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

Percentage Accurate: 77.0% → 100.0%

Time: 5.7s

Alternatives: 9

Speedup: 1.3×

• could not determine a ground truth

  1. x = -5.752000162422447e-282
  2. y = 0.4096229578863035

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: 77.0% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. +-commutative 77.4%

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

   2. remove-double-neg 77.4%

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

   3. unsub-neg 77.4%

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

   4. sub-neg 77.4%

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

   5. +-commutative 77.4%

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

   6. distribute-rgt-in 77.4%

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

   7. *-lft-identity 77.4%

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

   8. associate--l+ 87.3%

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

   9. associate--l- 100.0%

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

   10. sub-neg 100.0%

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

   11. +-inverses 100.0%

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

   12. metadata-eval 100.0%

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

   13. +-commutative 100.0%

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

   14. distribute-lft-neg-out 100.0%

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

   15. distribute-rgt-neg-in 100.0%

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

   16. neg-sub0 100.0%

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

   17. associate--r- 100.0%

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

   18. metadata-eval 100.0%

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

   19. +-commutative 100.0%

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

3. Simplified 100.0%

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

   1. distribute-lft-in 100.0%

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

   2. *-commutative 100.0%

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

   3. mul-1-neg 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 60.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{+118}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-292}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-267}:\\ \;\;\;\;-y\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-228}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.85e+118)
   (* x y)
   (if (<= x -1.05e-292)
     1.0
     (if (<= x 9e-267)
       (- y)
       (if (<= x 2.3e-228) 1.0 (if (<= x 1.0) (- y) (* x y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.85e+118) {
		tmp = x * y;
	} else if (x <= -1.05e-292) {
		tmp = 1.0;
	} else if (x <= 9e-267) {
		tmp = -y;
	} else if (x <= 2.3e-228) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = -y;
	} else {
		tmp = 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 (x <= (-2.85d+118)) then
        tmp = x * y
    else if (x <= (-1.05d-292)) then
        tmp = 1.0d0
    else if (x <= 9d-267) then
        tmp = -y
    else if (x <= 2.3d-228) then
        tmp = 1.0d0
    else if (x <= 1.0d0) then
        tmp = -y
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.85e+118) {
		tmp = x * y;
	} else if (x <= -1.05e-292) {
		tmp = 1.0;
	} else if (x <= 9e-267) {
		tmp = -y;
	} else if (x <= 2.3e-228) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = -y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.85e+118:
		tmp = x * y
	elif x <= -1.05e-292:
		tmp = 1.0
	elif x <= 9e-267:
		tmp = -y
	elif x <= 2.3e-228:
		tmp = 1.0
	elif x <= 1.0:
		tmp = -y
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.85e+118)
		tmp = Float64(x * y);
	elseif (x <= -1.05e-292)
		tmp = 1.0;
	elseif (x <= 9e-267)
		tmp = Float64(-y);
	elseif (x <= 2.3e-228)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = Float64(-y);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.85e+118)
		tmp = x * y;
	elseif (x <= -1.05e-292)
		tmp = 1.0;
	elseif (x <= 9e-267)
		tmp = -y;
	elseif (x <= 2.3e-228)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = -y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.85e+118], N[(x * y), $MachinePrecision], If[LessEqual[x, -1.05e-292], 1.0, If[LessEqual[x, 9e-267], (-y), If[LessEqual[x, 2.3e-228], 1.0, If[LessEqual[x, 1.0], (-y), N[(x * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.85000000000000001e118 or 1 < x

   1. Initial program 58.4%

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

      1. +-commutative 58.4%

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

      2. remove-double-neg 58.4%

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

      3. unsub-neg 58.4%

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

      4. sub-neg 58.4%

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

      5. +-commutative 58.4%

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

      6. distribute-rgt-in 58.4%

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

      7. *-lft-identity 58.4%

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

      8. associate--l+ 79.8%

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

      9. associate--l- 100.0%

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

      10. sub-neg 100.0%

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

      11. +-inverses 100.0%

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

      12. metadata-eval 100.0%

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

      13. +-commutative 100.0%

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

      14. distribute-lft-neg-out 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. neg-sub0 100.0%

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

      17. associate--r- 100.0%

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

      18. metadata-eval 100.0%

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

      19. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 78.0%

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

      1. *-commutative 78.0%

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

   9. Simplified 78.0%

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

   if -2.85000000000000001e118 < x < -1.04999999999999994e-292 or 8.9999999999999999e-267 < x < 2.2999999999999999e-228

   1. Initial program 88.6%

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

      1. +-commutative 88.6%

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

      2. remove-double-neg 88.6%

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

      3. unsub-neg 88.6%

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

      4. sub-neg 88.6%

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

      5. +-commutative 88.6%

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

      6. distribute-rgt-in 88.6%

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

      7. *-lft-identity 88.6%

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

      8. associate--l+ 88.7%

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

      9. associate--l- 100.0%

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

      10. sub-neg 100.0%

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

      11. +-inverses 100.0%

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

      12. metadata-eval 100.0%

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

      13. +-commutative 100.0%

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

      14. distribute-lft-neg-out 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. neg-sub0 100.0%

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

      17. associate--r- 100.0%

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

      18. metadata-eval 100.0%

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

      19. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 58.4%

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

   if -1.04999999999999994e-292 < x < 8.9999999999999999e-267 or 2.2999999999999999e-228 < x < 1

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. remove-double-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate--l+ 100.0%

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

      9. associate--l- 100.0%

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

      10. sub-neg 100.0%

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

      11. +-inverses 100.0%

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

      12. metadata-eval 100.0%

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

      13. +-commutative 100.0%

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

      14. distribute-lft-neg-out 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. neg-sub0 100.0%

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

      17. associate--r- 100.0%

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

      18. metadata-eval 100.0%

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

      19. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around inf 69.2%

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

   8. Taylor expanded in x around 0 66.6%

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

      1. neg-mul-1 66.6%

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

   10. Simplified 66.6%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{+118}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-292}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-267}:\\ \;\;\;\;-y\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-228}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - y \leq -1000000000000 \lor \neg \left(1 - y \leq 2\right):\\ \;\;\;\;y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (- 1.0 y) -1000000000000.0) (not (<= (- 1.0 y) 2.0)))
   (* y (+ x -1.0))
   (+ 1.0 (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 - y) <= -1000000000000.0) || !((1.0 - y) <= 2.0)) {
		tmp = y * (x + -1.0);
	} else {
		tmp = 1.0 + (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) <= (-1000000000000.0d0)) .or. (.not. ((1.0d0 - y) <= 2.0d0))) then
        tmp = y * (x + (-1.0d0))
    else
        tmp = 1.0d0 + (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((1.0 - y) <= -1000000000000.0) || !((1.0 - y) <= 2.0)) {
		tmp = y * (x + -1.0);
	} else {
		tmp = 1.0 + (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((1.0 - y) <= -1000000000000.0) or not ((1.0 - y) <= 2.0):
		tmp = y * (x + -1.0)
	else:
		tmp = 1.0 + (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(1.0 - y) <= -1000000000000.0) || !(Float64(1.0 - y) <= 2.0))
		tmp = Float64(y * Float64(x + -1.0));
	else
		tmp = Float64(1.0 + Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((1.0 - y) <= -1000000000000.0) || ~(((1.0 - y) <= 2.0)))
		tmp = y * (x + -1.0);
	else
		tmp = 1.0 + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(1.0 - y), $MachinePrecision], -1000000000000.0], N[Not[LessEqual[N[(1.0 - y), $MachinePrecision], 2.0]], $MachinePrecision]], N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -1e12 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. +-commutative 100.0%

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

      2. remove-double-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate--l+ 100.0%

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

      9. associate--l- 100.0%

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

      10. sub-neg 100.0%

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

      11. +-inverses 100.0%

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

      12. metadata-eval 100.0%

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

      13. +-commutative 100.0%

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

      14. distribute-lft-neg-out 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. neg-sub0 100.0%

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

      17. associate--r- 100.0%

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

      18. metadata-eval 100.0%

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

      19. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around inf 99.6%

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

   if -1e12 < (-.f64 #s(literal 1 binary64) y) < 2

   1. Initial program 49.7%

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

      1. +-commutative 49.7%

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

      2. remove-double-neg 49.7%

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

      3. unsub-neg 49.7%

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

      4. sub-neg 49.7%

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

      5. +-commutative 49.7%

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

      6. distribute-rgt-in 49.7%

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

      7. *-lft-identity 49.7%

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

      8. associate--l+ 71.8%

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

      9. associate--l- 100.0%

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

      10. sub-neg 100.0%

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

      11. +-inverses 100.0%

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

      12. metadata-eval 100.0%

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

      13. +-commutative 100.0%

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

      14. distribute-lft-neg-out 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. neg-sub0 100.0%

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

      17. associate--r- 100.0%

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

      18. metadata-eval 100.0%

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

      19. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.4%

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

      1. *-commutative 99.4%

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

   7. Simplified 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -1000000000000 \lor \neg \left(1 - y \leq 2\right):\\ \;\;\;\;y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - y \leq -1000000000000 \lor \neg \left(1 - y \leq 2\right):\\ \;\;\;\;x \cdot y - y\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (- 1.0 y) -1000000000000.0) (not (<= (- 1.0 y) 2.0)))
   (- (* x y) y)
   (+ 1.0 (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 - y) <= -1000000000000.0) || !((1.0 - y) <= 2.0)) {
		tmp = (x * y) - y;
	} else {
		tmp = 1.0 + (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) <= (-1000000000000.0d0)) .or. (.not. ((1.0d0 - y) <= 2.0d0))) then
        tmp = (x * y) - y
    else
        tmp = 1.0d0 + (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((1.0 - y) <= -1000000000000.0) || !((1.0 - y) <= 2.0)) {
		tmp = (x * y) - y;
	} else {
		tmp = 1.0 + (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((1.0 - y) <= -1000000000000.0) or not ((1.0 - y) <= 2.0):
		tmp = (x * y) - y
	else:
		tmp = 1.0 + (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(1.0 - y) <= -1000000000000.0) || !(Float64(1.0 - y) <= 2.0))
		tmp = Float64(Float64(x * y) - y);
	else
		tmp = Float64(1.0 + Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((1.0 - y) <= -1000000000000.0) || ~(((1.0 - y) <= 2.0)))
		tmp = (x * y) - y;
	else
		tmp = 1.0 + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(1.0 - y), $MachinePrecision], -1000000000000.0], N[Not[LessEqual[N[(1.0 - y), $MachinePrecision], 2.0]], $MachinePrecision]], N[(N[(x * y), $MachinePrecision] - y), $MachinePrecision], N[(1.0 + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -1e12 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. +-commutative 100.0%

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

      2. remove-double-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate--l+ 100.0%

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

      9. associate--l- 100.0%

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

      10. sub-neg 100.0%

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

      11. +-inverses 100.0%

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

      12. metadata-eval 100.0%

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

      13. +-commutative 100.0%

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

      14. distribute-lft-neg-out 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. neg-sub0 100.0%

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

      17. associate--r- 100.0%

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

      18. metadata-eval 100.0%

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

      19. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around inf 99.6%

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

      1. sub-neg 99.6%

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

      2. metadata-eval 99.6%

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

      3. distribute-rgt-in 99.7%

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

      4. *-commutative 99.7%

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

      5. neg-mul-1 99.7%

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

   9. Applied egg-rr 99.7%

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

   if -1e12 < (-.f64 #s(literal 1 binary64) y) < 2

   1. Initial program 49.7%

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

      1. +-commutative 49.7%

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

      2. remove-double-neg 49.7%

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

      3. unsub-neg 49.7%

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

      4. sub-neg 49.7%

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

      5. +-commutative 49.7%

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

      6. distribute-rgt-in 49.7%

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

      7. *-lft-identity 49.7%

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

      8. associate--l+ 71.8%

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

      9. associate--l- 100.0%

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

      10. sub-neg 100.0%

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

      11. +-inverses 100.0%

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

      12. metadata-eval 100.0%

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

      13. +-commutative 100.0%

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

      14. distribute-lft-neg-out 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. neg-sub0 100.0%

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

      17. associate--r- 100.0%

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

      18. metadata-eval 100.0%

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

      19. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.4%

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

      1. *-commutative 99.4%

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

   7. Simplified 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -1000000000000 \lor \neg \left(1 - y \leq 2\right):\\ \;\;\;\;x \cdot y - y\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-69} \lor \neg \left(y \leq 6.6 \cdot 10^{-15}\right):\\ \;\;\;\;y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6.5e-69) (not (<= y 6.6e-15))) (* y (+ x -1.0)) (- 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6.5e-69) || !(y <= 6.6e-15)) {
		tmp = y * (x + -1.0);
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-6.5d-69)) .or. (.not. (y <= 6.6d-15))) then
        tmp = y * (x + (-1.0d0))
    else
        tmp = 1.0d0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -6.5e-69) || !(y <= 6.6e-15)) {
		tmp = y * (x + -1.0);
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6.5e-69) or not (y <= 6.6e-15):
		tmp = y * (x + -1.0)
	else:
		tmp = 1.0 - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6.5e-69) || !(y <= 6.6e-15))
		tmp = Float64(y * Float64(x + -1.0));
	else
		tmp = Float64(1.0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -6.5e-69) || ~((y <= 6.6e-15)))
		tmp = y * (x + -1.0);
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6.5e-69], N[Not[LessEqual[y, 6.6e-15]], $MachinePrecision]], N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.49999999999999951e-69 or 6.6e-15 < y

   1. Initial program 94.6%

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

      1. +-commutative 94.6%

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

      2. remove-double-neg 94.6%

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

      3. unsub-neg 94.6%

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

      4. sub-neg 94.6%

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

      5. +-commutative 94.6%

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

      6. distribute-rgt-in 94.6%

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

      7. *-lft-identity 94.6%

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

      8. associate--l+ 100.0%

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

      9. associate--l- 100.0%

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

      10. sub-neg 100.0%

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

      11. +-inverses 100.0%

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

      12. metadata-eval 100.0%

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

      13. +-commutative 100.0%

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

      14. distribute-lft-neg-out 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. neg-sub0 100.0%

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

      17. associate--r- 100.0%

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

      18. metadata-eval 100.0%

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

      19. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around inf 96.1%

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

   if -6.49999999999999951e-69 < y < 6.6e-15

   1. Initial program 50.5%

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

      1. +-commutative 50.5%

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

      2. remove-double-neg 50.5%

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

      3. unsub-neg 50.5%

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

      4. sub-neg 50.5%

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

      5. +-commutative 50.5%

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

      6. distribute-rgt-in 50.5%

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

      7. *-lft-identity 50.5%

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

      8. associate--l+ 67.5%

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

      9. associate--l- 100.0%

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

      10. sub-neg 100.0%

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

      11. +-inverses 100.0%

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

      12. metadata-eval 100.0%

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

      13. +-commutative 100.0%

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

      14. distribute-lft-neg-out 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. neg-sub0 100.0%

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

      17. associate--r- 100.0%

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

      18. metadata-eval 100.0%

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

      19. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 79.9%

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

      1. neg-mul-1 79.9%

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

      2. unsub-neg 79.9%

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

   7. Simplified 79.9%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-69} \lor \neg \left(y \leq 6.6 \cdot 10^{-15}\right):\\ \;\;\;\;y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{+118} \lor \neg \left(x \leq 1.22 \cdot 10^{+64}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.85e+118) (not (<= x 1.22e+64))) (* x y) (- 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.85e+118) || !(x <= 1.22e+64)) {
		tmp = x * y;
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-2.85d+118)) .or. (.not. (x <= 1.22d+64))) then
        tmp = x * y
    else
        tmp = 1.0d0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2.85e+118) || !(x <= 1.22e+64)) {
		tmp = x * y;
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.85e+118) or not (x <= 1.22e+64):
		tmp = x * y
	else:
		tmp = 1.0 - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.85e+118) || !(x <= 1.22e+64))
		tmp = Float64(x * y);
	else
		tmp = Float64(1.0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.85e+118) || ~((x <= 1.22e+64)))
		tmp = x * y;
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.85e+118], N[Not[LessEqual[x, 1.22e+64]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(1.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.85000000000000001e118 or 1.21999999999999994e64 < x

   1. Initial program 59.0%

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

      1. +-commutative 59.0%

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

      2. remove-double-neg 59.0%

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

      3. unsub-neg 59.0%

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

      4. sub-neg 59.0%

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

      5. +-commutative 59.0%

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

      6. distribute-rgt-in 59.0%

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

      7. *-lft-identity 59.0%

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

      8. associate--l+ 84.1%

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

      9. associate--l- 100.0%

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

      10. sub-neg 100.0%

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

      11. +-inverses 100.0%

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

      12. metadata-eval 100.0%

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

      13. +-commutative 100.0%

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

      14. distribute-lft-neg-out 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. neg-sub0 100.0%

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

      17. associate--r- 100.0%

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

      18. metadata-eval 100.0%

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

      19. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 84.1%

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

      1. *-commutative 84.1%

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

   9. Simplified 84.1%

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

   if -2.85000000000000001e118 < x < 1.21999999999999994e64

   1. Initial program 89.4%

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

      1. +-commutative 89.4%

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

      2. remove-double-neg 89.4%

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

      3. unsub-neg 89.4%

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

      4. sub-neg 89.4%

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

      5. +-commutative 89.4%

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

      6. distribute-rgt-in 89.4%

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

      7. *-lft-identity 89.4%

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

      8. associate--l+ 89.4%

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

      9. associate--l- 100.0%

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

      10. sub-neg 100.0%

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

      11. +-inverses 100.0%

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

      12. metadata-eval 100.0%

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

      13. +-commutative 100.0%

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

      14. distribute-lft-neg-out 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. neg-sub0 100.0%

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

      17. associate--r- 100.0%

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

      18. metadata-eval 100.0%

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

      19. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 87.2%

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

      1. neg-mul-1 87.2%

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

      2. unsub-neg 87.2%

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

   7. Simplified 87.2%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{+118} \lor \neg \left(x \leq 1.22 \cdot 10^{+64}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 2.05 \cdot 10^{-10}\right):\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 2.05e-10))) (- y) 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 2.05e-10)) {
		tmp = -y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 2.05d-10))) then
        tmp = -y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 2.05e-10)) {
		tmp = -y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 2.05e-10):
		tmp = -y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 2.05e-10))
		tmp = Float64(-y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 2.05e-10)))
		tmp = -y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 2.05e-10]], $MachinePrecision]], (-y), 1.0]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 2.0499999999999999e-10 < y

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. remove-double-neg 99.6%

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

      3. unsub-neg 99.6%

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

      4. sub-neg 99.6%

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

      5. +-commutative 99.6%

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

      6. distribute-rgt-in 99.6%

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

      7. *-lft-identity 99.6%

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

      8. associate--l+ 100.0%

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

      9. associate--l- 100.0%

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

      10. sub-neg 100.0%

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

      11. +-inverses 100.0%

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

      12. metadata-eval 100.0%

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

      13. +-commutative 100.0%

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

      14. distribute-lft-neg-out 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. neg-sub0 100.0%

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

      17. associate--r- 100.0%

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

      18. metadata-eval 100.0%

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

      19. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around inf 99.6%

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

   8. Taylor expanded in x around 0 45.4%

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

      1. neg-mul-1 45.4%

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

   10. Simplified 45.4%

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

   if -1 < y < 2.0499999999999999e-10

   1. Initial program 49.7%

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

      1. +-commutative 49.7%

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

      2. remove-double-neg 49.7%

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

      3. unsub-neg 49.7%

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

      4. sub-neg 49.7%

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

      5. +-commutative 49.7%

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

      6. distribute-rgt-in 49.7%

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

      7. *-lft-identity 49.7%

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

      8. associate--l+ 71.5%

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

      9. associate--l- 100.0%

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

      10. sub-neg 100.0%

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

      11. +-inverses 100.0%

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

      12. metadata-eval 100.0%

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

      13. +-commutative 100.0%

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

      14. distribute-lft-neg-out 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. neg-sub0 100.0%

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

      17. associate--r- 100.0%

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

      18. metadata-eval 100.0%

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

      19. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 74.9%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 2.05 \cdot 10^{-10}\right):\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 77.4%

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

   1. +-commutative 77.4%

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

   2. remove-double-neg 77.4%

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

   3. unsub-neg 77.4%

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

   4. sub-neg 77.4%

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

   5. +-commutative 77.4%

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

   6. distribute-rgt-in 77.4%

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

   7. *-lft-identity 77.4%

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

   8. associate--l+ 87.3%

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

   9. associate--l- 100.0%

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

   10. sub-neg 100.0%

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

   11. +-inverses 100.0%

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

   12. metadata-eval 100.0%

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

   13. +-commutative 100.0%

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

   14. distribute-lft-neg-out 100.0%

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

   15. distribute-rgt-neg-in 100.0%

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

   16. neg-sub0 100.0%

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

   17. associate--r- 100.0%

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

   18. metadata-eval 100.0%

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

   19. +-commutative 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 9: 38.1% accurate, 9.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 77.4%

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

   1. +-commutative 77.4%

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

   2. remove-double-neg 77.4%

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

   3. unsub-neg 77.4%

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

   4. sub-neg 77.4%

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

   5. +-commutative 77.4%

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

   6. distribute-rgt-in 77.4%

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

   7. *-lft-identity 77.4%

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

   8. associate--l+ 87.3%

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

   9. associate--l- 100.0%

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

   10. sub-neg 100.0%

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

   11. +-inverses 100.0%

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

   12. metadata-eval 100.0%

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

   13. +-commutative 100.0%

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

   14. distribute-lft-neg-out 100.0%

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

   15. distribute-rgt-neg-in 100.0%

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

   16. neg-sub0 100.0%

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

   17. associate--r- 100.0%

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

   18. metadata-eval 100.0%

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

   19. +-commutative 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in y around 0 34.8%

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

6. Final simplification 34.8%

   \[\leadsto 1 \]
7. Add Preprocessing

Developer target: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024073 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.Vectors:renderPlotVectors from Chart-1.5.3"
  :precision binary64

  :alt
  (- (* y x) (- y 1.0))

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