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

Percentage Accurate: 78.2% → 100.0%

Time: 4.3s

Alternatives: 8

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: 78.2% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.9%

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

   1. +-commutative 74.9%

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

   2. sub-neg 74.9%

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

   3. +-commutative 74.9%

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

   4. distribute-rgt-in 75.0%

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

   5. *-lft-identity 75.0%

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

   6. associate-+l+ 88.8%

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

   7. neg-mul-1 88.8%

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

   8. *-commutative 88.8%

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

   9. associate-*l* 88.8%

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

   10. associate-+l- 100.0%

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

   11. +-inverses 100.0%

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

   12. metadata-eval 100.0%

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

   13. fma-def 100.0%

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

   14. mul-1-neg 100.0%

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

   15. neg-sub0 100.0%

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

   16. associate--r- 100.0%

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

   17. metadata-eval 100.0%

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

   18. +-commutative 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 59.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+121}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq -3 \cdot 10^{-108}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-130}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-54}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-38}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.08 \cdot 10^{+113} \lor \neg \left(y \leq 2.7 \cdot 10^{+172}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.3e+121)
   (- y)
   (if (<= y -3e-108)
     (* y x)
     (if (<= y 2e-130)
       1.0
       (if (<= y 5.1e-54)
         (* y x)
         (if (<= y 1.7e-38)
           1.0
           (if (or (<= y 1.08e+113) (not (<= y 2.7e+172))) (* y x) (- y))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.3e+121) {
		tmp = -y;
	} else if (y <= -3e-108) {
		tmp = y * x;
	} else if (y <= 2e-130) {
		tmp = 1.0;
	} else if (y <= 5.1e-54) {
		tmp = y * x;
	} else if (y <= 1.7e-38) {
		tmp = 1.0;
	} else if ((y <= 1.08e+113) || !(y <= 2.7e+172)) {
		tmp = y * x;
	} else {
		tmp = -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.3d+121)) then
        tmp = -y
    else if (y <= (-3d-108)) then
        tmp = y * x
    else if (y <= 2d-130) then
        tmp = 1.0d0
    else if (y <= 5.1d-54) then
        tmp = y * x
    else if (y <= 1.7d-38) then
        tmp = 1.0d0
    else if ((y <= 1.08d+113) .or. (.not. (y <= 2.7d+172))) then
        tmp = y * x
    else
        tmp = -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.3e+121) {
		tmp = -y;
	} else if (y <= -3e-108) {
		tmp = y * x;
	} else if (y <= 2e-130) {
		tmp = 1.0;
	} else if (y <= 5.1e-54) {
		tmp = y * x;
	} else if (y <= 1.7e-38) {
		tmp = 1.0;
	} else if ((y <= 1.08e+113) || !(y <= 2.7e+172)) {
		tmp = y * x;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.3e+121:
		tmp = -y
	elif y <= -3e-108:
		tmp = y * x
	elif y <= 2e-130:
		tmp = 1.0
	elif y <= 5.1e-54:
		tmp = y * x
	elif y <= 1.7e-38:
		tmp = 1.0
	elif (y <= 1.08e+113) or not (y <= 2.7e+172):
		tmp = y * x
	else:
		tmp = -y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.3e+121)
		tmp = Float64(-y);
	elseif (y <= -3e-108)
		tmp = Float64(y * x);
	elseif (y <= 2e-130)
		tmp = 1.0;
	elseif (y <= 5.1e-54)
		tmp = Float64(y * x);
	elseif (y <= 1.7e-38)
		tmp = 1.0;
	elseif ((y <= 1.08e+113) || !(y <= 2.7e+172))
		tmp = Float64(y * x);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.3e+121)
		tmp = -y;
	elseif (y <= -3e-108)
		tmp = y * x;
	elseif (y <= 2e-130)
		tmp = 1.0;
	elseif (y <= 5.1e-54)
		tmp = y * x;
	elseif (y <= 1.7e-38)
		tmp = 1.0;
	elseif ((y <= 1.08e+113) || ~((y <= 2.7e+172)))
		tmp = y * x;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.3e+121], (-y), If[LessEqual[y, -3e-108], N[(y * x), $MachinePrecision], If[LessEqual[y, 2e-130], 1.0, If[LessEqual[y, 5.1e-54], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.7e-38], 1.0, If[Or[LessEqual[y, 1.08e+113], N[Not[LessEqual[y, 2.7e+172]], $MachinePrecision]], N[(y * x), $MachinePrecision], (-y)]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2999999999999999e121 or 1.08e113 < y < 2.7e172

   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. sub-neg 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. *-lft-identity 100.0%

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

      5. associate-+l+ 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-lft-neg-out 100.0%

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

      8. sub-neg 100.0%

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

      9. associate--l+ 100.0%

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

      10. associate-+l- 100.0%

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

      11. +-inverses 100.0%

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

      12. metadata-eval 100.0%

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

      13. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 100.0%

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

   5. Taylor expanded in x around 0 68.2%

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

      1. neg-mul-1 68.2%

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

   7. Simplified 68.2%

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

   if -2.2999999999999999e121 < y < -2.99999999999999993e-108 or 2.0000000000000002e-130 < y < 5.1000000000000001e-54 or 1.7000000000000001e-38 < y < 1.08e113 or 2.7e172 < y

   1. Initial program 75.3%

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

      1. +-commutative 75.3%

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

      2. sub-neg 75.3%

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

      3. distribute-rgt-in 75.3%

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

      4. *-lft-identity 75.3%

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

      5. associate-+l+ 75.3%

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

      6. +-commutative 75.3%

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

      7. distribute-lft-neg-out 75.3%

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

      8. sub-neg 75.3%

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

      9. associate--l+ 97.9%

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

      10. associate-+l- 99.9%

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

      11. +-inverses 99.9%

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

      12. metadata-eval 99.9%

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

      13. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 68.2%

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

      1. *-commutative 68.2%

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

   6. Simplified 68.2%

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

   if -2.99999999999999993e-108 < y < 2.0000000000000002e-130 or 5.1000000000000001e-54 < y < 1.7000000000000001e-38

   1. Initial program 62.4%

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

      1. +-commutative 62.4%

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

      2. sub-neg 62.4%

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

      3. distribute-rgt-in 62.4%

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

      4. *-lft-identity 62.4%

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

      5. associate-+l+ 62.4%

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

      6. +-commutative 62.4%

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

      7. distribute-lft-neg-out 62.4%

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

      8. sub-neg 62.4%

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

      9. associate--l+ 72.3%

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

      10. associate-+l- 100.0%

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

      11. +-inverses 100.0%

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

      12. metadata-eval 100.0%

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

      13. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 88.6%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+121}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq -3 \cdot 10^{-108}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-130}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-54}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-38}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.08 \cdot 10^{+113} \lor \neg \left(y \leq 2.7 \cdot 10^{+172}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 3: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -620000000000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -620000000000.0) (not (<= x 1.0))) (+ 1.0 (* y x)) (- 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -620000000000.0) || !(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 <= (-620000000000.0d0)) .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 <= -620000000000.0) || !(x <= 1.0)) {
		tmp = 1.0 + (y * x);
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -620000000000.0) 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 <= -620000000000.0) || !(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 <= -620000000000.0) || ~((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, -620000000000.0], 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 < -6.2e11 or 1 < x

   1. Initial program 51.8%

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

      1. +-commutative 51.8%

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

      2. sub-neg 51.8%

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

      3. distribute-rgt-in 51.8%

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

      4. *-lft-identity 51.8%

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

      5. associate-+l+ 51.8%

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

      6. +-commutative 51.8%

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

      7. distribute-lft-neg-out 51.8%

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

      8. sub-neg 51.8%

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

      9. associate--l+ 78.4%

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

      10. associate-+l- 100.0%

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

      11. +-inverses 100.0%

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

      12. metadata-eval 100.0%

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

      13. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-lft-neg-out 99.8%

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

      3. *-commutative 99.8%

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

   6. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. cancel-sign-sub 99.8%

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

      3. *-commutative 99.8%

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

      4. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   if -6.2e11 < 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. sub-neg 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. *-lft-identity 100.0%

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

      5. associate-+l+ 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-lft-neg-out 100.0%

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

      8. sub-neg 100.0%

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

      9. associate--l+ 100.0%

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

      10. associate-+l- 100.0%

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

      11. +-inverses 100.0%

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

      12. metadata-eval 100.0%

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

      13. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.1%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -620000000000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]

Alternative 4: 87.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.42 \cdot 10^{+26} \lor \neg \left(x \leq 2.75 \cdot 10^{+20}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.42e+26) (not (<= x 2.75e+20))) (* y x) (- 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.42e+26) || !(x <= 2.75e+20)) {
		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 <= (-1.42d+26)) .or. (.not. (x <= 2.75d+20))) 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 <= -1.42e+26) || !(x <= 2.75e+20)) {
		tmp = y * x;
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.42e+26) or not (x <= 2.75e+20):
		tmp = y * x
	else:
		tmp = 1.0 - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.42e+26) || !(x <= 2.75e+20))
		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 <= -1.42e+26) || ~((x <= 2.75e+20)))
		tmp = y * x;
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.42e+26], N[Not[LessEqual[x, 2.75e+20]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(1.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.42e26 or 2.75e20 < x

   1. Initial program 52.4%

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

      1. +-commutative 52.4%

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

      2. sub-neg 52.4%

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

      3. distribute-rgt-in 52.5%

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

      4. *-lft-identity 52.5%

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

      5. associate-+l+ 52.5%

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

      6. +-commutative 52.5%

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

      7. distribute-lft-neg-out 52.5%

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

      8. sub-neg 52.5%

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

      9. associate--l+ 80.3%

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

      10. associate-+l- 100.0%

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

      11. +-inverses 100.0%

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

      12. metadata-eval 100.0%

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

      13. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 80.3%

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

      1. *-commutative 80.3%

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

   6. Simplified 80.3%

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

   if -1.42e26 < x < 2.75e20

   1. Initial program 97.1%

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

      1. +-commutative 97.1%

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

      2. sub-neg 97.1%

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

      3. distribute-rgt-in 97.1%

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

      4. *-lft-identity 97.1%

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

      5. associate-+l+ 97.1%

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

      6. +-commutative 97.1%

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

      7. distribute-lft-neg-out 97.1%

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

      8. sub-neg 97.1%

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

      9. associate--l+ 97.1%

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

      10. associate-+l- 100.0%

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

      11. +-inverses 100.0%

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

      12. metadata-eval 100.0%

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

      13. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 98.5%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.42 \cdot 10^{+26} \lor \neg \left(x \leq 2.75 \cdot 10^{+20}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]

Alternative 5: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.9%

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

   1. +-commutative 74.9%

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

   2. sub-neg 74.9%

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

   3. distribute-rgt-in 75.0%

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

   4. *-lft-identity 75.0%

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

   5. associate-+l+ 75.0%

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

   6. +-commutative 75.0%

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

   7. distribute-lft-neg-out 75.0%

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

   8. sub-neg 75.0%

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

   9. associate--l+ 88.8%

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

   10. associate-+l- 100.0%

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

   11. +-inverses 100.0%

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

   12. metadata-eval 100.0%

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

   13. *-commutative 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 6: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.9%

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

   1. +-commutative 74.9%

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

   2. sub-neg 74.9%

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

   3. distribute-rgt-in 75.0%

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

   4. *-lft-identity 75.0%

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

   5. associate-+l+ 75.0%

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

   6. +-commutative 75.0%

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

   7. distribute-lft-neg-out 75.0%

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

   8. sub-neg 75.0%

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

   9. associate--l+ 88.8%

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

   10. associate-+l- 100.0%

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

   11. +-inverses 100.0%

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

   12. metadata-eval 100.0%

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

   13. *-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 100.0%

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

   1. *-commutative 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

Alternative 7: 62.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 125\right):\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 125.0))) (- y) 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 125.0)) {
		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 <= 125.0d0))) 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 <= 125.0)) {
		tmp = -y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 125.0):
		tmp = -y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 125.0))
		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 <= 125.0)))
		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, 125.0]], $MachinePrecision]], (-y), 1.0]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 125 < 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. sub-neg 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. *-lft-identity 100.0%

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

      5. associate-+l+ 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-lft-neg-out 100.0%

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

      8. sub-neg 100.0%

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

      9. associate--l+ 100.0%

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

      10. associate-+l- 100.0%

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

      11. +-inverses 100.0%

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

      12. metadata-eval 100.0%

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

      13. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 98.8%

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

   5. Taylor expanded in x around 0 48.0%

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

      1. neg-mul-1 48.0%

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

   7. Simplified 48.0%

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

   if -1 < y < 125

   1. Initial program 53.5%

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

      1. +-commutative 53.5%

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

      2. sub-neg 53.5%

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

      3. distribute-rgt-in 53.6%

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

      4. *-lft-identity 53.6%

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

      5. associate-+l+ 53.6%

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

      6. +-commutative 53.6%

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

      7. distribute-lft-neg-out 53.6%

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

      8. sub-neg 53.6%

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

      9. associate--l+ 79.2%

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

      10. associate-+l- 100.0%

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

      11. +-inverses 100.0%

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

      12. metadata-eval 100.0%

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

      13. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 68.8%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 125\right):\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 38.1% 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 74.9%

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

   1. +-commutative 74.9%

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

   2. sub-neg 74.9%

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

   3. distribute-rgt-in 75.0%

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

   4. *-lft-identity 75.0%

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

   5. associate-+l+ 75.0%

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

   6. +-commutative 75.0%

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

   7. distribute-lft-neg-out 75.0%

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

   8. sub-neg 75.0%

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

   9. associate--l+ 88.8%

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

   10. associate-+l- 100.0%

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

   11. +-inverses 100.0%

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

   12. metadata-eval 100.0%

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

   13. *-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in y around 0 38.3%

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

5. Final simplification 38.3%

   \[\leadsto 1 \]

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 2023318 
(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))))