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

Percentage Accurate: 77.3% → 100.0%

Time: 6.3s

Alternatives: 6

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: 77.3% 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 74.1%

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

   1. +-commutative 74.1%

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

   2. remove-double-neg 74.1%

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

   3. unsub-neg 74.1%

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

   4. sub-neg 74.1%

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

   5. +-commutative 74.1%

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

   6. distribute-rgt-in 74.1%

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

   7. *-lft-identity 74.1%

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

   8. associate-+r- 74.1%

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

   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. Add Preprocessing

Alternative 2: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -250000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -250000.0) (not (<= x 1.0))) (+ 1.0 (* y x)) (- 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -250000.0) || !(x <= 1.0)) {
		tmp = 1.0 + (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 <= (-250000.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = 1.0d0 + (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 <= -250000.0) || !(x <= 1.0)) {
		tmp = 1.0 + (y * x);
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -250000.0) or not (x <= 1.0):
		tmp = 1.0 + (y * x)
	else:
		tmp = 1.0 - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -250000.0) || !(x <= 1.0))
		tmp = Float64(1.0 + 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 <= -250000.0) || ~((x <= 1.0)))
		tmp = 1.0 + (y * x);
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -250000.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5e5 or 1 < x

   1. Initial program 48.6%

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

      1. +-commutative 48.6%

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

      2. remove-double-neg 48.6%

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

      3. unsub-neg 48.6%

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

      4. sub-neg 48.6%

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

      5. +-commutative 48.6%

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

      6. distribute-rgt-in 48.6%

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

      7. *-lft-identity 48.6%

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

      8. associate-+r- 48.6%

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

      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 99.8%

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

      1. *-commutative 99.8%

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

   7. Simplified 99.8%

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

   if -2.5e5 < x < 1

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. remove-double-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate-+r- 100.0%

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

      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 99.3%

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

      1. neg-mul-1 99.3%

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

      2. unsub-neg 99.3%

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

   7. Simplified 99.3%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -250000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1360000\right):\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1360000.0))) (- y) 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1360000.0)) {
		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 ((y <= (-1.0d0)) .or. (.not. (y <= 1360000.0d0))) 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 ((y <= -1.0) || !(y <= 1360000.0)) {
		tmp = -y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 1360000.0):
		tmp = -y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1360000.0))
		tmp = Float64(-y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 1360000.0)))
		tmp = -y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1360000.0]], $MachinePrecision]], (-y), 1.0]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.36e6 < y

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. remove-double-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate-+r- 100.0%

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

      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 54.0%

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

      1. neg-mul-1 54.0%

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

      2. unsub-neg 54.0%

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

   7. Simplified 54.0%

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

   8. Taylor expanded in y around inf 53.7%

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

      1. neg-mul-1 53.7%

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

   10. Simplified 53.7%

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

   if -1 < y < 1.36e6

   1. Initial program 53.0%

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

      1. +-commutative 53.0%

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

      2. remove-double-neg 53.0%

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

      3. unsub-neg 53.0%

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

      4. sub-neg 53.0%

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

      5. +-commutative 53.0%

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

      6. distribute-rgt-in 53.0%

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

      7. *-lft-identity 53.0%

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

      8. associate-+r- 53.0%

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

      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 68.8%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1360000\right):\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.8% accurate, 3.0× 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 74.1%

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

   1. +-commutative 74.1%

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

   2. remove-double-neg 74.1%

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

   3. unsub-neg 74.1%

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

   4. sub-neg 74.1%

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

   5. +-commutative 74.1%

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

   6. distribute-rgt-in 74.1%

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

   7. *-lft-identity 74.1%

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

   8. associate-+r- 74.1%

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

   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 0 62.3%

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

   1. neg-mul-1 62.3%

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

   2. unsub-neg 62.3%

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

7. Simplified 62.3%

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

Alternative 5: 38.4% 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 74.1%

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

   1. +-commutative 74.1%

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

   2. remove-double-neg 74.1%

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

   3. unsub-neg 74.1%

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

   4. sub-neg 74.1%

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

   5. +-commutative 74.1%

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

   6. distribute-rgt-in 74.1%

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

   7. *-lft-identity 74.1%

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

   8. associate-+r- 74.1%

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

   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 y around 0 39.0%

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

Alternative 6: 2.5% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.0)

C

double code(double x, double y) {
	return 0.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.0d0
end function

Java

public static double code(double x, double y) {
	return 0.0;
}

Python

def code(x, y):
	return 0.0

Julia

function code(x, y)
	return 0.0
end

MATLAB

function tmp = code(x, y)
	tmp = 0.0;
end

Wolfram

code[x_, y_] := 0.0

Derivation

1. Initial program 74.1%

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

   1. +-commutative 74.1%

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

   2. remove-double-neg 74.1%

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

   3. unsub-neg 74.1%

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

   4. sub-neg 74.1%

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

   5. +-commutative 74.1%

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

   6. distribute-rgt-in 74.1%

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

   7. *-lft-identity 74.1%

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

   8. associate-+r- 74.1%

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

   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 0 62.3%

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

   1. neg-mul-1 62.3%

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

   2. unsub-neg 62.3%

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

7. Simplified 62.3%

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

8. Taylor expanded in y around inf 25.9%

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

   1. neg-mul-1 25.9%

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

10. Simplified 25.9%

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

    1. add-sqr-sqrt 10.9%

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

    2. sqrt-unprod 12.6%

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

    3. sqr-neg 12.6%

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

    4. sqrt-unprod 1.7%

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

    5. add-log-exp 3.5%

       \[\leadsto \color{blue}{\log \left(e^{\sqrt{y} \cdot \sqrt{y}}\right)} \]

    6. add-sqr-sqrt 7.6%

       \[\leadsto \log \left(e^{\color{blue}{y}}\right) \]

    7. add-sqr-sqrt 7.6%

       \[\leadsto \log \color{blue}{\left(\sqrt{e^{y}} \cdot \sqrt{e^{y}}\right)} \]

    8. sqrt-unprod 7.6%

       \[\leadsto \log \color{blue}{\left(\sqrt{e^{y} \cdot e^{y}}\right)} \]

    9. add-sqr-sqrt 3.5%

       \[\leadsto \log \left(\sqrt{e^{y} \cdot e^{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}}\right) \]

    10. sqrt-unprod 4.2%

        \[\leadsto \log \left(\sqrt{e^{y} \cdot e^{\color{blue}{\sqrt{y \cdot y}}}}\right) \]

    11. sqr-neg 4.2%

        \[\leadsto \log \left(\sqrt{e^{y} \cdot e^{\sqrt{\color{blue}{\left(-y\right) \cdot \left(-y\right)}}}}\right) \]

    12. sqrt-unprod 0.7%

        \[\leadsto \log \left(\sqrt{e^{y} \cdot e^{\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}}}\right) \]

    13. add-sqr-sqrt 1.6%

        \[\leadsto \log \left(\sqrt{e^{y} \cdot e^{\color{blue}{-y}}}\right) \]

    14. exp-neg 1.6%

        \[\leadsto \log \left(\sqrt{e^{y} \cdot \color{blue}{\frac{1}{e^{y}}}}\right) \]

    15. rgt-mult-inverse 2.5%

        \[\leadsto \log \left(\sqrt{\color{blue}{1}}\right) \]

    16. metadata-eval 2.5%

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

    17. metadata-eval 2.5%

        \[\leadsto \color{blue}{0} \]

12. Applied egg-rr 2.5%

    \[\leadsto \color{blue}{0} \]
13. 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 2024185 
(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))))