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

Percentage Accurate: 76.6% → 100.0%

Time: 3.3s

Alternatives: 9

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: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.6%

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

   1. +-commutative 74.6%

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

   2. sub-neg 74.6%

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

   3. +-commutative 74.6%

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

   4. distribute-rgt-in 74.6%

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

   5. *-lft-identity 74.6%

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

   6. associate-+r- 74.6%

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

   7. associate-+l- 100.0%

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

   8. +-inverses 100.0%

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

   9. --rgt-identity 100.0%

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

   10. +-commutative 100.0%

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

   11. distribute-lft-neg-out 100.0%

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

   12. distribute-rgt-neg-in 100.0%

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

   13. mul-1-neg 100.0%

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

   14. mul-1-neg 100.0%

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

   15. neg-sub0 100.0%

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

   16. associate--r- 100.0%

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

   17. metadata-eval 100.0%

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

   18. +-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. Taylor expanded in y around 0 100.0%

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

   1. distribute-lft-out-- 100.0%

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

   2. cancel-sign-sub-inv 100.0%

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

   3. *-rgt-identity 100.0%

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

   4. unsub-neg 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 61.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-6}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-65}:\\ \;\;\;\;-y\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-293}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-271}:\\ \;\;\;\;-y\\ \mathbf{elif}\;x \leq 1.42 \cdot 10^{-129}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1e-6)
   (* y x)
   (if (<= x -1.45e-65)
     (- y)
     (if (<= x -9.2e-293)
       1.0
       (if (<= x 2.6e-271)
         (- y)
         (if (<= x 1.42e-129) 1.0 (if (<= x 1.0) (- y) (* y x))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1e-6) {
		tmp = y * x;
	} else if (x <= -1.45e-65) {
		tmp = -y;
	} else if (x <= -9.2e-293) {
		tmp = 1.0;
	} else if (x <= 2.6e-271) {
		tmp = -y;
	} else if (x <= 1.42e-129) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = -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 <= (-1d-6)) then
        tmp = y * x
    else if (x <= (-1.45d-65)) then
        tmp = -y
    else if (x <= (-9.2d-293)) then
        tmp = 1.0d0
    else if (x <= 2.6d-271) then
        tmp = -y
    else if (x <= 1.42d-129) then
        tmp = 1.0d0
    else if (x <= 1.0d0) then
        tmp = -y
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1e-6) {
		tmp = y * x;
	} else if (x <= -1.45e-65) {
		tmp = -y;
	} else if (x <= -9.2e-293) {
		tmp = 1.0;
	} else if (x <= 2.6e-271) {
		tmp = -y;
	} else if (x <= 1.42e-129) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = -y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1e-6:
		tmp = y * x
	elif x <= -1.45e-65:
		tmp = -y
	elif x <= -9.2e-293:
		tmp = 1.0
	elif x <= 2.6e-271:
		tmp = -y
	elif x <= 1.42e-129:
		tmp = 1.0
	elif x <= 1.0:
		tmp = -y
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1e-6)
		tmp = Float64(y * x);
	elseif (x <= -1.45e-65)
		tmp = Float64(-y);
	elseif (x <= -9.2e-293)
		tmp = 1.0;
	elseif (x <= 2.6e-271)
		tmp = Float64(-y);
	elseif (x <= 1.42e-129)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = Float64(-y);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1e-6)
		tmp = y * x;
	elseif (x <= -1.45e-65)
		tmp = -y;
	elseif (x <= -9.2e-293)
		tmp = 1.0;
	elseif (x <= 2.6e-271)
		tmp = -y;
	elseif (x <= 1.42e-129)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = -y;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1e-6], N[(y * x), $MachinePrecision], If[LessEqual[x, -1.45e-65], (-y), If[LessEqual[x, -9.2e-293], 1.0, If[LessEqual[x, 2.6e-271], (-y), If[LessEqual[x, 1.42e-129], 1.0, If[LessEqual[x, 1.0], (-y), N[(y * x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.99999999999999955e-7 or 1 < x

   1. Initial program 53.9%

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

      1. +-commutative 53.9%

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

      2. sub-neg 53.9%

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

      3. +-commutative 53.9%

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

      4. distribute-rgt-in 53.9%

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

      5. *-lft-identity 53.9%

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

      6. associate-+r- 53.9%

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

      7. associate-+l- 100.0%

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

      8. +-inverses 100.0%

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

      9. --rgt-identity 100.0%

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

      10. +-commutative 100.0%

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

      11. distribute-lft-neg-out 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. mul-1-neg 100.0%

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

      14. mul-1-neg 100.0%

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

      15. neg-sub0 100.0%

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

      16. associate--r- 100.0%

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

      17. metadata-eval 100.0%

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

      18. +-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. Taylor expanded in x around inf 98.7%

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

   5. Taylor expanded in y around inf 74.9%

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

   if -9.99999999999999955e-7 < x < -1.4499999999999999e-65 or -9.1999999999999998e-293 < x < 2.6e-271 or 1.42e-129 < x < 1

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. distribute-rgt-in 99.9%

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

      5. *-lft-identity 99.9%

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

      6. associate-+r- 99.9%

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

      7. associate-+l- 99.9%

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

      8. +-inverses 99.9%

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

      9. --rgt-identity 99.9%

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

      10. +-commutative 99.9%

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

      11. distribute-lft-neg-out 99.9%

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

      12. distribute-rgt-neg-in 99.9%

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

      13. mul-1-neg 99.9%

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

      14. mul-1-neg 99.9%

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

      15. neg-sub0 99.9%

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

      16. associate--r- 99.9%

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

      17. metadata-eval 99.9%

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

      18. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 99.9%

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

      1. distribute-lft-out-- 100.0%

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

      2. cancel-sign-sub-inv 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. unsub-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around inf 71.6%

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

   8. Taylor expanded in x around 0 69.0%

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

      1. mul-1-neg 69.0%

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

   10. Simplified 69.0%

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

   if -1.4499999999999999e-65 < x < -9.1999999999999998e-293 or 2.6e-271 < x < 1.42e-129

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-in 100.0%

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

      5. *-lft-identity 100.0%

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

      6. associate-+r- 100.0%

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

      7. associate-+l- 100.0%

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

      8. +-inverses 100.0%

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

      9. --rgt-identity 100.0%

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

      10. +-commutative 100.0%

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

      11. distribute-lft-neg-out 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. mul-1-neg 100.0%

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

      14. mul-1-neg 100.0%

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

      15. neg-sub0 100.0%

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

      16. associate--r- 100.0%

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

      17. metadata-eval 100.0%

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

      18. +-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. Taylor expanded in y around 0 64.5%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-6}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-65}:\\ \;\;\;\;-y\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-293}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-271}:\\ \;\;\;\;-y\\ \mathbf{elif}\;x \leq 1.42 \cdot 10^{-129}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - y \leq -100 \lor \neg \left(1 - y \leq 1.0000002\right):\\ \;\;\;\;y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (- 1.0 y) -100.0) (not (<= (- 1.0 y) 1.0000002)))
   (* y (+ x -1.0))
   (+ 1.0 (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 - y) <= -100.0) || !((1.0 - y) <= 1.0000002)) {
		tmp = y * (x + -1.0);
	} else {
		tmp = 1.0 + (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 (((1.0d0 - y) <= (-100.0d0)) .or. (.not. ((1.0d0 - y) <= 1.0000002d0))) then
        tmp = y * (x + (-1.0d0))
    else
        tmp = 1.0d0 + (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((1.0 - y) <= -100.0) || !((1.0 - y) <= 1.0000002)) {
		tmp = y * (x + -1.0);
	} else {
		tmp = 1.0 + (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((1.0 - y) <= -100.0) or not ((1.0 - y) <= 1.0000002):
		tmp = y * (x + -1.0)
	else:
		tmp = 1.0 + (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(1.0 - y) <= -100.0) || !(Float64(1.0 - y) <= 1.0000002))
		tmp = Float64(y * Float64(x + -1.0));
	else
		tmp = Float64(1.0 + Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((1.0 - y) <= -100.0) || ~(((1.0 - y) <= 1.0000002)))
		tmp = y * (x + -1.0);
	else
		tmp = 1.0 + (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(1.0 - y), $MachinePrecision], -100.0], N[Not[LessEqual[N[(1.0 - y), $MachinePrecision], 1.0000002]], $MachinePrecision]], N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 y) < -100 or 1.00000019999999989 < (-.f64 1 y)

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-rgt-in 99.7%

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

      5. *-lft-identity 99.7%

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

      6. associate-+r- 99.7%

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

      7. associate-+l- 100.0%

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

      8. +-inverses 100.0%

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

      9. --rgt-identity 100.0%

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

      10. +-commutative 100.0%

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

      11. distribute-lft-neg-out 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. mul-1-neg 100.0%

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

      14. mul-1-neg 100.0%

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

      15. neg-sub0 100.0%

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

      16. associate--r- 100.0%

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

      17. metadata-eval 100.0%

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

      18. +-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. Taylor expanded in y around 0 100.0%

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

      1. distribute-lft-out-- 100.0%

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

      2. cancel-sign-sub-inv 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. unsub-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around inf 99.4%

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

   if -100 < (-.f64 1 y) < 1.00000019999999989

   1. Initial program 50.3%

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

      1. +-commutative 50.3%

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

      2. sub-neg 50.3%

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

      3. +-commutative 50.3%

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

      4. distribute-rgt-in 50.3%

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

      5. *-lft-identity 50.3%

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

      6. associate-+r- 50.3%

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

      7. associate-+l- 100.0%

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

      8. +-inverses 100.0%

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

      9. --rgt-identity 100.0%

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

      10. +-commutative 100.0%

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

      11. distribute-lft-neg-out 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. mul-1-neg 100.0%

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

      14. mul-1-neg 100.0%

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

      15. neg-sub0 100.0%

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

      16. associate--r- 100.0%

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

      17. metadata-eval 100.0%

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

      18. +-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. Taylor expanded in x around inf 98.5%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -100 \lor \neg \left(1 - y \leq 1.0000002\right):\\ \;\;\;\;y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot x\\ \end{array} \]

Alternative 4: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - y \leq -100 \lor \neg \left(1 - y \leq 1.0000002\right):\\ \;\;\;\;y \cdot x - y\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (- 1.0 y) -100.0) (not (<= (- 1.0 y) 1.0000002)))
   (- (* y x) y)
   (+ 1.0 (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 - y) <= -100.0) || !((1.0 - y) <= 1.0000002)) {
		tmp = (y * x) - y;
	} else {
		tmp = 1.0 + (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 (((1.0d0 - y) <= (-100.0d0)) .or. (.not. ((1.0d0 - y) <= 1.0000002d0))) then
        tmp = (y * x) - y
    else
        tmp = 1.0d0 + (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((1.0 - y) <= -100.0) || !((1.0 - y) <= 1.0000002)) {
		tmp = (y * x) - y;
	} else {
		tmp = 1.0 + (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((1.0 - y) <= -100.0) or not ((1.0 - y) <= 1.0000002):
		tmp = (y * x) - y
	else:
		tmp = 1.0 + (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(1.0 - y) <= -100.0) || !(Float64(1.0 - y) <= 1.0000002))
		tmp = Float64(Float64(y * x) - y);
	else
		tmp = Float64(1.0 + Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((1.0 - y) <= -100.0) || ~(((1.0 - y) <= 1.0000002)))
		tmp = (y * x) - y;
	else
		tmp = 1.0 + (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(1.0 - y), $MachinePrecision], -100.0], N[Not[LessEqual[N[(1.0 - y), $MachinePrecision], 1.0000002]], $MachinePrecision]], N[(N[(y * x), $MachinePrecision] - y), $MachinePrecision], N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 y) < -100 or 1.00000019999999989 < (-.f64 1 y)

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-rgt-in 99.7%

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

      5. *-lft-identity 99.7%

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

      6. associate-+r- 99.7%

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

      7. associate-+l- 100.0%

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

      8. +-inverses 100.0%

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

      9. --rgt-identity 100.0%

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

      10. +-commutative 100.0%

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

      11. distribute-lft-neg-out 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. mul-1-neg 100.0%

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

      14. mul-1-neg 100.0%

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

      15. neg-sub0 100.0%

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

      16. associate--r- 100.0%

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

      17. metadata-eval 100.0%

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

      18. +-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. Taylor expanded in y around 0 100.0%

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

      1. distribute-lft-out-- 100.0%

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

      2. cancel-sign-sub-inv 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. unsub-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around inf 99.4%

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

   8. Taylor expanded in x around 0 99.4%

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

      1. mul-1-neg 99.4%

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

      2. unsub-neg 99.4%

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

   10. Simplified 99.4%

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

   if -100 < (-.f64 1 y) < 1.00000019999999989

   1. Initial program 50.3%

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

      1. +-commutative 50.3%

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

      2. sub-neg 50.3%

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

      3. +-commutative 50.3%

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

      4. distribute-rgt-in 50.3%

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

      5. *-lft-identity 50.3%

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

      6. associate-+r- 50.3%

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

      7. associate-+l- 100.0%

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

      8. +-inverses 100.0%

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

      9. --rgt-identity 100.0%

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

      10. +-commutative 100.0%

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

      11. distribute-lft-neg-out 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. mul-1-neg 100.0%

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

      14. mul-1-neg 100.0%

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

      15. neg-sub0 100.0%

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

      16. associate--r- 100.0%

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

      17. metadata-eval 100.0%

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

      18. +-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. Taylor expanded in x around inf 98.5%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -100 \lor \neg \left(1 - y \leq 1.0000002\right):\\ \;\;\;\;y \cdot x - y\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot x\\ \end{array} \]

Alternative 5: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-13} \lor \neg \left(x \leq 56000000000\right):\\ \;\;\;\;y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.5e-13) (not (<= x 56000000000.0)))
   (* y (+ x -1.0))
   (- 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.5e-13) || !(x <= 56000000000.0)) {
		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 ((x <= (-3.5d-13)) .or. (.not. (x <= 56000000000.0d0))) 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 ((x <= -3.5e-13) || !(x <= 56000000000.0)) {
		tmp = y * (x + -1.0);
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.5e-13) or not (x <= 56000000000.0):
		tmp = y * (x + -1.0)
	else:
		tmp = 1.0 - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.5e-13) || !(x <= 56000000000.0))
		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 ((x <= -3.5e-13) || ~((x <= 56000000000.0)))
		tmp = y * (x + -1.0);
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.5e-13], N[Not[LessEqual[x, 56000000000.0]], $MachinePrecision]], N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5000000000000002e-13 or 5.6e10 < x

   1. Initial program 55.0%

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

      1. +-commutative 55.0%

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

      2. sub-neg 55.0%

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

      3. +-commutative 55.0%

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

      4. distribute-rgt-in 55.0%

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

      5. *-lft-identity 55.0%

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

      6. associate-+r- 55.0%

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

      7. associate-+l- 100.0%

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

      8. +-inverses 100.0%

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

      9. --rgt-identity 100.0%

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

      10. +-commutative 100.0%

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

      11. distribute-lft-neg-out 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. mul-1-neg 100.0%

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

      14. mul-1-neg 100.0%

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

      15. neg-sub0 100.0%

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

      16. associate--r- 100.0%

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

      17. metadata-eval 100.0%

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

      18. +-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. Taylor expanded in y around 0 100.0%

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

      1. distribute-lft-out-- 100.0%

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

      2. cancel-sign-sub-inv 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. unsub-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around inf 77.0%

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

   if -3.5000000000000002e-13 < x < 5.6e10

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-rgt-in 99.8%

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

      5. *-lft-identity 99.8%

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

      6. associate-+r- 99.8%

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

      7. associate-+l- 100.0%

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

      8. +-inverses 100.0%

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

      9. --rgt-identity 100.0%

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

      10. +-commutative 100.0%

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

      11. distribute-lft-neg-out 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. mul-1-neg 100.0%

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

      14. mul-1-neg 100.0%

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

      15. neg-sub0 100.0%

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

      16. associate--r- 100.0%

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

      17. metadata-eval 100.0%

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

      18. +-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. Taylor expanded in x around 0 99.3%

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

      1. neg-mul-1 99.3%

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

      2. unsub-neg 99.3%

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

   6. Simplified 99.3%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-13} \lor \neg \left(x \leq 56000000000\right):\\ \;\;\;\;y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]

Alternative 6: 85.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -105000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 35000000000:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -105000.0) (* y x) (if (<= x 35000000000.0) (- 1.0 y) (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -105000.0) {
		tmp = y * x;
	} else if (x <= 35000000000.0) {
		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 <= (-105000.0d0)) then
        tmp = y * x
    else if (x <= 35000000000.0d0) 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 <= -105000.0) {
		tmp = y * x;
	} else if (x <= 35000000000.0) {
		tmp = 1.0 - y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -105000.0:
		tmp = y * x
	elif x <= 35000000000.0:
		tmp = 1.0 - y
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -105000.0)
		tmp = Float64(y * x);
	elseif (x <= 35000000000.0)
		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 <= -105000.0)
		tmp = y * x;
	elseif (x <= 35000000000.0)
		tmp = 1.0 - y;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -105000.0], N[(y * x), $MachinePrecision], If[LessEqual[x, 35000000000.0], N[(1.0 - y), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -105000 or 3.5e10 < x

   1. Initial program 53.4%

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

      1. +-commutative 53.4%

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

      2. sub-neg 53.4%

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

      3. +-commutative 53.4%

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

      4. distribute-rgt-in 53.5%

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

      5. *-lft-identity 53.5%

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

      6. associate-+r- 53.5%

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

      7. associate-+l- 100.0%

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

      8. +-inverses 100.0%

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

      9. --rgt-identity 100.0%

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

      10. +-commutative 100.0%

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

      11. distribute-lft-neg-out 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. mul-1-neg 100.0%

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

      14. mul-1-neg 100.0%

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

      15. neg-sub0 100.0%

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

      16. associate--r- 100.0%

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

      17. metadata-eval 100.0%

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

      18. +-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. Taylor expanded in x around inf 99.0%

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

   5. Taylor expanded in y around inf 75.9%

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

   if -105000 < x < 3.5e10

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-rgt-in 99.7%

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

      5. *-lft-identity 99.7%

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

      6. associate-+r- 99.7%

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

      7. associate-+l- 100.0%

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

      8. +-inverses 100.0%

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

      9. --rgt-identity 100.0%

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

      10. +-commutative 100.0%

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

      11. distribute-lft-neg-out 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. mul-1-neg 100.0%

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

      14. mul-1-neg 100.0%

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

      15. neg-sub0 100.0%

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

      16. associate--r- 100.0%

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

      17. metadata-eval 100.0%

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

      18. +-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. Taylor expanded in x around 0 98.3%

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

      1. neg-mul-1 98.3%

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

      2. unsub-neg 98.3%

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

   6. Simplified 98.3%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -105000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 35000000000:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

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 74.6%

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

   1. +-commutative 74.6%

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

   2. sub-neg 74.6%

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

   3. +-commutative 74.6%

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

   4. distribute-rgt-in 74.6%

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

   5. *-lft-identity 74.6%

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

   6. associate-+r- 74.6%

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

   7. associate-+l- 100.0%

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

   8. +-inverses 100.0%

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

   9. --rgt-identity 100.0%

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

   10. +-commutative 100.0%

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

   11. distribute-lft-neg-out 100.0%

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

   12. distribute-rgt-neg-in 100.0%

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

   13. mul-1-neg 100.0%

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

   14. mul-1-neg 100.0%

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

   15. neg-sub0 100.0%

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

   16. associate--r- 100.0%

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

   17. metadata-eval 100.0%

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

   18. +-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. Final simplification 100.0%

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

Alternative 8: 62.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 0.016:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) (- y) (if (<= y 0.016) 1.0 (- y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = -y;
	} else if (y <= 0.016) {
		tmp = 1.0;
	} else {
		tmp = -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = -y
    else if (y <= 0.016d0) then
        tmp = 1.0d0
    else
        tmp = -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = -y;
	} else if (y <= 0.016) {
		tmp = 1.0;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = -y
	elif y <= 0.016:
		tmp = 1.0
	else:
		tmp = -y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(-y);
	elseif (y <= 0.016)
		tmp = 1.0;
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = -y;
	elseif (y <= 0.016)
		tmp = 1.0;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], (-y), If[LessEqual[y, 0.016], 1.0, (-y)]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.016 < y

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. distribute-rgt-in 99.9%

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

      5. *-lft-identity 99.9%

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

      6. associate-+r- 99.9%

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

      7. associate-+l- 100.0%

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

      8. +-inverses 100.0%

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

      9. --rgt-identity 100.0%

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

      10. +-commutative 100.0%

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

      11. distribute-lft-neg-out 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. mul-1-neg 100.0%

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

      14. mul-1-neg 100.0%

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

      15. neg-sub0 100.0%

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

      16. associate--r- 100.0%

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

      17. metadata-eval 100.0%

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

      18. +-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. Taylor expanded in y around 0 100.0%

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

      1. distribute-lft-out-- 100.0%

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

      2. cancel-sign-sub-inv 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. unsub-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around inf 99.4%

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

   8. Taylor expanded in x around 0 45.0%

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

      1. mul-1-neg 45.0%

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

   10. Simplified 45.0%

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

   if -1 < y < 0.016

   1. Initial program 50.0%

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

      1. +-commutative 50.0%

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

      2. sub-neg 50.0%

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

      3. +-commutative 50.0%

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

      4. distribute-rgt-in 50.1%

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

      5. *-lft-identity 50.1%

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

      6. associate-+r- 50.1%

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

      7. associate-+l- 100.0%

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

      8. +-inverses 100.0%

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

      9. --rgt-identity 100.0%

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

      10. +-commutative 100.0%

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

      11. distribute-lft-neg-out 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. mul-1-neg 100.0%

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

      14. mul-1-neg 100.0%

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

      15. neg-sub0 100.0%

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

      16. associate--r- 100.0%

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

      17. metadata-eval 100.0%

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

      18. +-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. Taylor expanded in y around 0 69.6%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 0.016:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 9: 38.4% accurate, 9.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 74.6%

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

   1. +-commutative 74.6%

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

   2. sub-neg 74.6%

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

   3. +-commutative 74.6%

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

   4. distribute-rgt-in 74.6%

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

   5. *-lft-identity 74.6%

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

   6. associate-+r- 74.6%

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

   7. associate-+l- 100.0%

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

   8. +-inverses 100.0%

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

   9. --rgt-identity 100.0%

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

   10. +-commutative 100.0%

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

   11. distribute-lft-neg-out 100.0%

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

   12. distribute-rgt-neg-in 100.0%

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

   13. mul-1-neg 100.0%

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

   14. mul-1-neg 100.0%

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

   15. neg-sub0 100.0%

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

   16. associate--r- 100.0%

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

   17. metadata-eval 100.0%

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

   18. +-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. Taylor expanded in y around 0 36.7%

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

5. Final simplification 36.7%

   \[\leadsto 1 \]

Developer target: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023268 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.Vectors:renderPlotVectors from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (- (* y x) (- y 1.0))

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