Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, A

Percentage Accurate: 100.0% → 100.0%

Time: 5.7s

Alternatives: 10

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673))

C

double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * (y - 1.0d0)) - (y * 0.5d0)) + 0.918938533204673d0
end function

Java

public static double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

Python

def code(x, y):
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673

Julia

function code(x, y)
	return Float64(Float64(Float64(x * Float64(y - 1.0)) - Float64(y * 0.5)) + 0.918938533204673)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
end

Wolfram

code[x_, y_] := N[(N[(N[(x * N[(y - 1.0), $MachinePrecision]), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision] + 0.918938533204673), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673))

C

double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * (y - 1.0d0)) - (y * 0.5d0)) + 0.918938533204673d0
end function

Java

public static double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

Python

def code(x, y):
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673

Julia

function code(x, y)
	return Float64(Float64(Float64(x * Float64(y - 1.0)) - Float64(y * 0.5)) + 0.918938533204673)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
end

Wolfram

code[x_, y_] := N[(N[(N[(x * N[(y - 1.0), $MachinePrecision]), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision] + 0.918938533204673), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \left(y + -1\right) - y \cdot 0.5\right) + 0.918938533204673 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (- (* x (+ y -1.0)) (* y 0.5)) 0.918938533204673))

C

double code(double x, double y) {
	return ((x * (y + -1.0)) - (y * 0.5)) + 0.918938533204673;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * (y + (-1.0d0))) - (y * 0.5d0)) + 0.918938533204673d0
end function

Java

public static double code(double x, double y) {
	return ((x * (y + -1.0)) - (y * 0.5)) + 0.918938533204673;
}

Python

def code(x, y):
	return ((x * (y + -1.0)) - (y * 0.5)) + 0.918938533204673

Julia

function code(x, y)
	return Float64(Float64(Float64(x * Float64(y + -1.0)) - Float64(y * 0.5)) + 0.918938533204673)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * (y + -1.0)) - (y * 0.5)) + 0.918938533204673;
end

Wolfram

code[x_, y_] := N[(N[(N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision] + 0.918938533204673), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \left(x \cdot \left(y + -1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
4. Add Preprocessing

Alternative 2: 50.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-13}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -4.7 \cdot 10^{-157}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 27000:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+108}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.5e-13)
   (* y -0.5)
   (if (<= y -4.7e-157)
     (- x)
     (if (<= y 27000.0)
       0.918938533204673
       (if (<= y 6.5e+108) (* y -0.5) (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.5e-13) {
		tmp = y * -0.5;
	} else if (y <= -4.7e-157) {
		tmp = -x;
	} else if (y <= 27000.0) {
		tmp = 0.918938533204673;
	} else if (y <= 6.5e+108) {
		tmp = y * -0.5;
	} 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 (y <= (-9.5d-13)) then
        tmp = y * (-0.5d0)
    else if (y <= (-4.7d-157)) then
        tmp = -x
    else if (y <= 27000.0d0) then
        tmp = 0.918938533204673d0
    else if (y <= 6.5d+108) then
        tmp = y * (-0.5d0)
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9.5e-13) {
		tmp = y * -0.5;
	} else if (y <= -4.7e-157) {
		tmp = -x;
	} else if (y <= 27000.0) {
		tmp = 0.918938533204673;
	} else if (y <= 6.5e+108) {
		tmp = y * -0.5;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9.5e-13:
		tmp = y * -0.5
	elif y <= -4.7e-157:
		tmp = -x
	elif y <= 27000.0:
		tmp = 0.918938533204673
	elif y <= 6.5e+108:
		tmp = y * -0.5
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.5e-13)
		tmp = Float64(y * -0.5);
	elseif (y <= -4.7e-157)
		tmp = Float64(-x);
	elseif (y <= 27000.0)
		tmp = 0.918938533204673;
	elseif (y <= 6.5e+108)
		tmp = Float64(y * -0.5);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -9.5e-13)
		tmp = y * -0.5;
	elseif (y <= -4.7e-157)
		tmp = -x;
	elseif (y <= 27000.0)
		tmp = 0.918938533204673;
	elseif (y <= 6.5e+108)
		tmp = y * -0.5;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.5e-13], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, -4.7e-157], (-x), If[LessEqual[y, 27000.0], 0.918938533204673, If[LessEqual[y, 6.5e+108], N[(y * -0.5), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -9.49999999999999991e-13 or 27000 < y < 6.4999999999999996e108

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 98.8%

      \[\leadsto \color{blue}{x \cdot y} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   6. Taylor expanded in y around inf 94.9%

      \[\leadsto x \cdot y - \color{blue}{0.5 \cdot y} \]
   7. Step-by-step derivation

      1. *-commutative 94.9%

         \[\leadsto x \cdot y - \color{blue}{y \cdot 0.5} \]

   8. Simplified 94.9%

      \[\leadsto x \cdot y - \color{blue}{y \cdot 0.5} \]

   9. Taylor expanded in x around 0 62.8%

      \[\leadsto \color{blue}{-0.5 \cdot y} \]
   10. Step-by-step derivation

       1. *-commutative 62.8%

          \[\leadsto \color{blue}{y \cdot -0.5} \]

   11. Simplified 62.8%

       \[\leadsto \color{blue}{y \cdot -0.5} \]

   if -9.49999999999999991e-13 < y < -4.7000000000000002e-157

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\left(0.918938533204673 + x \cdot \left(y - 1\right)\right) - 0.5 \cdot y} \]
   6. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \color{blue}{\left(0.918938533204673 + x \cdot \left(y - 1\right)\right) + \left(-0.5 \cdot y\right)} \]

      2. distribute-lft-out-- 100.0%

         \[\leadsto \left(0.918938533204673 + \color{blue}{\left(x \cdot y - x \cdot 1\right)}\right) + \left(-0.5 \cdot y\right) \]

      3. cancel-sign-sub-inv 100.0%

         \[\leadsto \left(0.918938533204673 + \color{blue}{\left(x \cdot y + \left(-x\right) \cdot 1\right)}\right) + \left(-0.5 \cdot y\right) \]

      4. *-rgt-identity 100.0%

         \[\leadsto \left(0.918938533204673 + \left(x \cdot y + \color{blue}{\left(-x\right)}\right)\right) + \left(-0.5 \cdot y\right) \]

      5. +-commutative 100.0%

         \[\leadsto \left(0.918938533204673 + \color{blue}{\left(\left(-x\right) + x \cdot y\right)}\right) + \left(-0.5 \cdot y\right) \]

      6. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(0.918938533204673 + \left(-x\right)\right) + x \cdot y\right)} + \left(-0.5 \cdot y\right) \]

      7. neg-mul-1 100.0%

         \[\leadsto \left(\left(0.918938533204673 + \color{blue}{-1 \cdot x}\right) + x \cdot y\right) + \left(-0.5 \cdot y\right) \]

      8. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(0.918938533204673 + -1 \cdot x\right) + \left(x \cdot y + \left(-0.5 \cdot y\right)\right)} \]

      9. distribute-lft-neg-in 100.0%

         \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \left(x \cdot y + \color{blue}{\left(-0.5\right) \cdot y}\right) \]

      10. cancel-sign-sub-inv 100.0%

          \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \color{blue}{\left(x \cdot y - 0.5 \cdot y\right)} \]

      11. distribute-rgt-out-- 100.0%

          \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \color{blue}{y \cdot \left(x - 0.5\right)} \]

      12. neg-mul-1 100.0%

          \[\leadsto \left(0.918938533204673 + \color{blue}{\left(-x\right)}\right) + y \cdot \left(x - 0.5\right) \]

      13. associate-+r+ 100.0%

          \[\leadsto \color{blue}{0.918938533204673 + \left(\left(-x\right) + y \cdot \left(x - 0.5\right)\right)} \]

      14. +-commutative 100.0%

          \[\leadsto 0.918938533204673 + \color{blue}{\left(y \cdot \left(x - 0.5\right) + \left(-x\right)\right)} \]

      15. unsub-neg 100.0%

          \[\leadsto 0.918938533204673 + \color{blue}{\left(y \cdot \left(x - 0.5\right) - x\right)} \]

      16. sub-neg 100.0%

          \[\leadsto 0.918938533204673 + \left(y \cdot \color{blue}{\left(x + \left(-0.5\right)\right)} - x\right) \]

      17. metadata-eval 100.0%

          \[\leadsto 0.918938533204673 + \left(y \cdot \left(x + \color{blue}{-0.5}\right) - x\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0.918938533204673 + \left(y \cdot \left(x + -0.5\right) - x\right)} \]

   8. Taylor expanded in x around inf 100.0%

      \[\leadsto 0.918938533204673 + \color{blue}{x \cdot \left(y - 1\right)} \]

   9. Taylor expanded in x around inf 63.1%

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

   10. Taylor expanded in y around 0 63.1%

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

       1. mul-1-neg 63.1%

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

   12. Simplified 63.1%

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

   if -4.7000000000000002e-157 < y < 27000

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 99.0%

      \[\leadsto \color{blue}{0.918938533204673 + -1 \cdot x} \]
   6. Step-by-step derivation

      1. neg-mul-1 99.0%

         \[\leadsto 0.918938533204673 + \color{blue}{\left(-x\right)} \]

      2. unsub-neg 99.0%

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   7. Simplified 99.0%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]

   8. Taylor expanded in x around 0 58.3%

      \[\leadsto \color{blue}{0.918938533204673} \]

   if 6.4999999999999996e108 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\left(0.918938533204673 + x \cdot \left(y - 1\right)\right) - 0.5 \cdot y} \]
   6. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \color{blue}{\left(0.918938533204673 + x \cdot \left(y - 1\right)\right) + \left(-0.5 \cdot y\right)} \]

      2. distribute-lft-out-- 100.0%

         \[\leadsto \left(0.918938533204673 + \color{blue}{\left(x \cdot y - x \cdot 1\right)}\right) + \left(-0.5 \cdot y\right) \]

      3. cancel-sign-sub-inv 100.0%

         \[\leadsto \left(0.918938533204673 + \color{blue}{\left(x \cdot y + \left(-x\right) \cdot 1\right)}\right) + \left(-0.5 \cdot y\right) \]

      4. *-rgt-identity 100.0%

         \[\leadsto \left(0.918938533204673 + \left(x \cdot y + \color{blue}{\left(-x\right)}\right)\right) + \left(-0.5 \cdot y\right) \]

      5. +-commutative 100.0%

         \[\leadsto \left(0.918938533204673 + \color{blue}{\left(\left(-x\right) + x \cdot y\right)}\right) + \left(-0.5 \cdot y\right) \]

      6. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(0.918938533204673 + \left(-x\right)\right) + x \cdot y\right)} + \left(-0.5 \cdot y\right) \]

      7. neg-mul-1 100.0%

         \[\leadsto \left(\left(0.918938533204673 + \color{blue}{-1 \cdot x}\right) + x \cdot y\right) + \left(-0.5 \cdot y\right) \]

      8. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(0.918938533204673 + -1 \cdot x\right) + \left(x \cdot y + \left(-0.5 \cdot y\right)\right)} \]

      9. distribute-lft-neg-in 100.0%

         \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \left(x \cdot y + \color{blue}{\left(-0.5\right) \cdot y}\right) \]

      10. cancel-sign-sub-inv 100.0%

          \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \color{blue}{\left(x \cdot y - 0.5 \cdot y\right)} \]

      11. distribute-rgt-out-- 100.0%

          \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \color{blue}{y \cdot \left(x - 0.5\right)} \]

      12. neg-mul-1 100.0%

          \[\leadsto \left(0.918938533204673 + \color{blue}{\left(-x\right)}\right) + y \cdot \left(x - 0.5\right) \]

      13. associate-+r+ 100.0%

          \[\leadsto \color{blue}{0.918938533204673 + \left(\left(-x\right) + y \cdot \left(x - 0.5\right)\right)} \]

      14. +-commutative 100.0%

          \[\leadsto 0.918938533204673 + \color{blue}{\left(y \cdot \left(x - 0.5\right) + \left(-x\right)\right)} \]

      15. unsub-neg 100.0%

          \[\leadsto 0.918938533204673 + \color{blue}{\left(y \cdot \left(x - 0.5\right) - x\right)} \]

      16. sub-neg 100.0%

          \[\leadsto 0.918938533204673 + \left(y \cdot \color{blue}{\left(x + \left(-0.5\right)\right)} - x\right) \]

      17. metadata-eval 100.0%

          \[\leadsto 0.918938533204673 + \left(y \cdot \left(x + \color{blue}{-0.5}\right) - x\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0.918938533204673 + \left(y \cdot \left(x + -0.5\right) - x\right)} \]

   8. Taylor expanded in x around inf 66.3%

      \[\leadsto 0.918938533204673 + \color{blue}{x \cdot \left(y - 1\right)} \]

   9. Taylor expanded in x around inf 66.7%

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

   10. Taylor expanded in y around inf 66.7%

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

Alternative 3: 73.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -26:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+110}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -26.0)
   (* y -0.5)
   (if (<= y 1.85)
     (- 0.918938533204673 x)
     (if (<= y 4.5e+110) (* y -0.5) (* x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -26.0) {
		tmp = y * -0.5;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 4.5e+110) {
		tmp = y * -0.5;
	} 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 (y <= (-26.0d0)) then
        tmp = y * (-0.5d0)
    else if (y <= 1.85d0) then
        tmp = 0.918938533204673d0 - x
    else if (y <= 4.5d+110) then
        tmp = y * (-0.5d0)
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -26.0) {
		tmp = y * -0.5;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 4.5e+110) {
		tmp = y * -0.5;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -26.0:
		tmp = y * -0.5
	elif y <= 1.85:
		tmp = 0.918938533204673 - x
	elif y <= 4.5e+110:
		tmp = y * -0.5
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -26.0)
		tmp = Float64(y * -0.5);
	elseif (y <= 1.85)
		tmp = Float64(0.918938533204673 - x);
	elseif (y <= 4.5e+110)
		tmp = Float64(y * -0.5);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -26.0)
		tmp = y * -0.5;
	elseif (y <= 1.85)
		tmp = 0.918938533204673 - x;
	elseif (y <= 4.5e+110)
		tmp = y * -0.5;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -26.0], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 1.85], N[(0.918938533204673 - x), $MachinePrecision], If[LessEqual[y, 4.5e+110], N[(y * -0.5), $MachinePrecision], N[(x * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -26 or 1.8500000000000001 < y < 4.5000000000000003e110

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 98.1%

      \[\leadsto \color{blue}{x \cdot y} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   6. Taylor expanded in y around inf 96.3%

      \[\leadsto x \cdot y - \color{blue}{0.5 \cdot y} \]
   7. Step-by-step derivation

      1. *-commutative 96.3%

         \[\leadsto x \cdot y - \color{blue}{y \cdot 0.5} \]

   8. Simplified 96.3%

      \[\leadsto x \cdot y - \color{blue}{y \cdot 0.5} \]

   9. Taylor expanded in x around 0 63.3%

      \[\leadsto \color{blue}{-0.5 \cdot y} \]
   10. Step-by-step derivation

       1. *-commutative 63.3%

          \[\leadsto \color{blue}{y \cdot -0.5} \]

   11. Simplified 63.3%

       \[\leadsto \color{blue}{y \cdot -0.5} \]

   if -26 < y < 1.8500000000000001

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 99.3%

      \[\leadsto \color{blue}{0.918938533204673 + -1 \cdot x} \]
   6. Step-by-step derivation

      1. neg-mul-1 99.3%

         \[\leadsto 0.918938533204673 + \color{blue}{\left(-x\right)} \]

      2. unsub-neg 99.3%

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   7. Simplified 99.3%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]

   if 4.5000000000000003e110 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\left(0.918938533204673 + x \cdot \left(y - 1\right)\right) - 0.5 \cdot y} \]
   6. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \color{blue}{\left(0.918938533204673 + x \cdot \left(y - 1\right)\right) + \left(-0.5 \cdot y\right)} \]

      2. distribute-lft-out-- 100.0%

         \[\leadsto \left(0.918938533204673 + \color{blue}{\left(x \cdot y - x \cdot 1\right)}\right) + \left(-0.5 \cdot y\right) \]

      3. cancel-sign-sub-inv 100.0%

         \[\leadsto \left(0.918938533204673 + \color{blue}{\left(x \cdot y + \left(-x\right) \cdot 1\right)}\right) + \left(-0.5 \cdot y\right) \]

      4. *-rgt-identity 100.0%

         \[\leadsto \left(0.918938533204673 + \left(x \cdot y + \color{blue}{\left(-x\right)}\right)\right) + \left(-0.5 \cdot y\right) \]

      5. +-commutative 100.0%

         \[\leadsto \left(0.918938533204673 + \color{blue}{\left(\left(-x\right) + x \cdot y\right)}\right) + \left(-0.5 \cdot y\right) \]

      6. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(0.918938533204673 + \left(-x\right)\right) + x \cdot y\right)} + \left(-0.5 \cdot y\right) \]

      7. neg-mul-1 100.0%

         \[\leadsto \left(\left(0.918938533204673 + \color{blue}{-1 \cdot x}\right) + x \cdot y\right) + \left(-0.5 \cdot y\right) \]

      8. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(0.918938533204673 + -1 \cdot x\right) + \left(x \cdot y + \left(-0.5 \cdot y\right)\right)} \]

      9. distribute-lft-neg-in 100.0%

         \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \left(x \cdot y + \color{blue}{\left(-0.5\right) \cdot y}\right) \]

      10. cancel-sign-sub-inv 100.0%

          \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \color{blue}{\left(x \cdot y - 0.5 \cdot y\right)} \]

      11. distribute-rgt-out-- 100.0%

          \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \color{blue}{y \cdot \left(x - 0.5\right)} \]

      12. neg-mul-1 100.0%

          \[\leadsto \left(0.918938533204673 + \color{blue}{\left(-x\right)}\right) + y \cdot \left(x - 0.5\right) \]

      13. associate-+r+ 100.0%

          \[\leadsto \color{blue}{0.918938533204673 + \left(\left(-x\right) + y \cdot \left(x - 0.5\right)\right)} \]

      14. +-commutative 100.0%

          \[\leadsto 0.918938533204673 + \color{blue}{\left(y \cdot \left(x - 0.5\right) + \left(-x\right)\right)} \]

      15. unsub-neg 100.0%

          \[\leadsto 0.918938533204673 + \color{blue}{\left(y \cdot \left(x - 0.5\right) - x\right)} \]

      16. sub-neg 100.0%

          \[\leadsto 0.918938533204673 + \left(y \cdot \color{blue}{\left(x + \left(-0.5\right)\right)} - x\right) \]

      17. metadata-eval 100.0%

          \[\leadsto 0.918938533204673 + \left(y \cdot \left(x + \color{blue}{-0.5}\right) - x\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0.918938533204673 + \left(y \cdot \left(x + -0.5\right) - x\right)} \]

   8. Taylor expanded in x around inf 66.3%

      \[\leadsto 0.918938533204673 + \color{blue}{x \cdot \left(y - 1\right)} \]

   9. Taylor expanded in x around inf 66.7%

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

   10. Taylor expanded in y around inf 66.7%

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

Alternative 4: 49.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.92:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 0.92:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+114}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.92)
   (- x)
   (if (<= x 0.92) 0.918938533204673 (if (<= x 9.5e+114) (* x y) (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.92) {
		tmp = -x;
	} else if (x <= 0.92) {
		tmp = 0.918938533204673;
	} else if (x <= 9.5e+114) {
		tmp = x * y;
	} else {
		tmp = -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 <= (-0.92d0)) then
        tmp = -x
    else if (x <= 0.92d0) then
        tmp = 0.918938533204673d0
    else if (x <= 9.5d+114) then
        tmp = x * y
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.92) {
		tmp = -x;
	} else if (x <= 0.92) {
		tmp = 0.918938533204673;
	} else if (x <= 9.5e+114) {
		tmp = x * y;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.92:
		tmp = -x
	elif x <= 0.92:
		tmp = 0.918938533204673
	elif x <= 9.5e+114:
		tmp = x * y
	else:
		tmp = -x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.92)
		tmp = Float64(-x);
	elseif (x <= 0.92)
		tmp = 0.918938533204673;
	elseif (x <= 9.5e+114)
		tmp = Float64(x * y);
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.92)
		tmp = -x;
	elseif (x <= 0.92)
		tmp = 0.918938533204673;
	elseif (x <= 9.5e+114)
		tmp = x * y;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.92], (-x), If[LessEqual[x, 0.92], 0.918938533204673, If[LessEqual[x, 9.5e+114], N[(x * y), $MachinePrecision], (-x)]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.92000000000000004 or 9.5000000000000001e114 < x

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\left(0.918938533204673 + x \cdot \left(y - 1\right)\right) - 0.5 \cdot y} \]
   6. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \color{blue}{\left(0.918938533204673 + x \cdot \left(y - 1\right)\right) + \left(-0.5 \cdot y\right)} \]

      2. distribute-lft-out-- 100.0%

         \[\leadsto \left(0.918938533204673 + \color{blue}{\left(x \cdot y - x \cdot 1\right)}\right) + \left(-0.5 \cdot y\right) \]

      3. cancel-sign-sub-inv 100.0%

         \[\leadsto \left(0.918938533204673 + \color{blue}{\left(x \cdot y + \left(-x\right) \cdot 1\right)}\right) + \left(-0.5 \cdot y\right) \]

      4. *-rgt-identity 100.0%

         \[\leadsto \left(0.918938533204673 + \left(x \cdot y + \color{blue}{\left(-x\right)}\right)\right) + \left(-0.5 \cdot y\right) \]

      5. +-commutative 100.0%

         \[\leadsto \left(0.918938533204673 + \color{blue}{\left(\left(-x\right) + x \cdot y\right)}\right) + \left(-0.5 \cdot y\right) \]

      6. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(0.918938533204673 + \left(-x\right)\right) + x \cdot y\right)} + \left(-0.5 \cdot y\right) \]

      7. neg-mul-1 100.0%

         \[\leadsto \left(\left(0.918938533204673 + \color{blue}{-1 \cdot x}\right) + x \cdot y\right) + \left(-0.5 \cdot y\right) \]

      8. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(0.918938533204673 + -1 \cdot x\right) + \left(x \cdot y + \left(-0.5 \cdot y\right)\right)} \]

      9. distribute-lft-neg-in 100.0%

         \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \left(x \cdot y + \color{blue}{\left(-0.5\right) \cdot y}\right) \]

      10. cancel-sign-sub-inv 100.0%

          \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \color{blue}{\left(x \cdot y - 0.5 \cdot y\right)} \]

      11. distribute-rgt-out-- 100.0%

          \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \color{blue}{y \cdot \left(x - 0.5\right)} \]

      12. neg-mul-1 100.0%

          \[\leadsto \left(0.918938533204673 + \color{blue}{\left(-x\right)}\right) + y \cdot \left(x - 0.5\right) \]

      13. associate-+r+ 100.0%

          \[\leadsto \color{blue}{0.918938533204673 + \left(\left(-x\right) + y \cdot \left(x - 0.5\right)\right)} \]

      14. +-commutative 100.0%

          \[\leadsto 0.918938533204673 + \color{blue}{\left(y \cdot \left(x - 0.5\right) + \left(-x\right)\right)} \]

      15. unsub-neg 100.0%

          \[\leadsto 0.918938533204673 + \color{blue}{\left(y \cdot \left(x - 0.5\right) - x\right)} \]

      16. sub-neg 100.0%

          \[\leadsto 0.918938533204673 + \left(y \cdot \color{blue}{\left(x + \left(-0.5\right)\right)} - x\right) \]

      17. metadata-eval 100.0%

          \[\leadsto 0.918938533204673 + \left(y \cdot \left(x + \color{blue}{-0.5}\right) - x\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0.918938533204673 + \left(y \cdot \left(x + -0.5\right) - x\right)} \]

   8. Taylor expanded in x around inf 100.0%

      \[\leadsto 0.918938533204673 + \color{blue}{x \cdot \left(y - 1\right)} \]

   9. Taylor expanded in x around inf 99.9%

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

   10. Taylor expanded in y around 0 60.1%

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

       1. mul-1-neg 60.1%

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

   12. Simplified 60.1%

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

   if -0.92000000000000004 < x < 0.92000000000000004

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 53.2%

      \[\leadsto \color{blue}{0.918938533204673 + -1 \cdot x} \]
   6. Step-by-step derivation

      1. neg-mul-1 53.2%

         \[\leadsto 0.918938533204673 + \color{blue}{\left(-x\right)} \]

      2. unsub-neg 53.2%

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   7. Simplified 53.2%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]

   8. Taylor expanded in x around 0 53.0%

      \[\leadsto \color{blue}{0.918938533204673} \]

   if 0.92000000000000004 < x < 9.5000000000000001e114

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\left(0.918938533204673 + x \cdot \left(y - 1\right)\right) - 0.5 \cdot y} \]
   6. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \color{blue}{\left(0.918938533204673 + x \cdot \left(y - 1\right)\right) + \left(-0.5 \cdot y\right)} \]

      2. distribute-lft-out-- 99.9%

         \[\leadsto \left(0.918938533204673 + \color{blue}{\left(x \cdot y - x \cdot 1\right)}\right) + \left(-0.5 \cdot y\right) \]

      3. cancel-sign-sub-inv 99.9%

         \[\leadsto \left(0.918938533204673 + \color{blue}{\left(x \cdot y + \left(-x\right) \cdot 1\right)}\right) + \left(-0.5 \cdot y\right) \]

      4. *-rgt-identity 99.9%

         \[\leadsto \left(0.918938533204673 + \left(x \cdot y + \color{blue}{\left(-x\right)}\right)\right) + \left(-0.5 \cdot y\right) \]

      5. +-commutative 99.9%

         \[\leadsto \left(0.918938533204673 + \color{blue}{\left(\left(-x\right) + x \cdot y\right)}\right) + \left(-0.5 \cdot y\right) \]

      6. associate-+r+ 99.9%

         \[\leadsto \color{blue}{\left(\left(0.918938533204673 + \left(-x\right)\right) + x \cdot y\right)} + \left(-0.5 \cdot y\right) \]

      7. neg-mul-1 99.9%

         \[\leadsto \left(\left(0.918938533204673 + \color{blue}{-1 \cdot x}\right) + x \cdot y\right) + \left(-0.5 \cdot y\right) \]

      8. associate-+r+ 99.9%

         \[\leadsto \color{blue}{\left(0.918938533204673 + -1 \cdot x\right) + \left(x \cdot y + \left(-0.5 \cdot y\right)\right)} \]

      9. distribute-lft-neg-in 99.9%

         \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \left(x \cdot y + \color{blue}{\left(-0.5\right) \cdot y}\right) \]

      10. cancel-sign-sub-inv 99.9%

          \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \color{blue}{\left(x \cdot y - 0.5 \cdot y\right)} \]

      11. distribute-rgt-out-- 99.9%

          \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \color{blue}{y \cdot \left(x - 0.5\right)} \]

      12. neg-mul-1 99.9%

          \[\leadsto \left(0.918938533204673 + \color{blue}{\left(-x\right)}\right) + y \cdot \left(x - 0.5\right) \]

      13. associate-+r+ 99.9%

          \[\leadsto \color{blue}{0.918938533204673 + \left(\left(-x\right) + y \cdot \left(x - 0.5\right)\right)} \]

      14. +-commutative 99.9%

          \[\leadsto 0.918938533204673 + \color{blue}{\left(y \cdot \left(x - 0.5\right) + \left(-x\right)\right)} \]

      15. unsub-neg 99.9%

          \[\leadsto 0.918938533204673 + \color{blue}{\left(y \cdot \left(x - 0.5\right) - x\right)} \]

      16. sub-neg 99.9%

          \[\leadsto 0.918938533204673 + \left(y \cdot \color{blue}{\left(x + \left(-0.5\right)\right)} - x\right) \]

      17. metadata-eval 99.9%

          \[\leadsto 0.918938533204673 + \left(y \cdot \left(x + \color{blue}{-0.5}\right) - x\right) \]

   7. Simplified 99.9%

      \[\leadsto \color{blue}{0.918938533204673 + \left(y \cdot \left(x + -0.5\right) - x\right)} \]

   8. Taylor expanded in x around inf 97.8%

      \[\leadsto 0.918938533204673 + \color{blue}{x \cdot \left(y - 1\right)} \]

   9. Taylor expanded in x around inf 97.8%

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

   10. Taylor expanded in y around inf 64.2%

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

Alternative 5: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.7 \lor \neg \left(x \leq 0.5\right):\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.7) (not (<= x 0.5)))
   (* x (+ y -1.0))
   (- 0.918938533204673 (* y 0.5))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.7) || !(x <= 0.5)) {
		tmp = x * (y + -1.0);
	} else {
		tmp = 0.918938533204673 - (y * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-0.7d0)) .or. (.not. (x <= 0.5d0))) then
        tmp = x * (y + (-1.0d0))
    else
        tmp = 0.918938533204673d0 - (y * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -0.7) || !(x <= 0.5)) {
		tmp = x * (y + -1.0);
	} else {
		tmp = 0.918938533204673 - (y * 0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.7) or not (x <= 0.5):
		tmp = x * (y + -1.0)
	else:
		tmp = 0.918938533204673 - (y * 0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.7) || !(x <= 0.5))
		tmp = Float64(x * Float64(y + -1.0));
	else
		tmp = Float64(0.918938533204673 - Float64(y * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -0.7) || ~((x <= 0.5)))
		tmp = x * (y + -1.0);
	else
		tmp = 0.918938533204673 - (y * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.7], N[Not[LessEqual[x, 0.5]], $MachinePrecision]], N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 - N[(y * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.69999999999999996 or 0.5 < x

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\left(0.918938533204673 + x \cdot \left(y - 1\right)\right) - 0.5 \cdot y} \]
   6. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \color{blue}{\left(0.918938533204673 + x \cdot \left(y - 1\right)\right) + \left(-0.5 \cdot y\right)} \]

      2. distribute-lft-out-- 100.0%

         \[\leadsto \left(0.918938533204673 + \color{blue}{\left(x \cdot y - x \cdot 1\right)}\right) + \left(-0.5 \cdot y\right) \]

      3. cancel-sign-sub-inv 100.0%

         \[\leadsto \left(0.918938533204673 + \color{blue}{\left(x \cdot y + \left(-x\right) \cdot 1\right)}\right) + \left(-0.5 \cdot y\right) \]

      4. *-rgt-identity 100.0%

         \[\leadsto \left(0.918938533204673 + \left(x \cdot y + \color{blue}{\left(-x\right)}\right)\right) + \left(-0.5 \cdot y\right) \]

      5. +-commutative 100.0%

         \[\leadsto \left(0.918938533204673 + \color{blue}{\left(\left(-x\right) + x \cdot y\right)}\right) + \left(-0.5 \cdot y\right) \]

      6. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(0.918938533204673 + \left(-x\right)\right) + x \cdot y\right)} + \left(-0.5 \cdot y\right) \]

      7. neg-mul-1 100.0%

         \[\leadsto \left(\left(0.918938533204673 + \color{blue}{-1 \cdot x}\right) + x \cdot y\right) + \left(-0.5 \cdot y\right) \]

      8. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(0.918938533204673 + -1 \cdot x\right) + \left(x \cdot y + \left(-0.5 \cdot y\right)\right)} \]

      9. distribute-lft-neg-in 100.0%

         \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \left(x \cdot y + \color{blue}{\left(-0.5\right) \cdot y}\right) \]

      10. cancel-sign-sub-inv 100.0%

          \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \color{blue}{\left(x \cdot y - 0.5 \cdot y\right)} \]

      11. distribute-rgt-out-- 100.0%

          \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \color{blue}{y \cdot \left(x - 0.5\right)} \]

      12. neg-mul-1 100.0%

          \[\leadsto \left(0.918938533204673 + \color{blue}{\left(-x\right)}\right) + y \cdot \left(x - 0.5\right) \]

      13. associate-+r+ 100.0%

          \[\leadsto \color{blue}{0.918938533204673 + \left(\left(-x\right) + y \cdot \left(x - 0.5\right)\right)} \]

      14. +-commutative 100.0%

          \[\leadsto 0.918938533204673 + \color{blue}{\left(y \cdot \left(x - 0.5\right) + \left(-x\right)\right)} \]

      15. unsub-neg 100.0%

          \[\leadsto 0.918938533204673 + \color{blue}{\left(y \cdot \left(x - 0.5\right) - x\right)} \]

      16. sub-neg 100.0%

          \[\leadsto 0.918938533204673 + \left(y \cdot \color{blue}{\left(x + \left(-0.5\right)\right)} - x\right) \]

      17. metadata-eval 100.0%

          \[\leadsto 0.918938533204673 + \left(y \cdot \left(x + \color{blue}{-0.5}\right) - x\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0.918938533204673 + \left(y \cdot \left(x + -0.5\right) - x\right)} \]

   8. Taylor expanded in x around inf 99.7%

      \[\leadsto 0.918938533204673 + \color{blue}{x \cdot \left(y - 1\right)} \]

   9. Taylor expanded in x around inf 99.6%

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

   if -0.69999999999999996 < x < 0.5

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.1%

      \[\leadsto \color{blue}{0.918938533204673 - 0.5 \cdot y} \]
   6. Step-by-step derivation

      1. *-commutative 99.1%

         \[\leadsto 0.918938533204673 - \color{blue}{y \cdot 0.5} \]

   7. Simplified 99.1%

      \[\leadsto \color{blue}{0.918938533204673 - y \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.7 \lor \neg \left(x \leq 0.5\right):\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.55) (not (<= y 1.0)))
   (* y (- x 0.5))
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.55) || !(y <= 1.0)) {
		tmp = y * (x - 0.5);
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.55d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = y * (x - 0.5d0)
    else
        tmp = 0.918938533204673d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.55) || !(y <= 1.0)) {
		tmp = y * (x - 0.5);
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.55) or not (y <= 1.0):
		tmp = y * (x - 0.5)
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.55) || !(y <= 1.0))
		tmp = Float64(y * Float64(x - 0.5));
	else
		tmp = Float64(0.918938533204673 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.55) || ~((y <= 1.0)))
		tmp = y * (x - 0.5);
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.55], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(y * N[(x - 0.5), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.55000000000000004 or 1 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 97.3%

      \[\leadsto \color{blue}{y \cdot \left(x - 0.5\right)} \]

   if -1.55000000000000004 < y < 1

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 99.3%

      \[\leadsto \color{blue}{0.918938533204673 + -1 \cdot x} \]
   6. Step-by-step derivation

      1. neg-mul-1 99.3%

         \[\leadsto 0.918938533204673 + \color{blue}{\left(-x\right)} \]

      2. unsub-neg 99.3%

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   7. Simplified 99.3%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y + -1\right)\\ \mathbf{if}\;x \leq -1.28 \cdot 10^{-8}:\\ \;\;\;\;t\_0 + 0.918938533204673\\ \mathbf{elif}\;x \leq 0.68:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (+ y -1.0))))
   (if (<= x -1.28e-8)
     (+ t_0 0.918938533204673)
     (if (<= x 0.68) (- 0.918938533204673 (* y 0.5)) t_0))))

C

double code(double x, double y) {
	double t_0 = x * (y + -1.0);
	double tmp;
	if (x <= -1.28e-8) {
		tmp = t_0 + 0.918938533204673;
	} else if (x <= 0.68) {
		tmp = 0.918938533204673 - (y * 0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (y + (-1.0d0))
    if (x <= (-1.28d-8)) then
        tmp = t_0 + 0.918938533204673d0
    else if (x <= 0.68d0) then
        tmp = 0.918938533204673d0 - (y * 0.5d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (y + -1.0);
	double tmp;
	if (x <= -1.28e-8) {
		tmp = t_0 + 0.918938533204673;
	} else if (x <= 0.68) {
		tmp = 0.918938533204673 - (y * 0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (y + -1.0)
	tmp = 0
	if x <= -1.28e-8:
		tmp = t_0 + 0.918938533204673
	elif x <= 0.68:
		tmp = 0.918938533204673 - (y * 0.5)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(y + -1.0))
	tmp = 0.0
	if (x <= -1.28e-8)
		tmp = Float64(t_0 + 0.918938533204673);
	elseif (x <= 0.68)
		tmp = Float64(0.918938533204673 - Float64(y * 0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (y + -1.0);
	tmp = 0.0;
	if (x <= -1.28e-8)
		tmp = t_0 + 0.918938533204673;
	elseif (x <= 0.68)
		tmp = 0.918938533204673 - (y * 0.5);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.28e-8], N[(t$95$0 + 0.918938533204673), $MachinePrecision], If[LessEqual[x, 0.68], N[(0.918938533204673 - N[(y * 0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.28000000000000005e-8

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\left(0.918938533204673 + x \cdot \left(y - 1\right)\right) - 0.5 \cdot y} \]
   6. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \color{blue}{\left(0.918938533204673 + x \cdot \left(y - 1\right)\right) + \left(-0.5 \cdot y\right)} \]

      2. distribute-lft-out-- 100.0%

         \[\leadsto \left(0.918938533204673 + \color{blue}{\left(x \cdot y - x \cdot 1\right)}\right) + \left(-0.5 \cdot y\right) \]

      3. cancel-sign-sub-inv 100.0%

         \[\leadsto \left(0.918938533204673 + \color{blue}{\left(x \cdot y + \left(-x\right) \cdot 1\right)}\right) + \left(-0.5 \cdot y\right) \]

      4. *-rgt-identity 100.0%

         \[\leadsto \left(0.918938533204673 + \left(x \cdot y + \color{blue}{\left(-x\right)}\right)\right) + \left(-0.5 \cdot y\right) \]

      5. +-commutative 100.0%

         \[\leadsto \left(0.918938533204673 + \color{blue}{\left(\left(-x\right) + x \cdot y\right)}\right) + \left(-0.5 \cdot y\right) \]

      6. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(0.918938533204673 + \left(-x\right)\right) + x \cdot y\right)} + \left(-0.5 \cdot y\right) \]

      7. neg-mul-1 100.0%

         \[\leadsto \left(\left(0.918938533204673 + \color{blue}{-1 \cdot x}\right) + x \cdot y\right) + \left(-0.5 \cdot y\right) \]

      8. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(0.918938533204673 + -1 \cdot x\right) + \left(x \cdot y + \left(-0.5 \cdot y\right)\right)} \]

      9. distribute-lft-neg-in 100.0%

         \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \left(x \cdot y + \color{blue}{\left(-0.5\right) \cdot y}\right) \]

      10. cancel-sign-sub-inv 100.0%

          \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \color{blue}{\left(x \cdot y - 0.5 \cdot y\right)} \]

      11. distribute-rgt-out-- 100.0%

          \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \color{blue}{y \cdot \left(x - 0.5\right)} \]

      12. neg-mul-1 100.0%

          \[\leadsto \left(0.918938533204673 + \color{blue}{\left(-x\right)}\right) + y \cdot \left(x - 0.5\right) \]

      13. associate-+r+ 100.0%

          \[\leadsto \color{blue}{0.918938533204673 + \left(\left(-x\right) + y \cdot \left(x - 0.5\right)\right)} \]

      14. +-commutative 100.0%

          \[\leadsto 0.918938533204673 + \color{blue}{\left(y \cdot \left(x - 0.5\right) + \left(-x\right)\right)} \]

      15. unsub-neg 100.0%

          \[\leadsto 0.918938533204673 + \color{blue}{\left(y \cdot \left(x - 0.5\right) - x\right)} \]

      16. sub-neg 100.0%

          \[\leadsto 0.918938533204673 + \left(y \cdot \color{blue}{\left(x + \left(-0.5\right)\right)} - x\right) \]

      17. metadata-eval 100.0%

          \[\leadsto 0.918938533204673 + \left(y \cdot \left(x + \color{blue}{-0.5}\right) - x\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0.918938533204673 + \left(y \cdot \left(x + -0.5\right) - x\right)} \]

   8. Taylor expanded in x around inf 100.0%

      \[\leadsto 0.918938533204673 + \color{blue}{x \cdot \left(y - 1\right)} \]

   if -1.28000000000000005e-8 < x < 0.680000000000000049

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.1%

      \[\leadsto \color{blue}{0.918938533204673 - 0.5 \cdot y} \]
   6. Step-by-step derivation

      1. *-commutative 99.1%

         \[\leadsto 0.918938533204673 - \color{blue}{y \cdot 0.5} \]

   7. Simplified 99.1%

      \[\leadsto \color{blue}{0.918938533204673 - y \cdot 0.5} \]

   if 0.680000000000000049 < x

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\left(0.918938533204673 + x \cdot \left(y - 1\right)\right) - 0.5 \cdot y} \]
   6. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \color{blue}{\left(0.918938533204673 + x \cdot \left(y - 1\right)\right) + \left(-0.5 \cdot y\right)} \]

      2. distribute-lft-out-- 99.9%

         \[\leadsto \left(0.918938533204673 + \color{blue}{\left(x \cdot y - x \cdot 1\right)}\right) + \left(-0.5 \cdot y\right) \]

      3. cancel-sign-sub-inv 99.9%

         \[\leadsto \left(0.918938533204673 + \color{blue}{\left(x \cdot y + \left(-x\right) \cdot 1\right)}\right) + \left(-0.5 \cdot y\right) \]

      4. *-rgt-identity 99.9%

         \[\leadsto \left(0.918938533204673 + \left(x \cdot y + \color{blue}{\left(-x\right)}\right)\right) + \left(-0.5 \cdot y\right) \]

      5. +-commutative 99.9%

         \[\leadsto \left(0.918938533204673 + \color{blue}{\left(\left(-x\right) + x \cdot y\right)}\right) + \left(-0.5 \cdot y\right) \]

      6. associate-+r+ 99.9%

         \[\leadsto \color{blue}{\left(\left(0.918938533204673 + \left(-x\right)\right) + x \cdot y\right)} + \left(-0.5 \cdot y\right) \]

      7. neg-mul-1 99.9%

         \[\leadsto \left(\left(0.918938533204673 + \color{blue}{-1 \cdot x}\right) + x \cdot y\right) + \left(-0.5 \cdot y\right) \]

      8. associate-+r+ 99.9%

         \[\leadsto \color{blue}{\left(0.918938533204673 + -1 \cdot x\right) + \left(x \cdot y + \left(-0.5 \cdot y\right)\right)} \]

      9. distribute-lft-neg-in 99.9%

         \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \left(x \cdot y + \color{blue}{\left(-0.5\right) \cdot y}\right) \]

      10. cancel-sign-sub-inv 99.9%

          \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \color{blue}{\left(x \cdot y - 0.5 \cdot y\right)} \]

      11. distribute-rgt-out-- 99.9%

          \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \color{blue}{y \cdot \left(x - 0.5\right)} \]

      12. neg-mul-1 99.9%

          \[\leadsto \left(0.918938533204673 + \color{blue}{\left(-x\right)}\right) + y \cdot \left(x - 0.5\right) \]

      13. associate-+r+ 99.9%

          \[\leadsto \color{blue}{0.918938533204673 + \left(\left(-x\right) + y \cdot \left(x - 0.5\right)\right)} \]

      14. +-commutative 99.9%

          \[\leadsto 0.918938533204673 + \color{blue}{\left(y \cdot \left(x - 0.5\right) + \left(-x\right)\right)} \]

      15. unsub-neg 99.9%

          \[\leadsto 0.918938533204673 + \color{blue}{\left(y \cdot \left(x - 0.5\right) - x\right)} \]

      16. sub-neg 99.9%

          \[\leadsto 0.918938533204673 + \left(y \cdot \color{blue}{\left(x + \left(-0.5\right)\right)} - x\right) \]

      17. metadata-eval 99.9%

          \[\leadsto 0.918938533204673 + \left(y \cdot \left(x + \color{blue}{-0.5}\right) - x\right) \]

   7. Simplified 99.9%

      \[\leadsto \color{blue}{0.918938533204673 + \left(y \cdot \left(x + -0.5\right) - x\right)} \]

   8. Taylor expanded in x around inf 99.3%

      \[\leadsto 0.918938533204673 + \color{blue}{x \cdot \left(y - 1\right)} \]

   9. Taylor expanded in x around inf 99.3%

      \[\leadsto \color{blue}{x \cdot \left(y - 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.28 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \left(y + -1\right) + 0.918938533204673\\ \mathbf{elif}\;x \leq 0.68:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.92 \lor \neg \left(x \leq 0.92\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.92) (not (<= x 0.92))) (- x) 0.918938533204673))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.92) || !(x <= 0.92)) {
		tmp = -x;
	} else {
		tmp = 0.918938533204673;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-0.92d0)) .or. (.not. (x <= 0.92d0))) then
        tmp = -x
    else
        tmp = 0.918938533204673d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -0.92) || !(x <= 0.92)) {
		tmp = -x;
	} else {
		tmp = 0.918938533204673;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.92) or not (x <= 0.92):
		tmp = -x
	else:
		tmp = 0.918938533204673
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.92) || !(x <= 0.92))
		tmp = Float64(-x);
	else
		tmp = 0.918938533204673;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -0.92) || ~((x <= 0.92)))
		tmp = -x;
	else
		tmp = 0.918938533204673;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.92], N[Not[LessEqual[x, 0.92]], $MachinePrecision]], (-x), 0.918938533204673]

Derivation

1. Split input into 2 regimes

2. if x < -0.92000000000000004 or 0.92000000000000004 < x

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\left(0.918938533204673 + x \cdot \left(y - 1\right)\right) - 0.5 \cdot y} \]
   6. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \color{blue}{\left(0.918938533204673 + x \cdot \left(y - 1\right)\right) + \left(-0.5 \cdot y\right)} \]

      2. distribute-lft-out-- 100.0%

         \[\leadsto \left(0.918938533204673 + \color{blue}{\left(x \cdot y - x \cdot 1\right)}\right) + \left(-0.5 \cdot y\right) \]

      3. cancel-sign-sub-inv 100.0%

         \[\leadsto \left(0.918938533204673 + \color{blue}{\left(x \cdot y + \left(-x\right) \cdot 1\right)}\right) + \left(-0.5 \cdot y\right) \]

      4. *-rgt-identity 100.0%

         \[\leadsto \left(0.918938533204673 + \left(x \cdot y + \color{blue}{\left(-x\right)}\right)\right) + \left(-0.5 \cdot y\right) \]

      5. +-commutative 100.0%

         \[\leadsto \left(0.918938533204673 + \color{blue}{\left(\left(-x\right) + x \cdot y\right)}\right) + \left(-0.5 \cdot y\right) \]

      6. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(0.918938533204673 + \left(-x\right)\right) + x \cdot y\right)} + \left(-0.5 \cdot y\right) \]

      7. neg-mul-1 100.0%

         \[\leadsto \left(\left(0.918938533204673 + \color{blue}{-1 \cdot x}\right) + x \cdot y\right) + \left(-0.5 \cdot y\right) \]

      8. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(0.918938533204673 + -1 \cdot x\right) + \left(x \cdot y + \left(-0.5 \cdot y\right)\right)} \]

      9. distribute-lft-neg-in 100.0%

         \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \left(x \cdot y + \color{blue}{\left(-0.5\right) \cdot y}\right) \]

      10. cancel-sign-sub-inv 100.0%

          \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \color{blue}{\left(x \cdot y - 0.5 \cdot y\right)} \]

      11. distribute-rgt-out-- 100.0%

          \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \color{blue}{y \cdot \left(x - 0.5\right)} \]

      12. neg-mul-1 100.0%

          \[\leadsto \left(0.918938533204673 + \color{blue}{\left(-x\right)}\right) + y \cdot \left(x - 0.5\right) \]

      13. associate-+r+ 100.0%

          \[\leadsto \color{blue}{0.918938533204673 + \left(\left(-x\right) + y \cdot \left(x - 0.5\right)\right)} \]

      14. +-commutative 100.0%

          \[\leadsto 0.918938533204673 + \color{blue}{\left(y \cdot \left(x - 0.5\right) + \left(-x\right)\right)} \]

      15. unsub-neg 100.0%

          \[\leadsto 0.918938533204673 + \color{blue}{\left(y \cdot \left(x - 0.5\right) - x\right)} \]

      16. sub-neg 100.0%

          \[\leadsto 0.918938533204673 + \left(y \cdot \color{blue}{\left(x + \left(-0.5\right)\right)} - x\right) \]

      17. metadata-eval 100.0%

          \[\leadsto 0.918938533204673 + \left(y \cdot \left(x + \color{blue}{-0.5}\right) - x\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0.918938533204673 + \left(y \cdot \left(x + -0.5\right) - x\right)} \]

   8. Taylor expanded in x around inf 99.7%

      \[\leadsto 0.918938533204673 + \color{blue}{x \cdot \left(y - 1\right)} \]

   9. Taylor expanded in x around inf 99.6%

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

   10. Taylor expanded in y around 0 56.3%

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

       1. mul-1-neg 56.3%

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

   12. Simplified 56.3%

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

   if -0.92000000000000004 < x < 0.92000000000000004

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 53.2%

      \[\leadsto \color{blue}{0.918938533204673 + -1 \cdot x} \]
   6. Step-by-step derivation

      1. neg-mul-1 53.2%

         \[\leadsto 0.918938533204673 + \color{blue}{\left(-x\right)} \]

      2. unsub-neg 53.2%

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   7. Simplified 53.2%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]

   8. Taylor expanded in x around 0 53.0%

      \[\leadsto \color{blue}{0.918938533204673} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.92 \lor \neg \left(x \leq 0.92\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 0.918938533204673 + \left(y \cdot \left(x + -0.5\right) - x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 0.918938533204673 (- (* y (+ x -0.5)) x)))

C

double code(double x, double y) {
	return 0.918938533204673 + ((y * (x + -0.5)) - x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.918938533204673d0 + ((y * (x + (-0.5d0))) - x)
end function

Java

public static double code(double x, double y) {
	return 0.918938533204673 + ((y * (x + -0.5)) - x);
}

Python

def code(x, y):
	return 0.918938533204673 + ((y * (x + -0.5)) - x)

Julia

function code(x, y)
	return Float64(0.918938533204673 + Float64(Float64(y * Float64(x + -0.5)) - x))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.918938533204673 + ((y * (x + -0.5)) - x);
end

Wolfram

code[x_, y_] := N[(0.918938533204673 + N[(N[(y * N[(x + -0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
2. Step-by-step derivation

   1. associate-+l- 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   2. sub-neg 100.0%

      \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. metadata-eval 100.0%

      \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 100.0%

   \[\leadsto \color{blue}{\left(0.918938533204673 + x \cdot \left(y - 1\right)\right) - 0.5 \cdot y} \]
6. Step-by-step derivation

   1. sub-neg 100.0%

      \[\leadsto \color{blue}{\left(0.918938533204673 + x \cdot \left(y - 1\right)\right) + \left(-0.5 \cdot y\right)} \]

   2. distribute-lft-out-- 100.0%

      \[\leadsto \left(0.918938533204673 + \color{blue}{\left(x \cdot y - x \cdot 1\right)}\right) + \left(-0.5 \cdot y\right) \]

   3. cancel-sign-sub-inv 100.0%

      \[\leadsto \left(0.918938533204673 + \color{blue}{\left(x \cdot y + \left(-x\right) \cdot 1\right)}\right) + \left(-0.5 \cdot y\right) \]

   4. *-rgt-identity 100.0%

      \[\leadsto \left(0.918938533204673 + \left(x \cdot y + \color{blue}{\left(-x\right)}\right)\right) + \left(-0.5 \cdot y\right) \]

   5. +-commutative 100.0%

      \[\leadsto \left(0.918938533204673 + \color{blue}{\left(\left(-x\right) + x \cdot y\right)}\right) + \left(-0.5 \cdot y\right) \]

   6. associate-+r+ 100.0%

      \[\leadsto \color{blue}{\left(\left(0.918938533204673 + \left(-x\right)\right) + x \cdot y\right)} + \left(-0.5 \cdot y\right) \]

   7. neg-mul-1 100.0%

      \[\leadsto \left(\left(0.918938533204673 + \color{blue}{-1 \cdot x}\right) + x \cdot y\right) + \left(-0.5 \cdot y\right) \]

   8. associate-+r+ 100.0%

      \[\leadsto \color{blue}{\left(0.918938533204673 + -1 \cdot x\right) + \left(x \cdot y + \left(-0.5 \cdot y\right)\right)} \]

   9. distribute-lft-neg-in 100.0%

      \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \left(x \cdot y + \color{blue}{\left(-0.5\right) \cdot y}\right) \]

   10. cancel-sign-sub-inv 100.0%

       \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \color{blue}{\left(x \cdot y - 0.5 \cdot y\right)} \]

   11. distribute-rgt-out-- 100.0%

       \[\leadsto \left(0.918938533204673 + -1 \cdot x\right) + \color{blue}{y \cdot \left(x - 0.5\right)} \]

   12. neg-mul-1 100.0%

       \[\leadsto \left(0.918938533204673 + \color{blue}{\left(-x\right)}\right) + y \cdot \left(x - 0.5\right) \]

   13. associate-+r+ 100.0%

       \[\leadsto \color{blue}{0.918938533204673 + \left(\left(-x\right) + y \cdot \left(x - 0.5\right)\right)} \]

   14. +-commutative 100.0%

       \[\leadsto 0.918938533204673 + \color{blue}{\left(y \cdot \left(x - 0.5\right) + \left(-x\right)\right)} \]

   15. unsub-neg 100.0%

       \[\leadsto 0.918938533204673 + \color{blue}{\left(y \cdot \left(x - 0.5\right) - x\right)} \]

   16. sub-neg 100.0%

       \[\leadsto 0.918938533204673 + \left(y \cdot \color{blue}{\left(x + \left(-0.5\right)\right)} - x\right) \]

   17. metadata-eval 100.0%

       \[\leadsto 0.918938533204673 + \left(y \cdot \left(x + \color{blue}{-0.5}\right) - x\right) \]

7. Simplified 100.0%

   \[\leadsto \color{blue}{0.918938533204673 + \left(y \cdot \left(x + -0.5\right) - x\right)} \]
8. Add Preprocessing

Alternative 10: 26.3% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ 0.918938533204673 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.918938533204673)

C

double code(double x, double y) {
	return 0.918938533204673;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.918938533204673d0
end function

Java

public static double code(double x, double y) {
	return 0.918938533204673;
}

Python

def code(x, y):
	return 0.918938533204673

Julia

function code(x, y)
	return 0.918938533204673
end

MATLAB

function tmp = code(x, y)
	tmp = 0.918938533204673;
end

Wolfram

code[x_, y_] := 0.918938533204673

Derivation

1. Initial program 100.0%

   \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
2. Step-by-step derivation

   1. associate-+l- 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   2. sub-neg 100.0%

      \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. metadata-eval 100.0%

      \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 54.7%

   \[\leadsto \color{blue}{0.918938533204673 + -1 \cdot x} \]
6. Step-by-step derivation

   1. neg-mul-1 54.7%

      \[\leadsto 0.918938533204673 + \color{blue}{\left(-x\right)} \]

   2. unsub-neg 54.7%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]

7. Simplified 54.7%

   \[\leadsto \color{blue}{0.918938533204673 - x} \]

8. Taylor expanded in x around 0 30.3%

   \[\leadsto \color{blue}{0.918938533204673} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024186 
(FPCore (x y)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673))