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

Percentage Accurate: 100.0% → 100.0%

Time: 4.8s

Alternatives: 10

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: 74.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+170}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{+81}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{+15}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.05:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+135}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.9e+170)
   (* y -0.5)
   (if (<= y -5.8e+81)
     (* x y)
     (if (<= y -2.3e+15)
       (* y -0.5)
       (if (<= y 1.05)
         (- 0.918938533204673 x)
         (if (<= y 2.25e+135) (* x y) (* y -0.5)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.9e+170) {
		tmp = y * -0.5;
	} else if (y <= -5.8e+81) {
		tmp = x * y;
	} else if (y <= -2.3e+15) {
		tmp = y * -0.5;
	} else if (y <= 1.05) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 2.25e+135) {
		tmp = x * y;
	} else {
		tmp = 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 (y <= (-1.9d+170)) then
        tmp = y * (-0.5d0)
    else if (y <= (-5.8d+81)) then
        tmp = x * y
    else if (y <= (-2.3d+15)) then
        tmp = y * (-0.5d0)
    else if (y <= 1.05d0) then
        tmp = 0.918938533204673d0 - x
    else if (y <= 2.25d+135) then
        tmp = x * y
    else
        tmp = y * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.9e+170) {
		tmp = y * -0.5;
	} else if (y <= -5.8e+81) {
		tmp = x * y;
	} else if (y <= -2.3e+15) {
		tmp = y * -0.5;
	} else if (y <= 1.05) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 2.25e+135) {
		tmp = x * y;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.9e+170:
		tmp = y * -0.5
	elif y <= -5.8e+81:
		tmp = x * y
	elif y <= -2.3e+15:
		tmp = y * -0.5
	elif y <= 1.05:
		tmp = 0.918938533204673 - x
	elif y <= 2.25e+135:
		tmp = x * y
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.9e+170)
		tmp = Float64(y * -0.5);
	elseif (y <= -5.8e+81)
		tmp = Float64(x * y);
	elseif (y <= -2.3e+15)
		tmp = Float64(y * -0.5);
	elseif (y <= 1.05)
		tmp = Float64(0.918938533204673 - x);
	elseif (y <= 2.25e+135)
		tmp = Float64(x * y);
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.9e+170)
		tmp = y * -0.5;
	elseif (y <= -5.8e+81)
		tmp = x * y;
	elseif (y <= -2.3e+15)
		tmp = y * -0.5;
	elseif (y <= 1.05)
		tmp = 0.918938533204673 - x;
	elseif (y <= 2.25e+135)
		tmp = x * y;
	else
		tmp = y * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.9e+170], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, -5.8e+81], N[(x * y), $MachinePrecision], If[LessEqual[y, -2.3e+15], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 1.05], N[(0.918938533204673 - x), $MachinePrecision], If[LessEqual[y, 2.25e+135], N[(x * y), $MachinePrecision], N[(y * -0.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.8999999999999999e170 or -5.7999999999999999e81 < y < -2.3e15 or 2.25000000000000004e135 < 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. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. 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 100.0%

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

   5. Taylor expanded in x around 0 64.0%

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

   if -1.8999999999999999e170 < y < -5.7999999999999999e81 or 1.05000000000000004 < y < 2.25000000000000004e135

   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. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. 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 97.5%

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

   5. Taylor expanded in x around inf 63.0%

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

   if -2.3e15 < y < 1.05000000000000004

   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. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. 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 0 97.1%

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

      1. neg-mul-1 97.1%

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

      2. sub-neg 97.1%

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

   6. Simplified 97.1%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+170}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{+81}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{+15}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.05:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+135}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]

Alternative 3: 98.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -34 \lor \neg \left(x \leq 700\right):\\ \;\;\;\;\left(0.918938533204673 + x \cdot y\right) - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + \left(0.918938533204673 - y \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -34.0) (not (<= x 700.0)))
   (- (+ 0.918938533204673 (* x y)) x)
   (+ (* x y) (- 0.918938533204673 (* y 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -34.0) || !(x <= 700.0)) {
		tmp = (0.918938533204673 + (x * y)) - x;
	} else {
		tmp = (x * y) + (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 <= (-34.0d0)) .or. (.not. (x <= 700.0d0))) then
        tmp = (0.918938533204673d0 + (x * y)) - x
    else
        tmp = (x * y) + (0.918938533204673d0 - (y * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -34.0) || !(x <= 700.0)) {
		tmp = (0.918938533204673 + (x * y)) - x;
	} else {
		tmp = (x * y) + (0.918938533204673 - (y * 0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -34.0) or not (x <= 700.0):
		tmp = (0.918938533204673 + (x * y)) - x
	else:
		tmp = (x * y) + (0.918938533204673 - (y * 0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -34.0) || !(x <= 700.0))
		tmp = Float64(Float64(0.918938533204673 + Float64(x * y)) - x);
	else
		tmp = Float64(Float64(x * y) + Float64(0.918938533204673 - Float64(y * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -34.0) || ~((x <= 700.0)))
		tmp = (0.918938533204673 + (x * y)) - x;
	else
		tmp = (x * y) + (0.918938533204673 - (y * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -34.0], N[Not[LessEqual[x, 700.0]], $MachinePrecision]], N[(N[(0.918938533204673 + N[(x * y), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], N[(N[(x * y), $MachinePrecision] + N[(0.918938533204673 - N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -34 or 700 < 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. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-rgt-in 100.0%

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

      5. associate-+r+ 100.0%

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

      6. associate-+l+ 100.0%

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

      7. distribute-rgt-neg-in 100.0%

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

      8. distribute-lft-out 100.0%

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

      9. fma-def 100.0%

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

      10. +-commutative 100.0%

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

      11. metadata-eval 100.0%

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

      12. +-commutative 100.0%

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

      13. cancel-sign-sub-inv 100.0%

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

      14. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

   5. Taylor expanded in x around inf 99.5%

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

   if -34 < x < 700

   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. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. 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 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -34 \lor \neg \left(x \leq 700\right):\\ \;\;\;\;\left(0.918938533204673 + x \cdot y\right) - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + \left(0.918938533204673 - y \cdot 0.5\right)\\ \end{array} \]

Alternative 4: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+15}:\\ \;\;\;\;x \cdot y + y \cdot -0.5\\ \mathbf{elif}\;y \leq 6500:\\ \;\;\;\;\left(0.918938533204673 + x \cdot y\right) - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.3e+15)
   (+ (* x y) (* y -0.5))
   (if (<= y 6500.0) (- (+ 0.918938533204673 (* x y)) x) (* y (- x 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.3e+15) {
		tmp = (x * y) + (y * -0.5);
	} else if (y <= 6500.0) {
		tmp = (0.918938533204673 + (x * y)) - x;
	} else {
		tmp = y * (x - 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 (y <= (-2.3d+15)) then
        tmp = (x * y) + (y * (-0.5d0))
    else if (y <= 6500.0d0) then
        tmp = (0.918938533204673d0 + (x * y)) - x
    else
        tmp = y * (x - 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.3e+15) {
		tmp = (x * y) + (y * -0.5);
	} else if (y <= 6500.0) {
		tmp = (0.918938533204673 + (x * y)) - x;
	} else {
		tmp = y * (x - 0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.3e+15:
		tmp = (x * y) + (y * -0.5)
	elif y <= 6500.0:
		tmp = (0.918938533204673 + (x * y)) - x
	else:
		tmp = y * (x - 0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.3e+15)
		tmp = Float64(Float64(x * y) + Float64(y * -0.5));
	elseif (y <= 6500.0)
		tmp = Float64(Float64(0.918938533204673 + Float64(x * y)) - x);
	else
		tmp = Float64(y * Float64(x - 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.3e+15)
		tmp = (x * y) + (y * -0.5);
	elseif (y <= 6500.0)
		tmp = (0.918938533204673 + (x * y)) - x;
	else
		tmp = y * (x - 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.3e+15], N[(N[(x * y), $MachinePrecision] + N[(y * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6500.0], N[(N[(0.918938533204673 + N[(x * y), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], N[(y * N[(x - 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.3e15

   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. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. 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 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   if -2.3e15 < y < 6500

   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. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-rgt-in 100.0%

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

      5. associate-+r+ 100.0%

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

      6. associate-+l+ 100.0%

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

      7. distribute-rgt-neg-in 100.0%

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

      8. distribute-lft-out 100.0%

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

      9. fma-def 100.0%

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

      10. +-commutative 100.0%

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

      11. metadata-eval 100.0%

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

      12. +-commutative 100.0%

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

      13. cancel-sign-sub-inv 100.0%

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

      14. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

   5. Taylor expanded in x around inf 98.9%

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

   if 6500 < 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. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. 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%

      \[\leadsto \color{blue}{y \cdot \left(x - 0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+15}:\\ \;\;\;\;x \cdot y + y \cdot -0.5\\ \mathbf{elif}\;y \leq 6500:\\ \;\;\;\;\left(0.918938533204673 + x \cdot y\right) - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \end{array} \]

Alternative 5: 97.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \lor \neg \left(y \leq 1.65\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.6) (not (<= y 1.65)))
   (* y (- x 0.5))
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.6) || !(y <= 1.65)) {
		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 <= (-1.6d0)) .or. (.not. (y <= 1.65d0))) 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 <= -1.6) || !(y <= 1.65)) {
		tmp = y * (x - 0.5);
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.6) or not (y <= 1.65):
		tmp = y * (x - 0.5)
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.6) || !(y <= 1.65))
		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 <= -1.6) || ~((y <= 1.65)))
		tmp = y * (x - 0.5);
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.6], N[Not[LessEqual[y, 1.65]], $MachinePrecision]], N[(y * N[(x - 0.5), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6000000000000001 or 1.6499999999999999 < 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. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. 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 99.0%

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

   if -1.6000000000000001 < y < 1.6499999999999999

   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. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. 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 0 97.9%

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

      1. neg-mul-1 97.9%

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

      2. sub-neg 97.9%

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

   6. Simplified 97.9%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \lor \neg \left(y \leq 1.65\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]

Alternative 6: 97.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.38:\\ \;\;\;\;x \cdot y + y \cdot -0.5\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.38)
   (+ (* x y) (* y -0.5))
   (if (<= y 1.0) (- 0.918938533204673 x) (* y (- x 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.38) {
		tmp = (x * y) + (y * -0.5);
	} else if (y <= 1.0) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = y * (x - 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 (y <= (-1.38d0)) then
        tmp = (x * y) + (y * (-0.5d0))
    else if (y <= 1.0d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = y * (x - 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.38) {
		tmp = (x * y) + (y * -0.5);
	} else if (y <= 1.0) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = y * (x - 0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.38:
		tmp = (x * y) + (y * -0.5)
	elif y <= 1.0:
		tmp = 0.918938533204673 - x
	else:
		tmp = y * (x - 0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.38)
		tmp = Float64(Float64(x * y) + Float64(y * -0.5));
	elseif (y <= 1.0)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = Float64(y * Float64(x - 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.38)
		tmp = (x * y) + (y * -0.5);
	elseif (y <= 1.0)
		tmp = 0.918938533204673 - x;
	else
		tmp = y * (x - 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.38], N[(N[(x * y), $MachinePrecision] + N[(y * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(0.918938533204673 - x), $MachinePrecision], N[(y * N[(x - 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.3799999999999999

   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. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. 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 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-lft-in 100.0%

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

      3. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   if -1.3799999999999999 < 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. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. 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 0 97.9%

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

      1. neg-mul-1 97.9%

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

      2. sub-neg 97.9%

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

   6. Simplified 97.9%

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

   if 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. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. 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%

      \[\leadsto \color{blue}{y \cdot \left(x - 0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.38:\\ \;\;\;\;x \cdot y + y \cdot -0.5\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \end{array} \]

Alternative 7: 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. sub-neg 100.0%

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

   2. +-commutative 100.0%

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

   3. sub-neg 100.0%

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

   4. distribute-rgt-in 100.0%

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

   5. associate-+r+ 100.0%

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

   6. associate-+l+ 100.0%

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

   7. distribute-rgt-neg-in 100.0%

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

   8. distribute-lft-out 100.0%

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

   9. fma-def 100.0%

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

   10. +-commutative 100.0%

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

   11. metadata-eval 100.0%

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

   12. +-commutative 100.0%

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

   13. cancel-sign-sub-inv 100.0%

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

   14. *-lft-identity 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in y around 0 100.0%

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

5. Final simplification 100.0%

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

Alternative 8: 49.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -13.2:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -13.2) (- x) (if (<= x 0.9) (* y -0.5) (- x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -13.2) {
		tmp = -x;
	} else if (x <= 0.9) {
		tmp = y * -0.5;
	} 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 <= (-13.2d0)) then
        tmp = -x
    else if (x <= 0.9d0) then
        tmp = y * (-0.5d0)
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -13.2) {
		tmp = -x;
	} else if (x <= 0.9) {
		tmp = y * -0.5;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -13.2:
		tmp = -x
	elif x <= 0.9:
		tmp = y * -0.5
	else:
		tmp = -x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -13.2)
		tmp = Float64(-x);
	elseif (x <= 0.9)
		tmp = Float64(y * -0.5);
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -13.2)
		tmp = -x;
	elseif (x <= 0.9)
		tmp = y * -0.5;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -13.2], (-x), If[LessEqual[x, 0.9], N[(y * -0.5), $MachinePrecision], (-x)]]

Derivation

1. Split input into 2 regimes

2. if x < -13.199999999999999 or 0.900000000000000022 < 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. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-rgt-in 100.0%

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

      5. associate-+r+ 100.0%

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

      6. associate-+l+ 100.0%

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

      7. distribute-rgt-neg-in 100.0%

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

      8. distribute-lft-out 100.0%

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

      9. fma-def 100.0%

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

      10. +-commutative 100.0%

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

      11. metadata-eval 100.0%

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

      12. +-commutative 100.0%

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

      13. cancel-sign-sub-inv 100.0%

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

      14. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

   5. Taylor expanded in x around inf 96.4%

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

   6. Taylor expanded in y around 0 47.7%

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

      1. neg-mul-1 47.7%

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

   8. Simplified 47.7%

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

   if -13.199999999999999 < x < 0.900000000000000022

   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. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. 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 61.0%

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

   5. Taylor expanded in x around 0 58.9%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -13.2:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 9: 49.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.124:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.124) (- x) (if (<= x 0.5) (* y -0.5) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.124) {
		tmp = -x;
	} else if (x <= 0.5) {
		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 (x <= (-0.124d0)) then
        tmp = -x
    else if (x <= 0.5d0) 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 (x <= -0.124) {
		tmp = -x;
	} else if (x <= 0.5) {
		tmp = y * -0.5;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.124:
		tmp = -x
	elif x <= 0.5:
		tmp = y * -0.5
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.124)
		tmp = Float64(-x);
	elseif (x <= 0.5)
		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 (x <= -0.124)
		tmp = -x;
	elseif (x <= 0.5)
		tmp = y * -0.5;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.124], (-x), If[LessEqual[x, 0.5], N[(y * -0.5), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.124

   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. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-rgt-in 100.0%

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

      5. associate-+r+ 100.0%

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

      6. associate-+l+ 100.0%

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

      7. distribute-rgt-neg-in 100.0%

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

      8. distribute-lft-out 100.0%

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

      9. fma-def 100.0%

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

      10. +-commutative 100.0%

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

      11. metadata-eval 100.0%

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

      12. +-commutative 100.0%

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

      13. cancel-sign-sub-inv 100.0%

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

      14. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

   5. Taylor expanded in x around inf 96.8%

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

   6. Taylor expanded in y around 0 57.0%

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

      1. neg-mul-1 57.0%

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

   8. Simplified 57.0%

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

   if -0.124 < 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. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. 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 61.0%

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

   5. Taylor expanded in x around 0 58.9%

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

   if 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. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. 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 63.5%

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

   5. Taylor expanded in x around inf 61.2%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.124:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 10: 26.2% 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. sub-neg 100.0%

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

   2. +-commutative 100.0%

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

   3. sub-neg 100.0%

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

   4. distribute-rgt-in 100.0%

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

   5. associate-+r+ 100.0%

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

   6. associate-+l+ 100.0%

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

   7. distribute-rgt-neg-in 100.0%

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

   8. distribute-lft-out 100.0%

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

   9. fma-def 100.0%

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

   10. +-commutative 100.0%

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

   11. metadata-eval 100.0%

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

   12. +-commutative 100.0%

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

   13. cancel-sign-sub-inv 100.0%

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

   14. *-lft-identity 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in y around 0 100.0%

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

5. Taylor expanded in x around inf 51.7%

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

6. Taylor expanded in y around 0 26.2%

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

   1. neg-mul-1 26.2%

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

8. Simplified 26.2%

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

9. Final simplification 26.2%

   \[\leadsto -x \]

Reproduce

herbie shell --seed 2023195 
(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))