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

Percentage Accurate: 100.0% → 100.0%

Time: 4.1s

Alternatives: 8

Speedup: 1.2×

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. Final simplification 100.0%

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

Alternative 2: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+18}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{elif}\;x \leq 118000000:\\ \;\;\;\;x \cdot y + \left(0.918938533204673 - y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.3e+18)
   (* x (+ y -1.0))
   (if (<= x 118000000.0)
     (+ (* x y) (- 0.918938533204673 (* y 0.5)))
     (- (* x y) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.3e+18) {
		tmp = x * (y + -1.0);
	} else if (x <= 118000000.0) {
		tmp = (x * y) + (0.918938533204673 - (y * 0.5));
	} else {
		tmp = (x * 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 <= (-2.3d+18)) then
        tmp = x * (y + (-1.0d0))
    else if (x <= 118000000.0d0) then
        tmp = (x * y) + (0.918938533204673d0 - (y * 0.5d0))
    else
        tmp = (x * y) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.3e+18) {
		tmp = x * (y + -1.0);
	} else if (x <= 118000000.0) {
		tmp = (x * y) + (0.918938533204673 - (y * 0.5));
	} else {
		tmp = (x * y) - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.3e+18:
		tmp = x * (y + -1.0)
	elif x <= 118000000.0:
		tmp = (x * y) + (0.918938533204673 - (y * 0.5))
	else:
		tmp = (x * y) - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.3e+18)
		tmp = Float64(x * Float64(y + -1.0));
	elseif (x <= 118000000.0)
		tmp = Float64(Float64(x * y) + Float64(0.918938533204673 - Float64(y * 0.5)));
	else
		tmp = Float64(Float64(x * y) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.3e+18)
		tmp = x * (y + -1.0);
	elseif (x <= 118000000.0)
		tmp = (x * y) + (0.918938533204673 - (y * 0.5));
	else
		tmp = (x * y) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.3e+18], N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 118000000.0], N[(N[(x * y), $MachinePrecision] + N[(0.918938533204673 - N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] - x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.3e18

   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. cancel-sign-sub-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-out-- 99.9%

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

      4. cancel-sign-sub-inv 99.9%

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

      5. *-rgt-identity 99.9%

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

      6. associate-+r+ 99.9%

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

      7. associate-+l+ 99.9%

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

      8. +-commutative 99.9%

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

      9. unsub-neg 99.9%

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

      10. associate-+r- 99.9%

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

      11. *-commutative 99.9%

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

      12. distribute-lft-neg-out 99.9%

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

      13. distribute-rgt-neg-in 99.9%

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

      14. distribute-lft-out 99.9%

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

      15. fma-def 99.9%

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

      16. +-commutative 99.9%

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

      17. sub-neg 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
   4. Step-by-step derivation

      1. fma-udef 99.9%

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

      2. *-un-lft-identity 99.9%

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

      3. *-un-lft-identity 99.9%

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

      4. sub-neg 99.9%

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

      5. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around inf 100.0%

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

   if -2.3e18 < x < 1.18e8

   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. Taylor expanded in y around inf 98.0%

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

   if 1.18e8 < x

   1. Initial program 99.9%

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

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-lft-out-- 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-+r+ 100.0%

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

      7. associate-+l+ 100.0%

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

      8. +-commutative 100.0%

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

      9. unsub-neg 100.0%

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

      10. associate-+r- 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-lft-neg-out 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. distribute-lft-out 100.0%

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

      15. fma-def 100.0%

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

      16. +-commutative 100.0%

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

      17. sub-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 99.3%

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

      1. sub-neg 99.3%

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

      2. metadata-eval 99.3%

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

      3. distribute-rgt-in 99.3%

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

      4. neg-mul-1 99.3%

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

      5. sub-neg 99.3%

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

   8. Simplified 99.3%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+18}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{elif}\;x \leq 118000000:\\ \;\;\;\;x \cdot y + \left(0.918938533204673 - y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - x\\ \end{array} \]

Alternative 3: 74.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1700000 \lor \neg \left(x \leq 10400\right):\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1700000.0) (not (<= x 10400.0)))
   (* x (+ y -1.0))
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1700000.0) || !(x <= 10400.0)) {
		tmp = x * (y + -1.0);
	} 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 ((x <= (-1700000.0d0)) .or. (.not. (x <= 10400.0d0))) then
        tmp = x * (y + (-1.0d0))
    else
        tmp = 0.918938533204673d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1700000.0) || !(x <= 10400.0)) {
		tmp = x * (y + -1.0);
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1700000.0) or not (x <= 10400.0):
		tmp = x * (y + -1.0)
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1700000.0) || !(x <= 10400.0))
		tmp = Float64(x * Float64(y + -1.0));
	else
		tmp = Float64(0.918938533204673 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1700000.0) || ~((x <= 10400.0)))
		tmp = x * (y + -1.0);
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1700000.0], N[Not[LessEqual[x, 10400.0]], $MachinePrecision]], N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7e6 or 10400 < 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. cancel-sign-sub-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-out-- 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-+r+ 100.0%

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

      7. associate-+l+ 100.0%

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

      8. +-commutative 100.0%

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

      9. unsub-neg 100.0%

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

      10. associate-+r- 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-lft-neg-out 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. distribute-lft-out 100.0%

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

      15. fma-def 100.0%

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

      16. +-commutative 100.0%

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

      17. sub-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 98.5%

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

   if -1.7e6 < x < 10400

   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. cancel-sign-sub-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-out-- 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-+r+ 100.0%

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

      7. associate-+l+ 100.0%

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

      8. +-commutative 100.0%

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

      9. unsub-neg 100.0%

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

      10. associate-+r- 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-lft-neg-out 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. distribute-lft-out 100.0%

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

      15. fma-def 100.0%

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

      16. +-commutative 100.0%

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

      17. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 53.4%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1700000 \lor \neg \left(x \leq 10400\right):\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]

Alternative 4: 98.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \lor \neg \left(y \leq 1.8\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.4) (not (<= y 1.8)))
   (* y (- x 0.5))
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.4) || !(y <= 1.8)) {
		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 <= (-1.4d0)) .or. (.not. (y <= 1.8d0))) 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 <= -1.4) || !(y <= 1.8)) {
		tmp = y * (x - 0.5);
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.4) or not (y <= 1.8):
		tmp = y * (x - 0.5)
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.4) || !(y <= 1.8))
		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 <= -1.4) || ~((y <= 1.8)))
		tmp = y * (x - 0.5);
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.4], N[Not[LessEqual[y, 1.8]], $MachinePrecision]], N[(y * N[(x - 0.5), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3999999999999999 or 1.80000000000000004 < 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. cancel-sign-sub-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-out-- 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-+r+ 100.0%

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

      7. associate-+l+ 100.0%

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

      8. +-commutative 100.0%

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

      9. unsub-neg 100.0%

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

      10. associate-+r- 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-lft-neg-out 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. distribute-lft-out 99.9%

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

      15. fma-def 99.9%

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

      16. +-commutative 99.9%

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

      17. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 98.3%

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

   if -1.3999999999999999 < y < 1.80000000000000004

   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. cancel-sign-sub-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-out-- 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-+r+ 100.0%

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

      7. associate-+l+ 100.0%

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

      8. +-commutative 100.0%

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

      9. unsub-neg 100.0%

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

      10. associate-+r- 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-lft-neg-out 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. distribute-lft-out 100.0%

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

      15. fma-def 100.0%

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

      16. +-commutative 100.0%

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

      17. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 98.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \lor \neg \left(y \leq 1.8\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]

Alternative 5: 100.0% accurate, 1.2× speedup

Math

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

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(Float64(0.918938533204673 + 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[(N[(0.918938533204673 + N[(y * N[(x + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $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. cancel-sign-sub-inv 100.0%

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

   2. +-commutative 100.0%

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

   3. distribute-lft-out-- 100.0%

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

   4. cancel-sign-sub-inv 100.0%

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

   5. *-rgt-identity 100.0%

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

   6. associate-+r+ 100.0%

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

   7. associate-+l+ 100.0%

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

   8. +-commutative 100.0%

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

   9. unsub-neg 100.0%

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

   10. associate-+r- 100.0%

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

   11. *-commutative 100.0%

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

   12. distribute-lft-neg-out 100.0%

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

   13. distribute-rgt-neg-in 100.0%

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

   14. distribute-lft-out 100.0%

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

   15. fma-def 100.0%

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

   16. +-commutative 100.0%

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

   17. sub-neg 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
4. Step-by-step derivation

   1. fma-udef 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. *-un-lft-identity 100.0%

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

   4. sub-neg 100.0%

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

   5. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

   \[\leadsto \left(0.918938533204673 + y \cdot \left(x + -0.5\right)\right) - x \]

Alternative 6: 49.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 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. cancel-sign-sub-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-out-- 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-+r+ 100.0%

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

      7. associate-+l+ 100.0%

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

      8. +-commutative 100.0%

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

      9. unsub-neg 100.0%

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

      10. associate-+r- 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-lft-neg-out 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. distribute-lft-out 99.9%

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

      15. fma-def 99.9%

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

      16. +-commutative 99.9%

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

      17. sub-neg 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
   4. Step-by-step derivation

      1. fma-udef 99.9%

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

      2. *-un-lft-identity 99.9%

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

      3. *-un-lft-identity 99.9%

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

      4. sub-neg 99.9%

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

      5. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around inf 51.7%

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

   7. Taylor expanded in y around inf 50.9%

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

   if -1 < 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. cancel-sign-sub-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-out-- 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-+r+ 100.0%

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

      7. associate-+l+ 100.0%

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

      8. +-commutative 100.0%

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

      9. unsub-neg 100.0%

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

      10. associate-+r- 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-lft-neg-out 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. distribute-lft-out 100.0%

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

      15. fma-def 100.0%

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

      16. +-commutative 100.0%

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

      17. sub-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 47.5%

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

   7. Taylor expanded in y around 0 46.0%

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

      1. neg-mul-1 46.0%

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

   9. Simplified 46.0%

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

4. Final simplification 48.5%

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

Alternative 7: 73.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -50:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.1:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -50.0) (* x y) (if (<= y 1.1) (- 0.918938533204673 x) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -50.0) {
		tmp = x * y;
	} else if (y <= 1.1) {
		tmp = 0.918938533204673 - x;
	} 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 <= (-50.0d0)) then
        tmp = x * y
    else if (y <= 1.1d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -50.0) {
		tmp = x * y;
	} else if (y <= 1.1) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -50.0:
		tmp = x * y
	elif y <= 1.1:
		tmp = 0.918938533204673 - x
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -50.0)
		tmp = Float64(x * y);
	elseif (y <= 1.1)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -50.0)
		tmp = x * y;
	elseif (y <= 1.1)
		tmp = 0.918938533204673 - x;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -50.0], N[(x * y), $MachinePrecision], If[LessEqual[y, 1.1], N[(0.918938533204673 - x), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -50 or 1.1000000000000001 < 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. cancel-sign-sub-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-out-- 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-+r+ 100.0%

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

      7. associate-+l+ 100.0%

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

      8. +-commutative 100.0%

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

      9. unsub-neg 100.0%

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

      10. associate-+r- 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-lft-neg-out 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. distribute-lft-out 99.9%

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

      15. fma-def 99.9%

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

      16. +-commutative 99.9%

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

      17. sub-neg 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
   4. Step-by-step derivation

      1. fma-udef 99.9%

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

      2. *-un-lft-identity 99.9%

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

      3. *-un-lft-identity 99.9%

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

      4. sub-neg 99.9%

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

      5. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around inf 51.7%

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

   7. Taylor expanded in y around inf 50.9%

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

   if -50 < y < 1.1000000000000001

   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. cancel-sign-sub-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-out-- 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-+r+ 100.0%

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

      7. associate-+l+ 100.0%

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

      8. +-commutative 100.0%

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

      9. unsub-neg 100.0%

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

      10. associate-+r- 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-lft-neg-out 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. distribute-lft-out 100.0%

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

      15. fma-def 100.0%

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

      16. +-commutative 100.0%

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

      17. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 98.0%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -50:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.1:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 8: 26.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ -x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -x

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := (-x)

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. cancel-sign-sub-inv 100.0%

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

   2. +-commutative 100.0%

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

   3. distribute-lft-out-- 100.0%

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

   4. cancel-sign-sub-inv 100.0%

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

   5. *-rgt-identity 100.0%

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

   6. associate-+r+ 100.0%

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

   7. associate-+l+ 100.0%

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

   8. +-commutative 100.0%

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

   9. unsub-neg 100.0%

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

   10. associate-+r- 100.0%

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

   11. *-commutative 100.0%

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

   12. distribute-lft-neg-out 100.0%

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

   13. distribute-rgt-neg-in 100.0%

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

   14. distribute-lft-out 100.0%

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

   15. fma-def 100.0%

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

   16. +-commutative 100.0%

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

   17. sub-neg 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
4. Step-by-step derivation

   1. fma-udef 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. *-un-lft-identity 100.0%

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

   4. sub-neg 100.0%

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

   5. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in x around inf 49.7%

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

7. Taylor expanded in y around 0 24.1%

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

   1. neg-mul-1 24.1%

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

9. Simplified 24.1%

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

10. Final simplification 24.1%

    \[\leadsto -x \]

Reproduce

herbie shell --seed 2023271 
(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))