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

Percentage Accurate: 100.0% → 100.0%

Time: 10.0s

Alternatives: 13

Speedup: 1.5×

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, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. associate-+l- N/A

      \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot \frac{1}{2} - \frac{918938533204673}{1000000000000000}\right)} \]

   2. sub-neg N/A

      \[\leadsto \color{blue}{x \cdot \left(y - 1\right) + \left(\mathsf{neg}\left(\left(y \cdot \frac{1}{2} - \frac{918938533204673}{1000000000000000}\right)\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(y - 1\right) \cdot x} + \left(\mathsf{neg}\left(\left(y \cdot \frac{1}{2} - \frac{918938533204673}{1000000000000000}\right)\right)\right) \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - 1, x, \mathsf{neg}\left(\left(y \cdot \frac{1}{2} - \frac{918938533204673}{1000000000000000}\right)\right)\right)} \]

   5. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{y + \left(\mathsf{neg}\left(1\right)\right)}, x, \mathsf{neg}\left(\left(y \cdot \frac{1}{2} - \frac{918938533204673}{1000000000000000}\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{y + \left(\mathsf{neg}\left(1\right)\right)}, x, \mathsf{neg}\left(\left(y \cdot \frac{1}{2} - \frac{918938533204673}{1000000000000000}\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(y + \color{blue}{-1}, x, \mathsf{neg}\left(\left(y \cdot \frac{1}{2} - \frac{918938533204673}{1000000000000000}\right)\right)\right) \]

   8. neg-sub0 N/A

      \[\leadsto \mathsf{fma}\left(y + -1, x, \color{blue}{0 - \left(y \cdot \frac{1}{2} - \frac{918938533204673}{1000000000000000}\right)}\right) \]

   9. associate-+l- N/A

      \[\leadsto \mathsf{fma}\left(y + -1, x, \color{blue}{\left(0 - y \cdot \frac{1}{2}\right) + \frac{918938533204673}{1000000000000000}}\right) \]

   10. neg-sub0 N/A

       \[\leadsto \mathsf{fma}\left(y + -1, x, \color{blue}{\left(\mathsf{neg}\left(y \cdot \frac{1}{2}\right)\right)} + \frac{918938533204673}{1000000000000000}\right) \]

   11. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{fma}\left(y + -1, x, \color{blue}{y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \frac{918938533204673}{1000000000000000}\right) \]

   12. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(y + -1, x, \color{blue}{\mathsf{fma}\left(y, \mathsf{neg}\left(\frac{1}{2}\right), \frac{918938533204673}{1000000000000000}\right)}\right) \]

   13. metadata-eval 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 74.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{elif}\;y \leq 1.1:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+118}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2e-6)
   (fma -0.5 y 0.918938533204673)
   (if (<= y 1.1)
     (- 0.918938533204673 x)
     (if (<= y 1.85e+118) (* y x) (fma -0.5 y 0.918938533204673)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2e-6) {
		tmp = fma(-0.5, y, 0.918938533204673);
	} else if (y <= 1.1) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 1.85e+118) {
		tmp = y * x;
	} else {
		tmp = fma(-0.5, y, 0.918938533204673);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2e-6)
		tmp = fma(-0.5, y, 0.918938533204673);
	elseif (y <= 1.1)
		tmp = Float64(0.918938533204673 - x);
	elseif (y <= 1.85e+118)
		tmp = Float64(y * x);
	else
		tmp = fma(-0.5, y, 0.918938533204673);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2e-6], N[(-0.5 * y + 0.918938533204673), $MachinePrecision], If[LessEqual[y, 1.1], N[(0.918938533204673 - x), $MachinePrecision], If[LessEqual[y, 1.85e+118], N[(y * x), $MachinePrecision], N[(-0.5 * y + 0.918938533204673), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.99999999999999991e-6 or 1.84999999999999993e118 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \frac{1}{2} \cdot y} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot y\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot y\right)\right) + \frac{918938533204673}{1000000000000000}} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot y} + \frac{918938533204673}{1000000000000000} \]

      4. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{-1}{2}} \cdot y + \frac{918938533204673}{1000000000000000} \]

      5. accelerator-lowering-fma.f64 60.2

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

   5. Simplified 60.2%

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

   if -1.99999999999999991e-6 < y < 1.1000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

      3. --lowering--.f64 97.4

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

   5. Simplified 97.4%

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

   if 1.1000000000000001 < y < 1.84999999999999993e118

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(x - \frac{1}{2}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto y \cdot \color{blue}{\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \]

      2. remove-double-neg N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot x}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot x + \frac{1}{2}\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} + -1 \cdot x\right)}\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)} \]

      7. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - y \cdot \left(\frac{1}{2} + -1 \cdot x\right)} \]

      8. --rgt-identity N/A

         \[\leadsto 0 - \color{blue}{\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right) - 0\right)} \]

      9. associate-+l- N/A

         \[\leadsto \color{blue}{\left(0 - y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right) + 0} \]

      10. neg-sub0 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)\right)} + 0 \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(\left(\frac{1}{2} + -1 \cdot x\right)\right)\right)} + 0 \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{neg}\left(\left(\frac{1}{2} + -1 \cdot x\right)\right), 0\right)} \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot x\right)\right)}, 0\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(-1 \cdot x\right)\right), 0\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-1}{2} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right), 0\right) \]

      16. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-1}{2} + \color{blue}{x}, 0\right) \]

      17. +-lowering-+.f64 98.7

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{-0.5 + x}, 0\right) \]

   5. Simplified 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.5 + x, 0\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{x}, 0\right) \]
   7. Step-by-step derivation

   8. Simplified 69.5%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{x}, 0\right) \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 69.5

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

   10. Applied egg-rr 69.5%

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

Alternative 3: 74.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -260:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.4:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+118}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -260.0)
   (* y -0.5)
   (if (<= y 1.4)
     (- 0.918938533204673 x)
     (if (<= y 1.1e+118) (* y x) (* y -0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -260.0) {
		tmp = y * -0.5;
	} else if (y <= 1.4) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 1.1e+118) {
		tmp = y * x;
	} 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 <= (-260.0d0)) then
        tmp = y * (-0.5d0)
    else if (y <= 1.4d0) then
        tmp = 0.918938533204673d0 - x
    else if (y <= 1.1d+118) then
        tmp = y * x
    else
        tmp = y * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -260.0) {
		tmp = y * -0.5;
	} else if (y <= 1.4) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 1.1e+118) {
		tmp = y * x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -260.0:
		tmp = y * -0.5
	elif y <= 1.4:
		tmp = 0.918938533204673 - x
	elif y <= 1.1e+118:
		tmp = y * x
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -260.0)
		tmp = Float64(y * -0.5);
	elseif (y <= 1.4)
		tmp = Float64(0.918938533204673 - x);
	elseif (y <= 1.1e+118)
		tmp = Float64(y * x);
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -260.0)
		tmp = y * -0.5;
	elseif (y <= 1.4)
		tmp = 0.918938533204673 - x;
	elseif (y <= 1.1e+118)
		tmp = y * x;
	else
		tmp = y * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -260.0], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 1.4], N[(0.918938533204673 - x), $MachinePrecision], If[LessEqual[y, 1.1e+118], N[(y * x), $MachinePrecision], N[(y * -0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -260 or 1.09999999999999993e118 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(x - \frac{1}{2}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto y \cdot \color{blue}{\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \]

      2. remove-double-neg N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot x}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot x + \frac{1}{2}\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} + -1 \cdot x\right)}\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)} \]

      7. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - y \cdot \left(\frac{1}{2} + -1 \cdot x\right)} \]

      8. --rgt-identity N/A

         \[\leadsto 0 - \color{blue}{\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right) - 0\right)} \]

      9. associate-+l- N/A

         \[\leadsto \color{blue}{\left(0 - y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right) + 0} \]

      10. neg-sub0 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)\right)} + 0 \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(\left(\frac{1}{2} + -1 \cdot x\right)\right)\right)} + 0 \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{neg}\left(\left(\frac{1}{2} + -1 \cdot x\right)\right), 0\right)} \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot x\right)\right)}, 0\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(-1 \cdot x\right)\right), 0\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-1}{2} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right), 0\right) \]

      16. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-1}{2} + \color{blue}{x}, 0\right) \]

      17. +-lowering-+.f64 99.8

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{-0.5 + x}, 0\right) \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.5 + x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{y \cdot \left(\frac{-1}{2} + x\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} + x\right) \cdot y} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} + x\right) \cdot y} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(x + \frac{-1}{2}\right)} \cdot y \]

      5. +-lowering-+.f64 99.8

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{2}} \cdot y \]
   9. Step-by-step derivation

   10. Simplified 60.4%

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

   if -260 < y < 1.3999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

      3. --lowering--.f64 96.3

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

   5. Simplified 96.3%

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

   if 1.3999999999999999 < y < 1.09999999999999993e118

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(x - \frac{1}{2}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto y \cdot \color{blue}{\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \]

      2. remove-double-neg N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot x}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot x + \frac{1}{2}\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} + -1 \cdot x\right)}\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)} \]

      7. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - y \cdot \left(\frac{1}{2} + -1 \cdot x\right)} \]

      8. --rgt-identity N/A

         \[\leadsto 0 - \color{blue}{\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right) - 0\right)} \]

      9. associate-+l- N/A

         \[\leadsto \color{blue}{\left(0 - y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right) + 0} \]

      10. neg-sub0 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)\right)} + 0 \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(\left(\frac{1}{2} + -1 \cdot x\right)\right)\right)} + 0 \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{neg}\left(\left(\frac{1}{2} + -1 \cdot x\right)\right), 0\right)} \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot x\right)\right)}, 0\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(-1 \cdot x\right)\right), 0\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-1}{2} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right), 0\right) \]

      16. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-1}{2} + \color{blue}{x}, 0\right) \]

      17. +-lowering-+.f64 98.7

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{-0.5 + x}, 0\right) \]

   5. Simplified 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.5 + x, 0\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{x}, 0\right) \]
   7. Step-by-step derivation

   8. Simplified 69.5%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{x}, 0\right) \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 69.5

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

   10. Applied egg-rr 69.5%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -260:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.4:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+118}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+29}:\\ \;\;\;\;\left(y + -1\right) \cdot x\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;0.918938533204673 - \mathsf{fma}\left(y, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y + -1, x, 0.918938533204673\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.4e+29)
   (* (+ y -1.0) x)
   (if (<= x 0.5)
     (- 0.918938533204673 (fma y 0.5 x))
     (fma (+ y -1.0) x 0.918938533204673))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.4e+29) {
		tmp = (y + -1.0) * x;
	} else if (x <= 0.5) {
		tmp = 0.918938533204673 - fma(y, 0.5, x);
	} else {
		tmp = fma((y + -1.0), x, 0.918938533204673);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.4e+29)
		tmp = Float64(Float64(y + -1.0) * x);
	elseif (x <= 0.5)
		tmp = Float64(0.918938533204673 - fma(y, 0.5, x));
	else
		tmp = fma(Float64(y + -1.0), x, 0.918938533204673);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.4e+29], N[(N[(y + -1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 0.5], N[(0.918938533204673 - N[(y * 0.5 + x), $MachinePrecision]), $MachinePrecision], N[(N[(y + -1.0), $MachinePrecision] * x + 0.918938533204673), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4e29

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y - 1, 0\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) \]

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 100.0

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y + -1, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-lowering-+.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   if -1.4e29 < x < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{918938533204673}{1000000000000000} + x \cdot \left(y - 1\right)\right) - \frac{1}{2} \cdot y} \]

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(y, \color{blue}{\frac{1}{2}}, x\right) \]
   7. Step-by-step derivation

   8. Simplified 98.2%

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

   if 0.5 < x

   1. Initial program 99.9%

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

      1. associate-+l- N/A

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot \frac{1}{2} - \frac{918938533204673}{1000000000000000}\right)} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) + \left(\mathsf{neg}\left(\left(y \cdot \frac{1}{2} - \frac{918938533204673}{1000000000000000}\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(y - 1\right) \cdot x} + \left(\mathsf{neg}\left(\left(y \cdot \frac{1}{2} - \frac{918938533204673}{1000000000000000}\right)\right)\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - 1, x, \mathsf{neg}\left(\left(y \cdot \frac{1}{2} - \frac{918938533204673}{1000000000000000}\right)\right)\right)} \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y + \left(\mathsf{neg}\left(1\right)\right)}, x, \mathsf{neg}\left(\left(y \cdot \frac{1}{2} - \frac{918938533204673}{1000000000000000}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y + \left(\mathsf{neg}\left(1\right)\right)}, x, \mathsf{neg}\left(\left(y \cdot \frac{1}{2} - \frac{918938533204673}{1000000000000000}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(y + \color{blue}{-1}, x, \mathsf{neg}\left(\left(y \cdot \frac{1}{2} - \frac{918938533204673}{1000000000000000}\right)\right)\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(y + -1, x, \color{blue}{0 - \left(y \cdot \frac{1}{2} - \frac{918938533204673}{1000000000000000}\right)}\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fma}\left(y + -1, x, \color{blue}{\left(0 - y \cdot \frac{1}{2}\right) + \frac{918938533204673}{1000000000000000}}\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(y + -1, x, \color{blue}{\left(\mathsf{neg}\left(y \cdot \frac{1}{2}\right)\right)} + \frac{918938533204673}{1000000000000000}\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(y + -1, x, \color{blue}{y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \frac{918938533204673}{1000000000000000}\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y + -1, x, \color{blue}{\mathsf{fma}\left(y, \mathsf{neg}\left(\frac{1}{2}\right), \frac{918938533204673}{1000000000000000}\right)}\right) \]

      13. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(y + -1, x, \color{blue}{\frac{918938533204673}{1000000000000000}}\right) \]
   6. Step-by-step derivation

   7. Simplified 100.0%

      \[\leadsto \mathsf{fma}\left(y + -1, x, \color{blue}{0.918938533204673}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+29}:\\ \;\;\;\;\left(y + -1\right) \cdot x\\ \mathbf{elif}\;x \leq 1300000:\\ \;\;\;\;0.918938533204673 - \mathsf{fma}\left(y, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.4e+29)
   (* (+ y -1.0) x)
   (if (<= x 1300000.0)
     (- 0.918938533204673 (fma y 0.5 x))
     (fma y x (- 0.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.4e+29) {
		tmp = (y + -1.0) * x;
	} else if (x <= 1300000.0) {
		tmp = 0.918938533204673 - fma(y, 0.5, x);
	} else {
		tmp = fma(y, x, (0.0 - x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.4e+29)
		tmp = Float64(Float64(y + -1.0) * x);
	elseif (x <= 1300000.0)
		tmp = Float64(0.918938533204673 - fma(y, 0.5, x));
	else
		tmp = fma(y, x, Float64(0.0 - x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.4e+29], N[(N[(y + -1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 1300000.0], N[(0.918938533204673 - N[(y * 0.5 + x), $MachinePrecision]), $MachinePrecision], N[(y * x + N[(0.0 - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4e29

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y - 1, 0\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) \]

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 100.0

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y + -1, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-lowering-+.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   if -1.4e29 < x < 1.3e6

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{918938533204673}{1000000000000000} + x \cdot \left(y - 1\right)\right) - \frac{1}{2} \cdot y} \]

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(y, \color{blue}{\frac{1}{2}}, x\right) \]
   7. Step-by-step derivation

   8. Simplified 98.2%

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

   if 1.3e6 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y - 1, 0\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) \]

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 99.7

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

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y + -1, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. neg-mul-1 N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{0 - x}\right) \]

      6. --lowering--.f64 99.7

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{0 - x}\right) \]

   7. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, 0 - x\right)} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 99.7

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

   9. Applied egg-rr 99.7%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+29}:\\ \;\;\;\;\left(y + -1\right) \cdot x\\ \mathbf{elif}\;x \leq 1300000:\\ \;\;\;\;0.918938533204673 - \mathsf{fma}\left(y, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.72:\\ \;\;\;\;\left(y + -1\right) \cdot x\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.72)
   (* (+ y -1.0) x)
   (if (<= x 0.65) (fma -0.5 y 0.918938533204673) (fma y x (- 0.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.72) {
		tmp = (y + -1.0) * x;
	} else if (x <= 0.65) {
		tmp = fma(-0.5, y, 0.918938533204673);
	} else {
		tmp = fma(y, x, (0.0 - x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.72)
		tmp = Float64(Float64(y + -1.0) * x);
	elseif (x <= 0.65)
		tmp = fma(-0.5, y, 0.918938533204673);
	else
		tmp = fma(y, x, Float64(0.0 - x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.72], N[(N[(y + -1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 0.65], N[(-0.5 * y + 0.918938533204673), $MachinePrecision], N[(y * x + N[(0.0 - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.71999999999999997

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y - 1, 0\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) \]

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 99.3

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

   5. Simplified 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y + -1, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-lowering-+.f64 99.3

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

   7. Applied egg-rr 99.3%

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

   if -0.71999999999999997 < x < 0.650000000000000022

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \frac{1}{2} \cdot y} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot y\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot y\right)\right) + \frac{918938533204673}{1000000000000000}} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot y} + \frac{918938533204673}{1000000000000000} \]

      4. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{-1}{2}} \cdot y + \frac{918938533204673}{1000000000000000} \]

      5. accelerator-lowering-fma.f64 97.8

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

   5. Simplified 97.8%

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

   if 0.650000000000000022 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y - 1, 0\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) \]

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 99.7

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

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y + -1, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. neg-mul-1 N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{0 - x}\right) \]

      6. --lowering--.f64 99.7

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{0 - x}\right) \]

   7. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, 0 - x\right)} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 99.7

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

   9. Applied egg-rr 99.7%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.72:\\ \;\;\;\;\left(y + -1\right) \cdot x\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + -1\right) \cdot x\\ \mathbf{if}\;x \leq -0.7:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.75:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (+ y -1.0) x)))
   (if (<= x -0.7) t_0 (if (<= x 0.75) (fma -0.5 y 0.918938533204673) t_0))))

C

double code(double x, double y) {
	double t_0 = (y + -1.0) * x;
	double tmp;
	if (x <= -0.7) {
		tmp = t_0;
	} else if (x <= 0.75) {
		tmp = fma(-0.5, y, 0.918938533204673);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y + -1.0) * x)
	tmp = 0.0
	if (x <= -0.7)
		tmp = t_0;
	elseif (x <= 0.75)
		tmp = fma(-0.5, y, 0.918938533204673);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + -1.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -0.7], t$95$0, If[LessEqual[x, 0.75], N[(-0.5 * y + 0.918938533204673), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.69999999999999996 or 0.75 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y - 1, 0\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) \]

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 99.5

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

   5. Simplified 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y + -1, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-lowering-+.f64 99.5

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

   7. Applied egg-rr 99.5%

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

   if -0.69999999999999996 < x < 0.75

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \frac{1}{2} \cdot y} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot y\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot y\right)\right) + \frac{918938533204673}{1000000000000000}} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot y} + \frac{918938533204673}{1000000000000000} \]

      4. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{-1}{2}} \cdot y + \frac{918938533204673}{1000000000000000} \]

      5. accelerator-lowering-fma.f64 97.8

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

   5. Simplified 97.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x + -0.5\right)\\ \mathbf{if}\;y \leq -1.45:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (+ x -0.5))))
   (if (<= y -1.45) t_0 (if (<= y 1.85) (- 0.918938533204673 x) t_0))))

C

double code(double x, double y) {
	double t_0 = y * (x + -0.5);
	double tmp;
	if (y <= -1.45) {
		tmp = t_0;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (x + (-0.5d0))
    if (y <= (-1.45d0)) then
        tmp = t_0
    else if (y <= 1.85d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (x + -0.5);
	double tmp;
	if (y <= -1.45) {
		tmp = t_0;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x + -0.5)
	tmp = 0
	if y <= -1.45:
		tmp = t_0
	elif y <= 1.85:
		tmp = 0.918938533204673 - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x + -0.5))
	tmp = 0.0
	if (y <= -1.45)
		tmp = t_0;
	elseif (y <= 1.85)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x + -0.5);
	tmp = 0.0;
	if (y <= -1.45)
		tmp = t_0;
	elseif (y <= 1.85)
		tmp = 0.918938533204673 - x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x + -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.45], t$95$0, If[LessEqual[y, 1.85], N[(0.918938533204673 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.44999999999999996 or 1.8500000000000001 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(x - \frac{1}{2}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto y \cdot \color{blue}{\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \]

      2. remove-double-neg N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot x}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot x + \frac{1}{2}\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} + -1 \cdot x\right)}\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)} \]

      7. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - y \cdot \left(\frac{1}{2} + -1 \cdot x\right)} \]

      8. --rgt-identity N/A

         \[\leadsto 0 - \color{blue}{\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right) - 0\right)} \]

      9. associate-+l- N/A

         \[\leadsto \color{blue}{\left(0 - y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right) + 0} \]

      10. neg-sub0 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)\right)} + 0 \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(\left(\frac{1}{2} + -1 \cdot x\right)\right)\right)} + 0 \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{neg}\left(\left(\frac{1}{2} + -1 \cdot x\right)\right), 0\right)} \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot x\right)\right)}, 0\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(-1 \cdot x\right)\right), 0\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-1}{2} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right), 0\right) \]

      16. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-1}{2} + \color{blue}{x}, 0\right) \]

      17. +-lowering-+.f64 99.0

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{-0.5 + x}, 0\right) \]

   5. Simplified 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.5 + x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{y \cdot \left(\frac{-1}{2} + x\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} + x\right) \cdot y} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} + x\right) \cdot y} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(x + \frac{-1}{2}\right)} \cdot y \]

      5. +-lowering-+.f64 99.0

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

   7. Applied egg-rr 99.0%

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

   if -1.44999999999999996 < y < 1.8500000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

      3. --lowering--.f64 96.9

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

   5. Simplified 96.9%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45:\\ \;\;\;\;y \cdot \left(x + -0.5\right)\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -260:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -260.0)
   (* y -0.5)
   (if (<= y 1.85) (- 0.918938533204673 x) (* y -0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -260.0) {
		tmp = y * -0.5;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} 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 <= (-260.0d0)) then
        tmp = y * (-0.5d0)
    else if (y <= 1.85d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = y * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -260.0) {
		tmp = y * -0.5;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -260.0:
		tmp = y * -0.5
	elif y <= 1.85:
		tmp = 0.918938533204673 - x
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -260.0)
		tmp = Float64(y * -0.5);
	elseif (y <= 1.85)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -260.0)
		tmp = y * -0.5;
	elseif (y <= 1.85)
		tmp = 0.918938533204673 - x;
	else
		tmp = y * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -260.0], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 1.85], N[(0.918938533204673 - x), $MachinePrecision], N[(y * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -260 or 1.8500000000000001 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(x - \frac{1}{2}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto y \cdot \color{blue}{\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \]

      2. remove-double-neg N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot x}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot x + \frac{1}{2}\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} + -1 \cdot x\right)}\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)} \]

      7. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - y \cdot \left(\frac{1}{2} + -1 \cdot x\right)} \]

      8. --rgt-identity N/A

         \[\leadsto 0 - \color{blue}{\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right) - 0\right)} \]

      9. associate-+l- N/A

         \[\leadsto \color{blue}{\left(0 - y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right) + 0} \]

      10. neg-sub0 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)\right)} + 0 \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(\left(\frac{1}{2} + -1 \cdot x\right)\right)\right)} + 0 \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{neg}\left(\left(\frac{1}{2} + -1 \cdot x\right)\right), 0\right)} \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot x\right)\right)}, 0\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(-1 \cdot x\right)\right), 0\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-1}{2} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right), 0\right) \]

      16. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-1}{2} + \color{blue}{x}, 0\right) \]

      17. +-lowering-+.f64 99.6

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{-0.5 + x}, 0\right) \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.5 + x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{y \cdot \left(\frac{-1}{2} + x\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} + x\right) \cdot y} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} + x\right) \cdot y} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(x + \frac{-1}{2}\right)} \cdot y \]

      5. +-lowering-+.f64 99.6

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

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{2}} \cdot y \]
   9. Step-by-step derivation

   10. Simplified 54.2%

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

   if -260 < y < 1.8500000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

      3. --lowering--.f64 96.3

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

   5. Simplified 96.3%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -260:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.92:\\ \;\;\;\;0 - x\\ \mathbf{elif}\;x \leq 0.92:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.92) (- 0.0 x) (if (<= x 0.92) 0.918938533204673 (- 0.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.92) {
		tmp = 0.0 - x;
	} else if (x <= 0.92) {
		tmp = 0.918938533204673;
	} else {
		tmp = 0.0 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.92d0)) then
        tmp = 0.0d0 - x
    else if (x <= 0.92d0) then
        tmp = 0.918938533204673d0
    else
        tmp = 0.0d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.92) {
		tmp = 0.0 - x;
	} else if (x <= 0.92) {
		tmp = 0.918938533204673;
	} else {
		tmp = 0.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.92:
		tmp = 0.0 - x
	elif x <= 0.92:
		tmp = 0.918938533204673
	else:
		tmp = 0.0 - x
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[x, -0.92], N[(0.0 - x), $MachinePrecision], If[LessEqual[x, 0.92], 0.918938533204673, N[(0.0 - x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.92000000000000004 or 0.92000000000000004 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

      3. --lowering--.f64 51.2

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

   5. Simplified 51.2%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - x} \]

      3. --lowering--.f64 50.8

         \[\leadsto \color{blue}{0 - x} \]

   8. Simplified 50.8%

      \[\leadsto \color{blue}{0 - x} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. neg-lowering-neg.f64 50.8

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

   10. Applied egg-rr 50.8%

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

   if -0.92000000000000004 < x < 0.92000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

      3. --lowering--.f64 55.7

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

   5. Simplified 55.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000}} \]
   7. Step-by-step derivation

   8. Simplified 55.5%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.92:\\ \;\;\;\;0 - x\\ \mathbf{elif}\;x \leq 0.92:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ 0.918938533204673 - \mathsf{fma}\left(y, 0.5 - x, x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 0.918938533204673 (fma y (- 0.5 x) x)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(\frac{918938533204673}{1000000000000000} + x \cdot \left(y - 1\right)\right) - \frac{1}{2} \cdot y} \]

5. Simplified 100.0%

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

Alternative 12: 51.2% accurate, 5.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (- 0.918938533204673 x))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 0.918938533204673 - x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

   2. unsub-neg N/A

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

   3. --lowering--.f64 53.8

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

5. Simplified 53.8%

   \[\leadsto \color{blue}{0.918938533204673 - x} \]
6. Add Preprocessing

Alternative 13: 26.7% accurate, 20.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 0.918938533204673)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 0.918938533204673

Julia

function code(x, y)
	return 0.918938533204673
end

MATLAB

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

Wolfram

code[x_, y_] := 0.918938533204673

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

   2. unsub-neg N/A

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

   3. --lowering--.f64 53.8

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

5. Simplified 53.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000}} \]
7. Step-by-step derivation

8. Simplified 32.8%

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

Reproduce

herbie shell --seed 2024196 
(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))