1 - y \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- 1.0 y) -20000000.0) (- y) (if (<= (- 1.0 y) 2.0) 1.0 (- y))))

C

double code(double x, double y) {
	double tmp;
	if ((1.0 - y) <= -20000000.0) {
		tmp = -y;
	} else if ((1.0 - y) <= 2.0) {
		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 ((1.0d0 - y) <= (-20000000.0d0)) then
        tmp = -y
    else if ((1.0d0 - y) <= 2.0d0) 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 ((1.0 - y) <= -20000000.0) {
		tmp = -y;
	} else if ((1.0 - y) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (1.0 - y) <= -20000000.0:
		tmp = -y
	elif (1.0 - y) <= 2.0:
		tmp = 1.0
	else:
		tmp = -y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(1.0 - y) <= -20000000.0)
		tmp = Float64(-y);
	elseif (Float64(1.0 - y) <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((1.0 - y) <= -20000000.0)
		tmp = -y;
	elseif ((1.0 - y) <= 2.0)
		tmp = 1.0;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 - y), $MachinePrecision], -20000000.0], (-y), If[LessEqual[N[(1.0 - y), $MachinePrecision], 2.0], 1.0, (-y)]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -2e7 or 2 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 100.0%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower--.f64 42.0

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

   5. Applied rewrites 42.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 41.3%

      \[\leadsto -y \]

   if -2e7 < (-.f64 #s(literal 1 binary64) y) < 2

   1. Initial program 54.7%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 73.8%

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

Alternative 5: 86.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+25}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+65}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.4e+25) (* y x) (if (<= x 5.2e+65) (- 1.0 y) (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.4e+25) {
		tmp = y * x;
	} else if (x <= 5.2e+65) {
		tmp = 1.0 - y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.4d+25)) then
        tmp = y * x
    else if (x <= 5.2d+65) then
        tmp = 1.0d0 - y
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.4e+25) {
		tmp = y * x;
	} else if (x <= 5.2e+65) {
		tmp = 1.0 - y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.4e+25:
		tmp = y * x
	elif x <= 5.2e+65:
		tmp = 1.0 - y
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.4e+25)
		tmp = Float64(y * x);
	elseif (x <= 5.2e+65)
		tmp = Float64(1.0 - y);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.4e+25)
		tmp = y * x;
	elseif (x <= 5.2e+65)
		tmp = 1.0 - y;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.4e+25], N[(y * x), $MachinePrecision], If[LessEqual[x, 5.2e+65], N[(1.0 - y), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4000000000000001e25 or 5.20000000000000005e65 < x

   1. Initial program 58.4%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. sub-neg N/A

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

      4. associate--r+ N/A

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

      5. metadata-eval N/A

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

      6. neg-sub0 N/A

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

      7. remove-double-neg N/A

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

      8. lower-*.f64 82.1

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

   5. Applied rewrites 82.1%

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

   if -1.4000000000000001e25 < x < 5.20000000000000005e65

   1. Initial program 92.9%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower--.f64 92.3

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

   5. Applied rewrites 92.3%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+25}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+65}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.4% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

   \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. lower--.f64 58.2

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

5. Applied rewrites 58.2%

   \[\leadsto \color{blue}{1 - y} \]
6. Add Preprocessing

Alternative 7: 38.2% accurate, 15.0× 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 77.2%

   \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 38.4%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Developer Target 1: 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 2024270 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.Vectors:renderPlotVectors from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* y x) (- y 1)))

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