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

Percentage Accurate: 100.0% → 100.0%

Time: 6.0s

Alternatives: 12

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(y - 1.0), $MachinePrecision] * x + N[(-0.5 * y + 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. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. fp-cancel-sub-sign-inv N/A

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

   5. associate-+l+ N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. distribute-lft-neg-out N/A

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

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

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. metadata-eval 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 74.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+209}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -0.165:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;x \leq 0.58:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+265}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -8.5e+209)
   (* x y)
   (if (<= x -0.165)
     (- 0.918938533204673 x)
     (if (<= x 0.58)
       (fma -0.5 y 0.918938533204673)
       (if (<= x 7.2e+265) (* x y) (- x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8.5e+209) {
		tmp = x * y;
	} else if (x <= -0.165) {
		tmp = 0.918938533204673 - x;
	} else if (x <= 0.58) {
		tmp = fma(-0.5, y, 0.918938533204673);
	} else if (x <= 7.2e+265) {
		tmp = x * y;
	} else {
		tmp = -x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8.5e+209)
		tmp = Float64(x * y);
	elseif (x <= -0.165)
		tmp = Float64(0.918938533204673 - x);
	elseif (x <= 0.58)
		tmp = fma(-0.5, y, 0.918938533204673);
	elseif (x <= 7.2e+265)
		tmp = Float64(x * y);
	else
		tmp = Float64(-x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8.5e+209], N[(x * y), $MachinePrecision], If[LessEqual[x, -0.165], N[(0.918938533204673 - x), $MachinePrecision], If[LessEqual[x, 0.58], N[(-0.5 * y + 0.918938533204673), $MachinePrecision], If[LessEqual[x, 7.2e+265], N[(x * y), $MachinePrecision], (-x)]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.50000000000000062e209 or 0.57999999999999996 < x < 7.20000000000000005e265

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 66.1

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

   5. Applied rewrites 66.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 65.5%

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

   if -8.50000000000000062e209 < x < -0.165000000000000008

   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. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{1} \cdot x \]

      3. *-lft-identity N/A

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

      4. lower--.f64 58.7

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

   5. Applied rewrites 58.7%

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

   if -0.165000000000000008 < x < 0.57999999999999996

   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. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if 7.20000000000000005e265 < 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. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{1} \cdot x \]

      3. *-lft-identity N/A

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

      4. lower--.f64 77.5

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 77.5%

      \[\leadsto -x \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 72.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+239}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -8 \cdot 10^{+169}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq -230:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.8e+239)
   (* x y)
   (if (<= y -8e+169)
     (* -0.5 y)
     (if (<= y -230.0)
       (* x y)
       (if (<= y 1.85) (- 0.918938533204673 x) (* -0.5 y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.8e+239) {
		tmp = x * y;
	} else if (y <= -8e+169) {
		tmp = -0.5 * y;
	} else if (y <= -230.0) {
		tmp = x * y;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = -0.5 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.8d+239)) then
        tmp = x * y
    else if (y <= (-8d+169)) then
        tmp = (-0.5d0) * y
    else if (y <= (-230.0d0)) then
        tmp = x * y
    else if (y <= 1.85d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = (-0.5d0) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.8e+239) {
		tmp = x * y;
	} else if (y <= -8e+169) {
		tmp = -0.5 * y;
	} else if (y <= -230.0) {
		tmp = x * y;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = -0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.8e+239:
		tmp = x * y
	elif y <= -8e+169:
		tmp = -0.5 * y
	elif y <= -230.0:
		tmp = x * y
	elif y <= 1.85:
		tmp = 0.918938533204673 - x
	else:
		tmp = -0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.8e+239)
		tmp = Float64(x * y);
	elseif (y <= -8e+169)
		tmp = Float64(-0.5 * y);
	elseif (y <= -230.0)
		tmp = Float64(x * y);
	elseif (y <= 1.85)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = Float64(-0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.8e+239)
		tmp = x * y;
	elseif (y <= -8e+169)
		tmp = -0.5 * y;
	elseif (y <= -230.0)
		tmp = x * y;
	elseif (y <= 1.85)
		tmp = 0.918938533204673 - x;
	else
		tmp = -0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.8e+239], N[(x * y), $MachinePrecision], If[LessEqual[y, -8e+169], N[(-0.5 * y), $MachinePrecision], If[LessEqual[y, -230.0], N[(x * y), $MachinePrecision], If[LessEqual[y, 1.85], N[(0.918938533204673 - x), $MachinePrecision], N[(-0.5 * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.8e239 or -7.99999999999999947e169 < y < -230

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 66.2%

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

   if -1.8e239 < y < -7.99999999999999947e169 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 65.2%

      \[\leadsto -0.5 \cdot y \]

   if -230 < 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. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{1} \cdot x \]

      3. *-lft-identity N/A

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

      4. lower--.f64 96.5

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

   5. Applied rewrites 96.5%

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

Alternative 4: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 3.2 \cdot 10^{-23}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5 + x, y, 0.918938533204673\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, -x\right) + 0.918938533204673\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 3.2e-23)))
   (fma (+ -0.5 x) y 0.918938533204673)
   (+ (fma -0.5 y (- x)) 0.918938533204673)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 3.2e-23)) {
		tmp = fma((-0.5 + x), y, 0.918938533204673);
	} else {
		tmp = fma(-0.5, y, -x) + 0.918938533204673;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 3.2e-23))
		tmp = fma(Float64(-0.5 + x), y, 0.918938533204673);
	else
		tmp = Float64(fma(-0.5, y, Float64(-x)) + 0.918938533204673);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 3.2e-23]], $MachinePrecision]], N[(N[(-0.5 + x), $MachinePrecision] * y + 0.918938533204673), $MachinePrecision], N[(N[(-0.5 * y + (-x)), $MachinePrecision] + 0.918938533204673), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 3.19999999999999976e-23 < 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

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.9%

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

   if -1 < y < 3.19999999999999976e-23

   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

      \[\leadsto \color{blue}{\left(x \cdot \left(y - 1\right) - \frac{1}{2} \cdot y\right)} + \frac{918938533204673}{1000000000000000} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-out-- N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. +-commutative N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. mul-1-neg N/A

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

      11. associate-+l- N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. mul-1-neg N/A

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

      17. distribute-lft-out-- N/A

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

      18. fp-cancel-sign-sub-inv N/A

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

      19. +-commutative N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.5%

      \[\leadsto \mathsf{fma}\left(-0.5, y, -x\right) + 0.918938533204673 \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 3.2 \cdot 10^{-23}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5 + x, y, 0.918938533204673\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, -x\right) + 0.918938533204673\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -39000 \lor \neg \left(x \leq 4000000000\right):\\ \;\;\;\;\left(y - 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 + x, y, 0.918938533204673\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -39000.0) (not (<= x 4000000000.0)))
   (* (- y 1.0) x)
   (fma (+ -0.5 x) y 0.918938533204673)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -39000.0) || !(x <= 4000000000.0)) {
		tmp = (y - 1.0) * x;
	} else {
		tmp = fma((-0.5 + x), y, 0.918938533204673);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -39000.0) || !(x <= 4000000000.0))
		tmp = Float64(Float64(y - 1.0) * x);
	else
		tmp = fma(Float64(-0.5 + x), y, 0.918938533204673);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -39000.0], N[Not[LessEqual[x, 4000000000.0]], $MachinePrecision]], N[(N[(y - 1.0), $MachinePrecision] * x), $MachinePrecision], N[(N[(-0.5 + x), $MachinePrecision] * y + 0.918938533204673), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -39000 or 4e9 < 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 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 99.0%

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

   if -39000 < x < 4e9

   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. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.9%

      \[\leadsto \mathsf{fma}\left(-0.5 + x, y, 0.918938533204673\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -39000 \lor \neg \left(x \leq 4000000000\right):\\ \;\;\;\;\left(y - 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 + x, y, 0.918938533204673\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.75 \lor \neg \left(x \leq 0.58\right):\\ \;\;\;\;\left(y - 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.75) (not (<= x 0.58)))
   (* (- y 1.0) x)
   (fma -0.5 y 0.918938533204673)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.75) || !(x <= 0.58)) {
		tmp = (y - 1.0) * x;
	} else {
		tmp = fma(-0.5, y, 0.918938533204673);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.75) || !(x <= 0.58))
		tmp = Float64(Float64(y - 1.0) * x);
	else
		tmp = fma(-0.5, y, 0.918938533204673);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.75], N[Not[LessEqual[x, 0.58]], $MachinePrecision]], N[(N[(y - 1.0), $MachinePrecision] * x), $MachinePrecision], N[(-0.5 * y + 0.918938533204673), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.75 or 0.57999999999999996 < 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 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 98.8%

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

   if -0.75 < x < 0.57999999999999996

   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. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 97.7

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

   5. Applied rewrites 97.7%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.75 \lor \neg \left(x \leq 0.58\right):\\ \;\;\;\;\left(y - 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -230 \lor \neg \left(y \leq 1.85\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -230.0) (not (<= y 1.85))) (* x y) (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -230.0) || !(y <= 1.85)) {
		tmp = x * y;
	} 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 <= (-230.0d0)) .or. (.not. (y <= 1.85d0))) then
        tmp = x * y
    else
        tmp = 0.918938533204673d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -230.0) || !(y <= 1.85)) {
		tmp = x * y;
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -230.0) or not (y <= 1.85):
		tmp = x * y
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -230.0) || !(y <= 1.85))
		tmp = Float64(x * y);
	else
		tmp = Float64(0.918938533204673 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -230.0) || ~((y <= 1.85)))
		tmp = x * y;
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -230.0], N[Not[LessEqual[y, 1.85]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -230 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 47.4%

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

   if -230 < 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. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{1} \cdot x \]

      3. *-lft-identity N/A

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

      4. lower--.f64 96.5

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

   5. Applied rewrites 96.5%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -230 \lor \neg \left(y \leq 1.85\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0007 \lor \neg \left(x \leq 4000000000\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.0007) (not (<= x 4000000000.0))) (- x) 0.918938533204673))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.0007) || !(x <= 4000000000.0)) {
		tmp = -x;
	} else {
		tmp = 0.918938533204673;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-0.0007d0)) .or. (.not. (x <= 4000000000.0d0))) then
        tmp = -x
    else
        tmp = 0.918938533204673d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -0.0007) || !(x <= 4000000000.0)) {
		tmp = -x;
	} else {
		tmp = 0.918938533204673;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.0007) or not (x <= 4000000000.0):
		tmp = -x
	else:
		tmp = 0.918938533204673
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.0007) || !(x <= 4000000000.0))
		tmp = Float64(-x);
	else
		tmp = 0.918938533204673;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -0.0007) || ~((x <= 4000000000.0)))
		tmp = -x;
	else
		tmp = 0.918938533204673;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.0007], N[Not[LessEqual[x, 4000000000.0]], $MachinePrecision]], (-x), 0.918938533204673]

Derivation

1. Split input into 2 regimes

2. if x < -6.99999999999999993e-4 or 4e9 < 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. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{1} \cdot x \]

      3. *-lft-identity N/A

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

      4. lower--.f64 48.0

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

   5. Applied rewrites 48.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 47.1%

      \[\leadsto -x \]

   if -6.99999999999999993e-4 < x < 4e9

   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. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{1} \cdot x \]

      3. *-lft-identity N/A

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

      4. lower--.f64 53.0

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

   5. Applied rewrites 53.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 52.3%

      \[\leadsto 0.918938533204673 \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0007 \lor \neg \left(x \leq 4000000000\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(x \cdot \left(y - 1\right) - \frac{1}{2} \cdot y\right)} + \frac{918938533204673}{1000000000000000} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. fp-cancel-sub-sign-inv N/A

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

   3. mul-1-neg N/A

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

   4. distribute-lft-out-- N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. fp-cancel-sign-sub-inv N/A

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

   8. +-commutative N/A

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

   9. fp-cancel-sign-sub-inv N/A

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

   10. mul-1-neg N/A

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

   11. associate-+l- N/A

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

   12. mul-1-neg N/A

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

   13. *-commutative N/A

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

   14. associate-*r* N/A

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

   15. *-commutative N/A

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

   16. mul-1-neg N/A

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

   17. distribute-lft-out-- N/A

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

   18. fp-cancel-sign-sub-inv N/A

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

   19. +-commutative N/A

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

5. Applied rewrites 100.0%

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

Alternative 10: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(-0.5 + x), $MachinePrecision] * y + N[(0.918938533204673 - 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. Applied rewrites 100.0%

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

Alternative 11: 48.7% 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. fp-cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

      \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{1} \cdot x \]

   3. *-lft-identity N/A

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

   4. lower--.f64 50.8

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

5. Applied rewrites 50.8%

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

Alternative 12: 25.4% 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. fp-cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

      \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{1} \cdot x \]

   3. *-lft-identity N/A

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

   4. lower--.f64 50.8

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

5. Applied rewrites 50.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 31.0%

   \[\leadsto 0.918938533204673 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024326 
(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))