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

Percentage Accurate: 76.8% → 100.0%

Time: 5.2s

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: 76.8% 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 83.2%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 100.0%

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

Alternative 2: 86.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(1 - x\right) \cdot \left(1 - y\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+134} \lor \neg \left(t\_0 \leq 10000000\right):\\ \;\;\;\;\left(-1 + x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (* (- 1.0 x) (- 1.0 y)))))
   (if (or (<= t_0 -5e+134) (not (<= t_0 10000000.0)))
     (* (+ -1.0 x) y)
     (- 1.0 y))))

C

double code(double x, double y) {
	double t_0 = x + ((1.0 - x) * (1.0 - y));
	double tmp;
	if ((t_0 <= -5e+134) || !(t_0 <= 10000000.0)) {
		tmp = (-1.0 + x) * y;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = x + ((1.0d0 - x) * (1.0d0 - y))
    if ((t_0 <= (-5d+134)) .or. (.not. (t_0 <= 10000000.0d0))) then
        tmp = ((-1.0d0) + x) * y
    else
        tmp = 1.0d0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + ((1.0 - x) * (1.0 - y));
	double tmp;
	if ((t_0 <= -5e+134) || !(t_0 <= 10000000.0)) {
		tmp = (-1.0 + x) * y;
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + ((1.0 - x) * (1.0 - y))
	tmp = 0
	if (t_0 <= -5e+134) or not (t_0 <= 10000000.0):
		tmp = (-1.0 + x) * y
	else:
		tmp = 1.0 - y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(Float64(1.0 - x) * Float64(1.0 - y)))
	tmp = 0.0
	if ((t_0 <= -5e+134) || !(t_0 <= 10000000.0))
		tmp = Float64(Float64(-1.0 + x) * y);
	else
		tmp = Float64(1.0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + ((1.0 - x) * (1.0 - y));
	tmp = 0.0;
	if ((t_0 <= -5e+134) || ~((t_0 <= 10000000.0)))
		tmp = (-1.0 + x) * y;
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(N[(1.0 - x), $MachinePrecision] * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e+134], N[Not[LessEqual[t$95$0, 10000000.0]], $MachinePrecision]], N[(N[(-1.0 + x), $MachinePrecision] * y), $MachinePrecision], N[(1.0 - y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 1 binary64) y))) < -4.99999999999999981e134 or 1e7 < (+.f64 x (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 1 binary64) y)))

   1. Initial program 99.9%

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

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

      6. metadata-eval N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -4.99999999999999981e134 < (+.f64 x (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 1 binary64) y))) < 1e7

   1. Initial program 68.4%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 86.1

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

   5. Applied rewrites 86.1%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(1 - x\right) \cdot \left(1 - y\right) \leq -5 \cdot 10^{+134} \lor \neg \left(x + \left(1 - x\right) \cdot \left(1 - y\right) \leq 10000000\right):\\ \;\;\;\;\left(-1 + x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(1 - x\right) \cdot \left(1 - y\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -y\right)\\ \mathbf{elif}\;t\_0 \leq 10000000:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (* (- 1.0 x) (- 1.0 y)))))
   (if (<= t_0 -5e+134)
     (fma y x (- y))
     (if (<= t_0 10000000.0) (- 1.0 y) (* (+ -1.0 x) y)))))

C

double code(double x, double y) {
	double t_0 = x + ((1.0 - x) * (1.0 - y));
	double tmp;
	if (t_0 <= -5e+134) {
		tmp = fma(y, x, -y);
	} else if (t_0 <= 10000000.0) {
		tmp = 1.0 - y;
	} else {
		tmp = (-1.0 + x) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x + Float64(Float64(1.0 - x) * Float64(1.0 - y)))
	tmp = 0.0
	if (t_0 <= -5e+134)
		tmp = fma(y, x, Float64(-y));
	elseif (t_0 <= 10000000.0)
		tmp = Float64(1.0 - y);
	else
		tmp = Float64(Float64(-1.0 + x) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(N[(1.0 - x), $MachinePrecision] * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+134], N[(y * x + (-y)), $MachinePrecision], If[LessEqual[t$95$0, 10000000.0], N[(1.0 - y), $MachinePrecision], N[(N[(-1.0 + x), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 1 binary64) y))) < -4.99999999999999981e134

   1. Initial program 100.0%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   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. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--r- N/A

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

      5. metadata-eval N/A

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

      6. +-lft-identity N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 59.5

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

   5. Applied rewrites 59.5%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. distribute-lft-out-- N/A

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

      3. *-rgt-identity N/A

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

      4. mul-1-neg N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -4.99999999999999981e134 < (+.f64 x (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 1 binary64) y))) < 1e7

   1. Initial program 68.4%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 86.1

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

   5. Applied rewrites 86.1%

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

   if 1e7 < (+.f64 x (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 1 binary64) y)))

   1. Initial program 99.7%

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

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

      6. metadata-eval N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(-1 + x\right) \cdot y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 61.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - y \leq -0.2 \lor \neg \left(1 - y \leq 2\right):\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (- 1.0 y) -0.2) (not (<= (- 1.0 y) 2.0))) (- y) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 - y) <= -0.2) || !((1.0 - y) <= 2.0)) {
		tmp = -y;
	} else {
		tmp = 1.0;
	}
	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) <= (-0.2d0)) .or. (.not. ((1.0d0 - y) <= 2.0d0))) then
        tmp = -y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((1.0 - y) <= -0.2) || !((1.0 - y) <= 2.0)) {
		tmp = -y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((1.0 - y) <= -0.2) or not ((1.0 - y) <= 2.0):
		tmp = -y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(1.0 - y) <= -0.2) || !(Float64(1.0 - y) <= 2.0))
		tmp = Float64(-y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((1.0 - y) <= -0.2) || ~(((1.0 - y) <= 2.0)))
		tmp = -y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(1.0 - y), $MachinePrecision], -0.2], N[Not[LessEqual[N[(1.0 - y), $MachinePrecision], 2.0]], $MachinePrecision]], (-y), 1.0]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -0.20000000000000001 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 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. *-lft-identity N/A

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

      6. metadata-eval N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower-+.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 51.6%

      \[\leadsto -y \]

   if -0.20000000000000001 < (-.f64 #s(literal 1 binary64) y) < 2

   1. Initial program 65.2%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   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. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--r- N/A

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

      5. metadata-eval N/A

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

      6. +-lft-identity N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 18.0

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

   5. Applied rewrites 18.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 83.3%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -0.2 \lor \neg \left(1 - y \leq 2\right):\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+51} \lor \neg \left(x \leq 2 \cdot 10^{+22}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -9.2e+51) (not (<= x 2e+22))) (* y x) (- 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -9.2e+51) || !(x <= 2e+22)) {
		tmp = y * x;
	} 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 <= (-9.2d+51)) .or. (.not. (x <= 2d+22))) then
        tmp = y * x
    else
        tmp = 1.0d0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -9.2e+51) || !(x <= 2e+22)) {
		tmp = y * x;
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -9.2e+51) or not (x <= 2e+22):
		tmp = y * x
	else:
		tmp = 1.0 - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -9.2e+51) || !(x <= 2e+22))
		tmp = Float64(y * x);
	else
		tmp = Float64(1.0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -9.2e+51) || ~((x <= 2e+22)))
		tmp = y * x;
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -9.2e+51], N[Not[LessEqual[x, 2e+22]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(1.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.2000000000000002e51 or 2e22 < x

   1. Initial program 62.7%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   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. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--r- N/A

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

      5. metadata-eval N/A

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

      6. +-lft-identity N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 79.7

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

   5. Applied rewrites 79.7%

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

   if -9.2000000000000002e51 < x < 2e22

   1. Initial program 96.8%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+51} \lor \neg \left(x \leq 2 \cdot 10^{+22}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.7% 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 83.2%

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

3. Taylor expanded in x around 0

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

   1. lower--.f64 67.9

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

5. Applied rewrites 67.9%

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

Alternative 7: 38.1% 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 83.2%

   \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
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. fp-cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. associate--r- N/A

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

   5. metadata-eval N/A

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

   6. +-lft-identity N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 34.0

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

5. Applied rewrites 34.0%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 41.5%

   \[\leadsto \color{blue}{1} \]
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 2024329 
(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))))