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

Percentage Accurate: 78.0% → 100.0%

Time: 6.7s

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: 78.0% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.1%

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

3. Taylor expanded in x around 0

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

   1. cancel-sign-sub-inv N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. sub-neg N/A

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

   5. associate--r+ N/A

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

   6. metadata-eval N/A

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

   7. neg-sub0 N/A

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

   8. remove-double-neg N/A

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

   9. distribute-lft-neg-out N/A

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

   10. distribute-rgt-neg-out N/A

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

   11. associate--r+ N/A

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

   12. +-commutative N/A

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

   13. associate--r+ N/A

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

   14. sub-neg N/A

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

   15. distribute-lft-neg-out N/A

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

   16. +-commutative N/A

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

   17. distribute-lft-neg-out N/A

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

   18. distribute-rgt-neg-in N/A

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

   19. remove-double-neg N/A

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

   20. *-commutative N/A

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

5. Applied rewrites 100.0%

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

Alternative 2: 88.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \left(-1 + y\right) \cdot \left(1 - x\right)\\ t_1 := \mathsf{fma}\left(y, x, -y\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2000000000000:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (* (+ -1.0 y) (- 1.0 x)))) (t_1 (fma y x (- y))))
   (if (<= t_0 0.0) t_1 (if (<= t_0 2000000000000.0) (- 1.0 y) t_1))))

C

double code(double x, double y) {
	double t_0 = x - ((-1.0 + y) * (1.0 - x));
	double t_1 = fma(y, x, -y);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2000000000000.0) {
		tmp = 1.0 - y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x - Float64(Float64(-1.0 + y) * Float64(1.0 - x)))
	t_1 = fma(y, x, Float64(-y))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2000000000000.0)
		tmp = Float64(1.0 - y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(N[(-1.0 + y), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * x + (-y)), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 2000000000000.0], N[(1.0 - y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 1 binary64) y))) < 0.0 or 2e12 < (+.f64 x (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 1 binary64) y)))

   1. Initial program 67.6%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   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 84.8

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

   5. Applied rewrites 84.8%

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

   7. Applied rewrites 84.8%

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

   if 0.0 < (+.f64 x (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 1 binary64) y))) < 2e12

   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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \left(-1 + y\right) \cdot \left(1 - x\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(y, x, -y\right)\\ \mathbf{elif}\;x - \left(-1 + y\right) \cdot \left(1 - x\right) \leq 2000000000000:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \left(-1 + y\right) \cdot \left(1 - x\right)\\ t_1 := \left(x - 1\right) \cdot y\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2000000000000:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (* (+ -1.0 y) (- 1.0 x)))) (t_1 (* (- x 1.0) y)))
   (if (<= t_0 0.0) t_1 (if (<= t_0 2000000000000.0) (- 1.0 y) t_1))))

C

double code(double x, double y) {
	double t_0 = x - ((-1.0 + y) * (1.0 - x));
	double t_1 = (x - 1.0) * y;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2000000000000.0) {
		tmp = 1.0 - y;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = x - (((-1.0d0) + y) * (1.0d0 - x))
    t_1 = (x - 1.0d0) * y
    if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 2000000000000.0d0) then
        tmp = 1.0d0 - y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x - ((-1.0 + y) * (1.0 - x));
	double t_1 = (x - 1.0) * y;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2000000000000.0) {
		tmp = 1.0 - y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x - ((-1.0 + y) * (1.0 - x))
	t_1 = (x - 1.0) * y
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 2000000000000.0:
		tmp = 1.0 - y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x - Float64(Float64(-1.0 + y) * Float64(1.0 - x)))
	t_1 = Float64(Float64(x - 1.0) * y)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2000000000000.0)
		tmp = Float64(1.0 - y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x - ((-1.0 + y) * (1.0 - x));
	t_1 = (x - 1.0) * y;
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2000000000000.0)
		tmp = 1.0 - y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(N[(-1.0 + y), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - 1.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 2000000000000.0], N[(1.0 - y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 1 binary64) y))) < 0.0 or 2e12 < (+.f64 x (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 1 binary64) y)))

   1. Initial program 67.6%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   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 84.8

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

   5. Applied rewrites 84.8%

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

   if 0.0 < (+.f64 x (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 1 binary64) y))) < 2e12

   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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \left(-1 + y\right) \cdot \left(1 - x\right) \leq 0:\\ \;\;\;\;\left(x - 1\right) \cdot y\\ \mathbf{elif}\;x - \left(-1 + y\right) \cdot \left(1 - x\right) \leq 2000000000000:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;\left(x - 1\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+29}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+21}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.7e+29) (* y x) (if (<= x 3e+21) (- 1.0 y) (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.7e+29) {
		tmp = y * x;
	} else if (x <= 3e+21) {
		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 <= (-4.7d+29)) then
        tmp = y * x
    else if (x <= 3d+21) 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 <= -4.7e+29) {
		tmp = y * x;
	} else if (x <= 3e+21) {
		tmp = 1.0 - y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.7e+29:
		tmp = y * x
	elif x <= 3e+21:
		tmp = 1.0 - y
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.7e+29)
		tmp = Float64(y * x);
	elseif (x <= 3e+21)
		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 <= -4.7e+29)
		tmp = y * x;
	elseif (x <= 3e+21)
		tmp = 1.0 - y;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.7e+29], N[(y * x), $MachinePrecision], If[LessEqual[x, 3e+21], N[(1.0 - y), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.7000000000000002e29 or 3e21 < x

   1. Initial program 48.4%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   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. *-commutative N/A

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

      9. lower-*.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if -4.7000000000000002e29 < x < 3e21

   1. Initial program 97.3%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 95.6

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

   5. Applied rewrites 95.6%

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

Alternative 5: 61.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = -y
	elif y <= 1.0:
		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 <= 1.0)
		tmp = 1.0;
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   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 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 55.3%

      \[\leadsto -y \]

   if -1 < y < 1

   1. Initial program 53.3%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   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. *-commutative N/A

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

      9. lower-*.f64 30.2

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

   5. Applied rewrites 30.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 70.7%

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

Alternative 6: 62.3% 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 76.1%

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

3. Taylor expanded in x around 0

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

   1. lower--.f64 64.1

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

5. Applied rewrites 64.1%

   \[\leadsto \color{blue}{1 - y} \]
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 76.1%

   \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
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. *-commutative N/A

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

   9. lower-*.f64 37.2

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

5. Applied rewrites 37.2%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 37.4%

   \[\leadsto \color{blue}{1} \]
9. 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 2024298 
(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))))