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

Percentage Accurate: 77.5% → 100.0%

Time: 5.4s

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: 77.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.3× 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 78.1%

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

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

   2. associate-+r+ N/A

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 100.0%

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

6. Applied egg-rr 100.0%

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

Alternative 2: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot x - y\\ \mathbf{if}\;1 - y \leq -50000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;1 - y \leq 1.01:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (* y x) y)))
   (if (<= (- 1.0 y) -50000.0)
     t_0
     (if (<= (- 1.0 y) 1.01) (+ 1.0 (* y x)) t_0))))

C

double code(double x, double y) {
	double t_0 = (y * x) - y;
	double tmp;
	if ((1.0 - y) <= -50000.0) {
		tmp = t_0;
	} else if ((1.0 - y) <= 1.01) {
		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) - y
    if ((1.0d0 - y) <= (-50000.0d0)) then
        tmp = t_0
    else if ((1.0d0 - y) <= 1.01d0) 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) - y;
	double tmp;
	if ((1.0 - y) <= -50000.0) {
		tmp = t_0;
	} else if ((1.0 - y) <= 1.01) {
		tmp = 1.0 + (y * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * x) - y
	tmp = 0
	if (1.0 - y) <= -50000.0:
		tmp = t_0
	elif (1.0 - y) <= 1.01:
		tmp = 1.0 + (y * x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) - y)
	tmp = 0.0
	if (Float64(1.0 - y) <= -50000.0)
		tmp = t_0;
	elseif (Float64(1.0 - y) <= 1.01)
		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) - y;
	tmp = 0.0;
	if ((1.0 - y) <= -50000.0)
		tmp = t_0;
	elseif ((1.0 - y) <= 1.01)
		tmp = 1.0 + (y * x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[N[(1.0 - y), $MachinePrecision], -50000.0], t$95$0, If[LessEqual[N[(1.0 - y), $MachinePrecision], 1.01], 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) < -5e4 or 1.01000000000000001 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 99.9%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   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 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 97.4%

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

   7. Simplified 97.4%

      \[\leadsto \color{blue}{y \cdot \left(x + -1\right)} \]
   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 97.5%

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

   9. Applied egg-rr 97.5%

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

   if -5e4 < (-.f64 #s(literal 1 binary64) y) < 1.01000000000000001

   1. Initial program 52.2%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   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 98.4%

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

   7. Simplified 98.4%

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

Alternative 3: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x + -1\right)\\ \mathbf{if}\;1 - y \leq -50000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;1 - y \leq 1.01:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (+ x -1.0))))
   (if (<= (- 1.0 y) -50000.0)
     t_0
     (if (<= (- 1.0 y) 1.01) (+ 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) <= -50000.0) {
		tmp = t_0;
	} else if ((1.0 - y) <= 1.01) {
		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) <= (-50000.0d0)) then
        tmp = t_0
    else if ((1.0d0 - y) <= 1.01d0) 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) <= -50000.0) {
		tmp = t_0;
	} else if ((1.0 - y) <= 1.01) {
		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) <= -50000.0:
		tmp = t_0
	elif (1.0 - y) <= 1.01:
		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) <= -50000.0)
		tmp = t_0;
	elseif (Float64(1.0 - y) <= 1.01)
		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) <= -50000.0)
		tmp = t_0;
	elseif ((1.0 - y) <= 1.01)
		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], -50000.0], t$95$0, If[LessEqual[N[(1.0 - y), $MachinePrecision], 1.01], 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) < -5e4 or 1.01000000000000001 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 99.9%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   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 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 97.4%

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

   7. Simplified 97.4%

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

   if -5e4 < (-.f64 #s(literal 1 binary64) y) < 1.01000000000000001

   1. Initial program 52.2%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   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 98.4%

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

   7. Simplified 98.4%

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

Alternative 4: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{-8}:\\ \;\;\;\;y \cdot \left(x + -1\right)\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+18}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.1e-8) (* y (+ x -1.0)) (if (<= x 6.5e+18) (- 1.0 y) (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.1e-8) {
		tmp = y * (x + -1.0);
	} else if (x <= 6.5e+18) {
		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 <= (-5.1d-8)) then
        tmp = y * (x + (-1.0d0))
    else if (x <= 6.5d+18) 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 <= -5.1e-8) {
		tmp = y * (x + -1.0);
	} else if (x <= 6.5e+18) {
		tmp = 1.0 - y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.1e-8:
		tmp = y * (x + -1.0)
	elif x <= 6.5e+18:
		tmp = 1.0 - y
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.1e-8)
		tmp = Float64(y * Float64(x + -1.0));
	elseif (x <= 6.5e+18)
		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 <= -5.1e-8)
		tmp = y * (x + -1.0);
	elseif (x <= 6.5e+18)
		tmp = 1.0 - y;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.1e-8], N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.5e+18], N[(1.0 - y), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.10000000000000001e-8

   1. Initial program 57.1%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   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 99.9%

      \[\leadsto \color{blue}{1 + y \cdot \left(x + -1\right)} \]
   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 82.9%

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

   7. Simplified 82.9%

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

   if -5.10000000000000001e-8 < x < 6.5e18

   1. Initial program 98.6%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   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 98.1%

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

   7. Simplified 98.1%

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

   if 6.5e18 < x

   1. Initial program 52.7%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   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 79.7%

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

   7. Simplified 79.7%

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

Alternative 5: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -52:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+18}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -52.0) (* y x) (if (<= x 4.4e+18) (- 1.0 y) (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -52.0) {
		tmp = y * x;
	} else if (x <= 4.4e+18) {
		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 <= (-52.0d0)) then
        tmp = y * x
    else if (x <= 4.4d+18) 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 <= -52.0) {
		tmp = y * x;
	} else if (x <= 4.4e+18) {
		tmp = 1.0 - y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -52.0:
		tmp = y * x
	elif x <= 4.4e+18:
		tmp = 1.0 - y
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -52.0)
		tmp = Float64(y * x);
	elseif (x <= 4.4e+18)
		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 <= -52.0)
		tmp = y * x;
	elseif (x <= 4.4e+18)
		tmp = 1.0 - y;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -52.0], N[(y * x), $MachinePrecision], If[LessEqual[x, 4.4e+18], N[(1.0 - y), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -52 or 4.4e18 < x

   1. Initial program 52.5%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   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 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 80.2%

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

   7. Simplified 80.2%

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

   if -52 < x < 4.4e18

   1. Initial program 98.6%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   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 95.2%

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

   7. Simplified 95.2%

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

Alternative 6: 62.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{-8}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+22}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.1e-8) (* y x) (if (<= x 1.45e+22) 1.0 (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.1e-8) {
		tmp = y * x;
	} else if (x <= 1.45e+22) {
		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 <= (-5.1d-8)) then
        tmp = y * x
    else if (x <= 1.45d+22) 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 <= -5.1e-8) {
		tmp = y * x;
	} else if (x <= 1.45e+22) {
		tmp = 1.0;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.1e-8:
		tmp = y * x
	elif x <= 1.45e+22:
		tmp = 1.0
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.1e-8)
		tmp = Float64(y * x);
	elseif (x <= 1.45e+22)
		tmp = 1.0;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.1e-8)
		tmp = y * x;
	elseif (x <= 1.45e+22)
		tmp = 1.0;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.1e-8], N[(y * x), $MachinePrecision], If[LessEqual[x, 1.45e+22], 1.0, N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.10000000000000001e-8 or 1.45e22 < x

   1. Initial program 55.2%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   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 99.9%

      \[\leadsto \color{blue}{1 + y \cdot \left(x + -1\right)} \]
   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 76.3%

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

   7. Simplified 76.3%

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

   if -5.10000000000000001e-8 < x < 1.45e22

   1. Initial program 98.6%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   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 45.0%

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

Alternative 7: 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 78.1%

   \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
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 8: 38.6% 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 78.1%

   \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
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 33.0%

   \[\leadsto \color{blue}{1} \]
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 2024140 
(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))))