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

Percentage Accurate: 77.6% → 100.0%

Time: 6.1s

Alternatives: 5

Speedup: 1.3×

Specification

Math

\[\begin{array}{l} \\ x + \left(1 - x\right) \cdot \left(1 - y\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.6% 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, 1.3× speedup

Math

\[\begin{array}{l} \\ 1 + y \cdot \left(x + -1\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

   1. +-commutative 75.9%

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

   2. remove-double-neg 75.9%

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

   3. unsub-neg 75.9%

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

   4. sub-neg 75.9%

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

   5. +-commutative 75.9%

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

   6. distribute-rgt-in 75.9%

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

   7. *-lft-identity 75.9%

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

   8. associate-+r- 75.9%

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

   9. associate--l- 100.0%

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

   10. sub-neg 100.0%

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

   11. +-inverses 100.0%

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

   12. --rgt-identity 100.0%

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

   13. +-commutative 100.0%

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

   14. distribute-lft-neg-out 100.0%

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

   15. distribute-rgt-neg-in 100.0%

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

   16. neg-sub0 100.0%

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

   17. associate--r- 100.0%

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

   18. metadata-eval 100.0%

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

   19. +-commutative 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 86.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+48}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-7}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.4e+48) (* y x) (if (<= x 3.8e-7) (- 1.0 y) (* y (+ x -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.4e+48) {
		tmp = y * x;
	} else if (x <= 3.8e-7) {
		tmp = 1.0 - y;
	} else {
		tmp = y * (x + -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 (x <= (-4.4d+48)) then
        tmp = y * x
    else if (x <= 3.8d-7) then
        tmp = 1.0d0 - y
    else
        tmp = y * (x + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.4e+48) {
		tmp = y * x;
	} else if (x <= 3.8e-7) {
		tmp = 1.0 - y;
	} else {
		tmp = y * (x + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.4e+48:
		tmp = y * x
	elif x <= 3.8e-7:
		tmp = 1.0 - y
	else:
		tmp = y * (x + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.4e+48)
		tmp = Float64(y * x);
	elseif (x <= 3.8e-7)
		tmp = Float64(1.0 - y);
	else
		tmp = Float64(y * Float64(x + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.4e+48)
		tmp = y * x;
	elseif (x <= 3.8e-7)
		tmp = 1.0 - y;
	else
		tmp = y * (x + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.4e+48], N[(y * x), $MachinePrecision], If[LessEqual[x, 3.8e-7], N[(1.0 - y), $MachinePrecision], N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.3999999999999999e48

   1. Initial program 50.6%

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

      1. +-commutative 50.6%

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

      2. remove-double-neg 50.6%

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

      3. unsub-neg 50.6%

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

      4. sub-neg 50.6%

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

      5. +-commutative 50.6%

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

      6. distribute-rgt-in 50.6%

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

      7. *-lft-identity 50.6%

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

      8. associate-+r- 50.6%

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

      9. associate--l- 100.0%

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

      10. sub-neg 100.0%

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

      11. +-inverses 100.0%

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

      12. --rgt-identity 100.0%

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

      13. +-commutative 100.0%

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

      14. distribute-lft-neg-out 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. neg-sub0 100.0%

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

      17. associate--r- 100.0%

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

      18. metadata-eval 100.0%

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

      19. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 76.5%

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

      1. *-commutative 76.5%

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

   7. Simplified 76.5%

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

   if -4.3999999999999999e48 < x < 3.80000000000000015e-7

   1. Initial program 95.7%

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

      1. +-commutative 95.7%

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

      2. remove-double-neg 95.7%

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

      3. unsub-neg 95.7%

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

      4. sub-neg 95.7%

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

      5. +-commutative 95.7%

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

      6. distribute-rgt-in 95.7%

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

      7. *-lft-identity 95.7%

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

      8. associate-+r- 95.7%

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

      9. associate--l- 100.0%

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

      10. sub-neg 100.0%

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

      11. +-inverses 100.0%

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

      12. --rgt-identity 100.0%

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

      13. +-commutative 100.0%

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

      14. distribute-lft-neg-out 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. neg-sub0 100.0%

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

      17. associate--r- 100.0%

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

      18. metadata-eval 100.0%

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

      19. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 96.6%

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

      1. mul-1-neg 96.6%

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

      2. unsub-neg 96.6%

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

   7. Simplified 96.6%

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

   if 3.80000000000000015e-7 < x

   1. Initial program 57.6%

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

      1. +-commutative 57.6%

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

      2. remove-double-neg 57.6%

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

      3. unsub-neg 57.6%

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

      4. sub-neg 57.6%

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

      5. +-commutative 57.6%

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

      6. distribute-rgt-in 57.7%

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

      7. *-lft-identity 57.7%

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

      8. associate-+r- 57.7%

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

      9. associate--l- 100.0%

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

      10. sub-neg 100.0%

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

      11. +-inverses 100.0%

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

      12. --rgt-identity 100.0%

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

      13. +-commutative 100.0%

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

      14. distribute-lft-neg-out 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. neg-sub0 100.0%

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

      17. associate--r- 100.0%

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

      18. metadata-eval 100.0%

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

      19. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 83.1%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+48}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-7}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+48} \lor \neg \left(x \leq 85000\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.4e+48) (not (<= x 85000.0))) (* y x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.4e+48) || !(x <= 85000.0)) {
		tmp = y * x;
	} 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 ((x <= (-2.4d+48)) .or. (.not. (x <= 85000.0d0))) then
        tmp = y * x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2.4e+48) || !(x <= 85000.0)) {
		tmp = y * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.4e+48) or not (x <= 85000.0):
		tmp = y * x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.4e+48) || !(x <= 85000.0))
		tmp = Float64(y * x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.4e+48) || ~((x <= 85000.0)))
		tmp = y * x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.4e+48], N[Not[LessEqual[x, 85000.0]], $MachinePrecision]], N[(y * x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -2.4000000000000001e48 or 85000 < x

   1. Initial program 53.1%

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

      1. +-commutative 53.1%

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

      2. remove-double-neg 53.1%

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

      3. unsub-neg 53.1%

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

      4. sub-neg 53.1%

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

      5. +-commutative 53.1%

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

      6. distribute-rgt-in 53.1%

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

      7. *-lft-identity 53.1%

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

      8. associate-+r- 53.1%

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

      9. associate--l- 100.0%

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

      10. sub-neg 100.0%

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

      11. +-inverses 100.0%

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

      12. --rgt-identity 100.0%

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

      13. +-commutative 100.0%

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

      14. distribute-lft-neg-out 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. neg-sub0 100.0%

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

      17. associate--r- 100.0%

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

      18. metadata-eval 100.0%

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

      19. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 79.1%

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

      1. *-commutative 79.1%

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

   7. Simplified 79.1%

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

   if -2.4000000000000001e48 < x < 85000

   1. Initial program 95.7%

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

      1. +-commutative 95.7%

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

      2. remove-double-neg 95.7%

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

      3. unsub-neg 95.7%

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

      4. sub-neg 95.7%

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

      5. +-commutative 95.7%

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

      6. distribute-rgt-in 95.7%

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

      7. *-lft-identity 95.7%

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

      8. associate-+r- 95.7%

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

      9. associate--l- 100.0%

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

      10. sub-neg 100.0%

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

      11. +-inverses 100.0%

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

      12. --rgt-identity 100.0%

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

      13. +-commutative 100.0%

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

      14. distribute-lft-neg-out 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. neg-sub0 100.0%

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

      17. associate--r- 100.0%

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

      18. metadata-eval 100.0%

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

      19. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 48.3%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+48} \lor \neg \left(x \leq 85000\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+48} \lor \neg \left(x \leq 85000\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.4e+48) (not (<= x 85000.0))) (* y x) (- 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.4e+48) || !(x <= 85000.0)) {
		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 <= (-2.4d+48)) .or. (.not. (x <= 85000.0d0))) 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 <= -2.4e+48) || !(x <= 85000.0)) {
		tmp = y * x;
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.4e+48) or not (x <= 85000.0):
		tmp = y * x
	else:
		tmp = 1.0 - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.4e+48) || !(x <= 85000.0))
		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 <= -2.4e+48) || ~((x <= 85000.0)))
		tmp = y * x;
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.4e+48], N[Not[LessEqual[x, 85000.0]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(1.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4000000000000001e48 or 85000 < x

   1. Initial program 53.1%

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

      1. +-commutative 53.1%

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

      2. remove-double-neg 53.1%

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

      3. unsub-neg 53.1%

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

      4. sub-neg 53.1%

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

      5. +-commutative 53.1%

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

      6. distribute-rgt-in 53.1%

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

      7. *-lft-identity 53.1%

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

      8. associate-+r- 53.1%

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

      9. associate--l- 100.0%

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

      10. sub-neg 100.0%

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

      11. +-inverses 100.0%

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

      12. --rgt-identity 100.0%

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

      13. +-commutative 100.0%

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

      14. distribute-lft-neg-out 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. neg-sub0 100.0%

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

      17. associate--r- 100.0%

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

      18. metadata-eval 100.0%

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

      19. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 79.1%

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

      1. *-commutative 79.1%

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

   7. Simplified 79.1%

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

   if -2.4000000000000001e48 < x < 85000

   1. Initial program 95.7%

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

      1. +-commutative 95.7%

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

      2. remove-double-neg 95.7%

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

      3. unsub-neg 95.7%

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

      4. sub-neg 95.7%

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

      5. +-commutative 95.7%

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

      6. distribute-rgt-in 95.7%

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

      7. *-lft-identity 95.7%

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

      8. associate-+r- 95.7%

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

      9. associate--l- 100.0%

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

      10. sub-neg 100.0%

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

      11. +-inverses 100.0%

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

      12. --rgt-identity 100.0%

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

      13. +-commutative 100.0%

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

      14. distribute-lft-neg-out 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. neg-sub0 100.0%

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

      17. associate--r- 100.0%

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

      18. metadata-eval 100.0%

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

      19. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 95.8%

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

      1. mul-1-neg 95.8%

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

      2. unsub-neg 95.8%

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

   7. Simplified 95.8%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+48} \lor \neg \left(x \leq 85000\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 37.4% accurate, 9.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 75.9%

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

   1. +-commutative 75.9%

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

   2. remove-double-neg 75.9%

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

   3. unsub-neg 75.9%

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

   4. sub-neg 75.9%

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

   5. +-commutative 75.9%

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

   6. distribute-rgt-in 75.9%

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

   7. *-lft-identity 75.9%

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

   8. associate-+r- 75.9%

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

   9. associate--l- 100.0%

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

   10. sub-neg 100.0%

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

   11. +-inverses 100.0%

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

   12. --rgt-identity 100.0%

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

   13. +-commutative 100.0%

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

   14. distribute-lft-neg-out 100.0%

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

   15. distribute-rgt-neg-in 100.0%

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

   16. neg-sub0 100.0%

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

   17. associate--r- 100.0%

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

   18. metadata-eval 100.0%

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

   19. +-commutative 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in y around 0 35.2%

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

6. Final simplification 35.2%

   \[\leadsto 1 \]
7. Add Preprocessing

Developer target: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024018 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.Vectors:renderPlotVectors from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (- (* y x) (- y 1.0))

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