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

Percentage Accurate: 77.9% → 100.0%

Time: 5.0s

Alternatives: 7

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: 77.9% 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

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

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(1.0 + Float64(Float64(y * x) - y))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

   1. sub-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. metadata-eval N/A

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

   7. associate-+l+ N/A

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

   8. associate-+r+ N/A

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

   9. *-lft-identity N/A

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

   10. neg-mul-1 N/A

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

   11. distribute-rgt1-in N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. associate-*r* N/A

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

   15. neg-mul-1 N/A

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

   16. *-lft-identity N/A

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

   17. mul0-lft N/A

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

   18. +-lft-identity N/A

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

   19. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(1 - x\right)\right)}\right) \]

   20. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(\left(1 - x\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

   21. neg-mul-1 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(\left(1 - x\right) \cdot \left(-1 \cdot \color{blue}{y}\right)\right)\right) \]

   22. associate-*r* N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\left(1 - x\right) \cdot -1\right) \cdot \color{blue}{y}\right)\right) \]

   23. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(\left(1 - x\right) \cdot -1\right)}\right)\right) \]

   24. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 - x\right) \cdot -1\right)}\right)\right) \]

3. Simplified 100.0%

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

   1. distribute-rgt-in N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot y + \color{blue}{-1 \cdot y}\right)\right) \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot y + \left(\mathsf{neg}\left(y\right)\right)\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot y - \color{blue}{y}\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(\left(x \cdot y\right), \color{blue}{y}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(\left(y \cdot x\right), y\right)\right) \]

   6. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, x\right), y\right)\right) \]

6. Applied egg-rr 100.0%

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

Alternative 2: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - y \leq -1 \cdot 10^{+15}:\\ \;\;\;\;y \cdot \left(x + -1\right)\\ \mathbf{elif}\;1 - y \leq 2:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- 1.0 y) -1e+15)
   (* y (+ x -1.0))
   (if (<= (- 1.0 y) 2.0) (+ 1.0 (* y x)) (- (* y x) y))))

C

double code(double x, double y) {
	double tmp;
	if ((1.0 - y) <= -1e+15) {
		tmp = y * (x + -1.0);
	} else if ((1.0 - y) <= 2.0) {
		tmp = 1.0 + (y * x);
	} else {
		tmp = (y * x) - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((1.0d0 - y) <= (-1d+15)) then
        tmp = y * (x + (-1.0d0))
    else if ((1.0d0 - y) <= 2.0d0) then
        tmp = 1.0d0 + (y * x)
    else
        tmp = (y * x) - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((1.0 - y) <= -1e+15) {
		tmp = y * (x + -1.0);
	} else if ((1.0 - y) <= 2.0) {
		tmp = 1.0 + (y * x);
	} else {
		tmp = (y * x) - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (1.0 - y) <= -1e+15:
		tmp = y * (x + -1.0)
	elif (1.0 - y) <= 2.0:
		tmp = 1.0 + (y * x)
	else:
		tmp = (y * x) - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(1.0 - y) <= -1e+15)
		tmp = Float64(y * Float64(x + -1.0));
	elseif (Float64(1.0 - y) <= 2.0)
		tmp = Float64(1.0 + Float64(y * x));
	else
		tmp = Float64(Float64(y * x) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((1.0 - y) <= -1e+15)
		tmp = y * (x + -1.0);
	elseif ((1.0 - y) <= 2.0)
		tmp = 1.0 + (y * x);
	else
		tmp = (y * x) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 - y), $MachinePrecision], -1e+15], N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 - y), $MachinePrecision], 2.0], N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] - y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -1e15

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. associate-+r+ N/A

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

      9. *-lft-identity N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt1-in N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. associate-*r* N/A

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

      15. neg-mul-1 N/A

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

      16. *-lft-identity N/A

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

      17. mul0-lft N/A

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

      18. +-lft-identity N/A

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

      19. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(1 - x\right)\right)}\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(1 - x\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      21. neg-mul-1 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(1 - x\right) \cdot \left(-1 \cdot \color{blue}{y}\right)\right)\right) \]

      22. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\left(1 - x\right) \cdot -1\right) \cdot \color{blue}{y}\right)\right) \]

      23. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(\left(1 - x\right) \cdot -1\right)}\right)\right) \]

      24. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 - x\right) \cdot -1\right)}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(x - 1\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(x + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(x + -1\right)\right) \]

      4. +-lowering-+.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{-1}\right)\right) \]

   7. Simplified 99.9%

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

   if -1e15 < (-.f64 #s(literal 1 binary64) y) < 2

   1. Initial program 50.4%

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. associate-+r+ N/A

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

      9. *-lft-identity N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt1-in N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. associate-*r* N/A

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

      15. neg-mul-1 N/A

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

      16. *-lft-identity N/A

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

      17. mul0-lft N/A

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

      18. +-lft-identity N/A

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

      19. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(1 - x\right)\right)}\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(1 - x\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      21. neg-mul-1 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(1 - x\right) \cdot \left(-1 \cdot \color{blue}{y}\right)\right)\right) \]

      22. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\left(1 - x\right) \cdot -1\right) \cdot \color{blue}{y}\right)\right) \]

      23. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(\left(1 - x\right) \cdot -1\right)}\right)\right) \]

      24. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 - x\right) \cdot -1\right)}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot y\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{x}\right)\right) \]

      2. *-lowering-*.f64 98.2%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{x}\right)\right) \]

   7. Simplified 98.2%

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

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

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. associate-+r+ N/A

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

      9. *-lft-identity N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt1-in N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. associate-*r* N/A

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

      15. neg-mul-1 N/A

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

      16. *-lft-identity N/A

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

      17. mul0-lft N/A

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

      18. +-lft-identity N/A

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

      19. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(1 - x\right)\right)}\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(1 - x\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      21. neg-mul-1 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(1 - x\right) \cdot \left(-1 \cdot \color{blue}{y}\right)\right)\right) \]

      22. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\left(1 - x\right) \cdot -1\right) \cdot \color{blue}{y}\right)\right) \]

      23. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(\left(1 - x\right) \cdot -1\right)}\right)\right) \]

      24. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 - x\right) \cdot -1\right)}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(x - 1\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(x + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(x + -1\right)\right) \]

      4. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{-1}\right)\right) \]

   7. Simplified 100.0%

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. neg-mul-1 N/A

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

      4. sub-neg N/A

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

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(y \cdot x\right), \color{blue}{y}\right) \]

      6. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, x\right), y\right) \]

   9. Applied egg-rr 100.0%

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

Alternative 3: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x + -1\right)\\ \mathbf{if}\;1 - y \leq -1 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;1 - y \leq 2:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (+ x -1.0))))
   (if (<= (- 1.0 y) -1e+15) t_0 (if (<= (- 1.0 y) 2.0) (+ 1.0 (* y x)) t_0))))

C

double code(double x, double y) {
	double t_0 = y * (x + -1.0);
	double tmp;
	if ((1.0 - y) <= -1e+15) {
		tmp = t_0;
	} else if ((1.0 - y) <= 2.0) {
		tmp = 1.0 + (y * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (x + (-1.0d0))
    if ((1.0d0 - y) <= (-1d+15)) then
        tmp = t_0
    else if ((1.0d0 - y) <= 2.0d0) then
        tmp = 1.0d0 + (y * x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (x + -1.0);
	double tmp;
	if ((1.0 - y) <= -1e+15) {
		tmp = t_0;
	} else if ((1.0 - y) <= 2.0) {
		tmp = 1.0 + (y * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x + -1.0)
	tmp = 0
	if (1.0 - y) <= -1e+15:
		tmp = t_0
	elif (1.0 - y) <= 2.0:
		tmp = 1.0 + (y * x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x + -1.0))
	tmp = 0.0
	if (Float64(1.0 - y) <= -1e+15)
		tmp = t_0;
	elseif (Float64(1.0 - y) <= 2.0)
		tmp = Float64(1.0 + Float64(y * x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x + -1.0);
	tmp = 0.0;
	if ((1.0 - y) <= -1e+15)
		tmp = t_0;
	elseif ((1.0 - y) <= 2.0)
		tmp = 1.0 + (y * x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 - y), $MachinePrecision], -1e+15], t$95$0, If[LessEqual[N[(1.0 - y), $MachinePrecision], 2.0], N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -1e15 or 2 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. associate-+r+ N/A

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

      9. *-lft-identity N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt1-in N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. associate-*r* N/A

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

      15. neg-mul-1 N/A

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

      16. *-lft-identity N/A

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

      17. mul0-lft N/A

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

      18. +-lft-identity N/A

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

      19. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(1 - x\right)\right)}\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(1 - x\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      21. neg-mul-1 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(1 - x\right) \cdot \left(-1 \cdot \color{blue}{y}\right)\right)\right) \]

      22. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\left(1 - x\right) \cdot -1\right) \cdot \color{blue}{y}\right)\right) \]

      23. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(\left(1 - x\right) \cdot -1\right)}\right)\right) \]

      24. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 - x\right) \cdot -1\right)}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(x - 1\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(x + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(x + -1\right)\right) \]

      4. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{-1}\right)\right) \]

   7. Simplified 100.0%

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

   if -1e15 < (-.f64 #s(literal 1 binary64) y) < 2

   1. Initial program 50.4%

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. associate-+r+ N/A

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

      9. *-lft-identity N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt1-in N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. associate-*r* N/A

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

      15. neg-mul-1 N/A

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

      16. *-lft-identity N/A

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

      17. mul0-lft N/A

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

      18. +-lft-identity N/A

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

      19. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(1 - x\right)\right)}\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(1 - x\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      21. neg-mul-1 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(1 - x\right) \cdot \left(-1 \cdot \color{blue}{y}\right)\right)\right) \]

      22. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\left(1 - x\right) \cdot -1\right) \cdot \color{blue}{y}\right)\right) \]

      23. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(\left(1 - x\right) \cdot -1\right)}\right)\right) \]

      24. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 - x\right) \cdot -1\right)}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot y\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{x}\right)\right) \]

      2. *-lowering-*.f64 98.2%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{x}\right)\right) \]

   7. Simplified 98.2%

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

Alternative 4: 85.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{+115}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{+17}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.9e+115) (* y x) (if (<= x 1.06e+17) (- 1.0 y) (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.9e+115) {
		tmp = y * x;
	} else if (x <= 1.06e+17) {
		tmp = 1.0 - y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.9d+115)) then
        tmp = y * x
    else if (x <= 1.06d+17) then
        tmp = 1.0d0 - y
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.9e+115) {
		tmp = y * x;
	} else if (x <= 1.06e+17) {
		tmp = 1.0 - y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.9e+115:
		tmp = y * x
	elif x <= 1.06e+17:
		tmp = 1.0 - y
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.9e+115)
		tmp = Float64(y * x);
	elseif (x <= 1.06e+17)
		tmp = Float64(1.0 - y);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.9e+115)
		tmp = y * x;
	elseif (x <= 1.06e+17)
		tmp = 1.0 - y;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.9e+115], N[(y * x), $MachinePrecision], If[LessEqual[x, 1.06e+17], N[(1.0 - y), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.90000000000000006e115 or 1.06e17 < x

   1. Initial program 58.9%

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. associate-+r+ N/A

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

      9. *-lft-identity N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt1-in N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. associate-*r* N/A

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

      15. neg-mul-1 N/A

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

      16. *-lft-identity N/A

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

      17. mul0-lft N/A

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

      18. +-lft-identity N/A

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

      19. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(1 - x\right)\right)}\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(1 - x\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      21. neg-mul-1 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(1 - x\right) \cdot \left(-1 \cdot \color{blue}{y}\right)\right)\right) \]

      22. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\left(1 - x\right) \cdot -1\right) \cdot \color{blue}{y}\right)\right) \]

      23. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(\left(1 - x\right) \cdot -1\right)}\right)\right) \]

      24. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 - x\right) \cdot -1\right)}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 85.7%

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{x}\right) \]

   7. Simplified 85.7%

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

   if -3.90000000000000006e115 < x < 1.06e17

   1. Initial program 87.3%

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. associate-+r+ N/A

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

      9. *-lft-identity N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt1-in N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. associate-*r* N/A

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

      15. neg-mul-1 N/A

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

      16. *-lft-identity N/A

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

      17. mul0-lft N/A

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

      18. +-lft-identity N/A

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

      19. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(1 - x\right)\right)}\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(1 - x\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      21. neg-mul-1 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(1 - x\right) \cdot \left(-1 \cdot \color{blue}{y}\right)\right)\right) \]

      22. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\left(1 - x\right) \cdot -1\right) \cdot \color{blue}{y}\right)\right) \]

      23. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(\left(1 - x\right) \cdot -1\right)}\right)\right) \]

      24. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 - x\right) \cdot -1\right)}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto 1 + \left(\mathsf{neg}\left(y\right)\right) \]

      2. unsub-neg N/A

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

      3. --lowering--.f64 94.0%

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{y}\right) \]

   7. Simplified 94.0%

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

Alternative 5: 62.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{+85}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 1.92 \cdot 10^{+25}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9.6e+85) (* y x) (if (<= x 1.92e+25) 1.0 (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9.6e+85) {
		tmp = y * x;
	} else if (x <= 1.92e+25) {
		tmp = 1.0;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-9.6d+85)) then
        tmp = y * x
    else if (x <= 1.92d+25) then
        tmp = 1.0d0
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -9.6e+85) {
		tmp = y * x;
	} else if (x <= 1.92e+25) {
		tmp = 1.0;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9.6e+85:
		tmp = y * x
	elif x <= 1.92e+25:
		tmp = 1.0
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9.6e+85)
		tmp = Float64(y * x);
	elseif (x <= 1.92e+25)
		tmp = 1.0;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9.6e+85)
		tmp = y * x;
	elseif (x <= 1.92e+25)
		tmp = 1.0;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9.6e+85], N[(y * x), $MachinePrecision], If[LessEqual[x, 1.92e+25], 1.0, N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.59999999999999986e85 or 1.9200000000000001e25 < x

   1. Initial program 57.7%

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. associate-+r+ N/A

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

      9. *-lft-identity N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt1-in N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. associate-*r* N/A

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

      15. neg-mul-1 N/A

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

      16. *-lft-identity N/A

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

      17. mul0-lft N/A

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

      18. +-lft-identity N/A

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

      19. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(1 - x\right)\right)}\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(1 - x\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      21. neg-mul-1 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(1 - x\right) \cdot \left(-1 \cdot \color{blue}{y}\right)\right)\right) \]

      22. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\left(1 - x\right) \cdot -1\right) \cdot \color{blue}{y}\right)\right) \]

      23. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(\left(1 - x\right) \cdot -1\right)}\right)\right) \]

      24. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 - x\right) \cdot -1\right)}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 83.9%

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{x}\right) \]

   7. Simplified 83.9%

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

   if -9.59999999999999986e85 < x < 1.9200000000000001e25

   1. Initial program 89.5%

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. associate-+r+ N/A

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

      9. *-lft-identity N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt1-in N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. associate-*r* N/A

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

      15. neg-mul-1 N/A

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

      16. *-lft-identity N/A

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

      17. mul0-lft N/A

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

      18. +-lft-identity N/A

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

      19. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(1 - x\right)\right)}\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(1 - x\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      21. neg-mul-1 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(1 - x\right) \cdot \left(-1 \cdot \color{blue}{y}\right)\right)\right) \]

      22. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\left(1 - x\right) \cdot -1\right) \cdot \color{blue}{y}\right)\right) \]

      23. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(\left(1 - x\right) \cdot -1\right)}\right)\right) \]

      24. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 - x\right) \cdot -1\right)}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 53.0%

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

Alternative 6: 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 75.0%

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

   1. sub-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. metadata-eval N/A

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

   7. associate-+l+ N/A

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

   8. associate-+r+ N/A

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

   9. *-lft-identity N/A

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

   10. neg-mul-1 N/A

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

   11. distribute-rgt1-in N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. associate-*r* N/A

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

   15. neg-mul-1 N/A

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

   16. *-lft-identity N/A

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

   17. mul0-lft N/A

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

   18. +-lft-identity N/A

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

   19. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(1 - x\right)\right)}\right) \]

   20. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(\left(1 - x\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

   21. neg-mul-1 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(\left(1 - x\right) \cdot \left(-1 \cdot \color{blue}{y}\right)\right)\right) \]

   22. associate-*r* N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\left(1 - x\right) \cdot -1\right) \cdot \color{blue}{y}\right)\right) \]

   23. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(\left(1 - x\right) \cdot -1\right)}\right)\right) \]

   24. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 - x\right) \cdot -1\right)}\right)\right) \]

3. Simplified 100.0%

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

Alternative 7: 37.9% 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 75.0%

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

   1. sub-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. metadata-eval N/A

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

   7. associate-+l+ N/A

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

   8. associate-+r+ N/A

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

   9. *-lft-identity N/A

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

   10. neg-mul-1 N/A

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

   11. distribute-rgt1-in N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. associate-*r* N/A

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

   15. neg-mul-1 N/A

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

   16. *-lft-identity N/A

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

   17. mul0-lft N/A

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

   18. +-lft-identity N/A

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

   19. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(1 - x\right)\right)}\right) \]

   20. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(\left(1 - x\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

   21. neg-mul-1 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(\left(1 - x\right) \cdot \left(-1 \cdot \color{blue}{y}\right)\right)\right) \]

   22. associate-*r* N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\left(1 - x\right) \cdot -1\right) \cdot \color{blue}{y}\right)\right) \]

   23. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(\left(1 - x\right) \cdot -1\right)}\right)\right) \]

   24. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 - x\right) \cdot -1\right)}\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in y around 0

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

7. Simplified 36.6%

   \[\leadsto \color{blue}{1} \]
8. 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 2024138 
(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))))