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

Percentage Accurate: 78.5% → 100.0%

Time: 10.3s

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: 78.5% 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

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

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 Float64(fma(y, x, 1.0) - y)
end

Wolfram

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

Derivation

1. Initial program 79.6%

   \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
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. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. associate--r- N/A

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

   7. metadata-eval N/A

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

   8. +-lft-identity N/A

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

   9. lower-fma.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 2: 88.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(1 - x\right) \cdot \left(1 - y\right)\\ t_1 := \mathsf{fma}\left(y, x, -y\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (* (- 1.0 x) (- 1.0 y)))) (t_1 (fma y x (- y))))
   (if (<= t_0 0.0) t_1 (if (<= t_0 2.0) (- 1.0 y) t_1))))

C

double code(double x, double y) {
	double t_0 = x + ((1.0 - x) * (1.0 - y));
	double t_1 = fma(y, x, -y);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x + Float64(Float64(1.0 - x) * Float64(1.0 - y)))
	t_1 = fma(y, x, Float64(-y))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		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 - x), $MachinePrecision] * N[(1.0 - y), $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, 2.0], 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 2 < (+.f64 x (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 1 binary64) y)))

   1. Initial program 72.9%

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

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

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

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

      10. distribute-lft-neg-in N/A

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

      11. remove-double-neg N/A

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

      12. lower-*.f64 88.4

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

   5. Applied rewrites 88.4%

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

   7. Applied rewrites 88.4%

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

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

   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 99.8

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

   5. Applied rewrites 99.8%

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

Alternative 3: 88.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(1 - x\right) \cdot \left(1 - y\right)\\ t_1 := x \cdot y - y\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (* (- 1.0 x) (- 1.0 y)))) (t_1 (- (* x y) y)))
   (if (<= t_0 0.0) t_1 (if (<= t_0 2.0) (- 1.0 y) t_1))))

C

double code(double x, double y) {
	double t_0 = x + ((1.0 - x) * (1.0 - y));
	double t_1 = (x * y) - y;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		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 - x) * (1.0d0 - y))
    t_1 = (x * y) - y
    if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 2.0d0) 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 - x) * (1.0 - y));
	double t_1 = (x * y) - y;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + ((1.0 - x) * (1.0 - y))
	t_1 = (x * y) - y
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 2.0:
		tmp = 1.0 - y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(Float64(1.0 - x) * Float64(1.0 - y)))
	t_1 = Float64(Float64(x * y) - y)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		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 - x) * (1.0 - y));
	t_1 = (x * y) - y;
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		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 - x), $MachinePrecision] * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 2.0], 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 2 < (+.f64 x (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 1 binary64) y)))

   1. Initial program 72.9%

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

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

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

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

      10. distribute-lft-neg-in N/A

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

      11. remove-double-neg N/A

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

      12. lower-*.f64 88.4

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

   5. Applied rewrites 88.4%

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

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

   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 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(1 - x\right) \cdot \left(1 - y\right) \leq 0:\\ \;\;\;\;x \cdot y - y\\ \mathbf{elif}\;x + \left(1 - x\right) \cdot \left(1 - y\right) \leq 2:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(1 - x\right) \cdot \left(1 - y\right)\\ \mathbf{if}\;t\_0 \leq -1000000000:\\ \;\;\;\;-y\\ \mathbf{elif}\;t\_0 \leq 100000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (* (- 1.0 x) (- 1.0 y)))))
   (if (<= t_0 -1000000000.0) (- y) (if (<= t_0 100000000.0) 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 <= -1000000000.0) {
		tmp = -y;
	} else if (t_0 <= 100000000.0) {
		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) :: t_0
    real(8) :: tmp
    t_0 = x + ((1.0d0 - x) * (1.0d0 - y))
    if (t_0 <= (-1000000000.0d0)) then
        tmp = -y
    else if (t_0 <= 100000000.0d0) then
        tmp = 1.0d0
    else
        tmp = -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 <= -1000000000.0) {
		tmp = -y;
	} else if (t_0 <= 100000000.0) {
		tmp = 1.0;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + ((1.0 - x) * (1.0 - y))
	tmp = 0
	if t_0 <= -1000000000.0:
		tmp = -y
	elif t_0 <= 100000000.0:
		tmp = 1.0
	else:
		tmp = -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 <= -1000000000.0)
		tmp = Float64(-y);
	elseif (t_0 <= 100000000.0)
		tmp = 1.0;
	else
		tmp = Float64(-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 <= -1000000000.0)
		tmp = -y;
	elseif (t_0 <= 100000000.0)
		tmp = 1.0;
	else
		tmp = -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[LessEqual[t$95$0, -1000000000.0], (-y), If[LessEqual[t$95$0, 100000000.0], 1.0, (-y)]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 1 binary64) y))) < -1e9 or 1e8 < (+.f64 x (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 1 binary64) y)))

   1. Initial program 99.8%

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

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

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

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

      10. distribute-lft-neg-in N/A

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

      11. remove-double-neg N/A

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

      12. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 48.0%

      \[\leadsto -y \]

   if -1e9 < (+.f64 x (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 1 binary64) y))) < 1e8

   1. Initial program 56.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 69.0%

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

Alternative 5: 87.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -280000:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 2700000000:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -280000.0) (* x y) (if (<= x 2700000000.0) (- 1.0 y) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -280000.0) {
		tmp = x * y;
	} else if (x <= 2700000000.0) {
		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 <= (-280000.0d0)) then
        tmp = x * y
    else if (x <= 2700000000.0d0) 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 <= -280000.0) {
		tmp = x * y;
	} else if (x <= 2700000000.0) {
		tmp = 1.0 - y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -280000.0:
		tmp = x * y
	elif x <= 2700000000.0:
		tmp = 1.0 - y
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -280000.0)
		tmp = Float64(x * y);
	elseif (x <= 2700000000.0)
		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 <= -280000.0)
		tmp = x * y;
	elseif (x <= 2700000000.0)
		tmp = 1.0 - y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -280000.0], N[(x * y), $MachinePrecision], If[LessEqual[x, 2700000000.0], N[(1.0 - y), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.8e5 or 2.7e9 < x

   1. Initial program 59.5%

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate--r- N/A

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

      5. metadata-eval N/A

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

      6. +-lft-identity N/A

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

      7. lower-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

   if -2.8e5 < x < 2.7e9

   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 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -280000:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 2700000000:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.2% 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 79.6%

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

3. Taylor expanded in x around 0

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

   1. lower--.f64 58.7

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

5. Applied rewrites 58.7%

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

Alternative 7: 38.5% 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 79.6%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 33.3%

   \[\leadsto \color{blue}{1} \]
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 2024220 
(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))))