Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B

Percentage Accurate: 100.0% → 100.0%

Time: 7.1s

Alternatives: 7

Speedup: 1.2×

Specification

Math

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

FPCore

(FPCore (x y) :precision binary64 (+ (+ (* x y) x) y))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return ((x * y) + x) + y

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (+ (+ (* x y) x) y))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return ((x * y) + x) + y

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.2× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (fma (+ 1.0 x) y x))

C

assert(x < y);
double code(double x, double y) {
	return fma((1.0 + x), y, x);
}

Julia

x, y = sort([x, y])
function code(x, y)
	return fma(Float64(1.0 + x), y, x)
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(1.0 + x), $MachinePrecision] * y + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+r+ N/A

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

   5. lift-*.f64 N/A

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

   6. distribute-rgt1-in N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(1 + x, y, x\right) \]
6. Add Preprocessing

Alternative 2: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;\left(y \cdot x + x\right) + y \leq -5 \cdot 10^{-184}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= (+ (+ (* y x) x) y) -5e-184) (fma y x x) (fma y x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if ((((y * x) + x) + y) <= -5e-184) {
		tmp = fma(y, x, x);
	} else {
		tmp = fma(y, x, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (Float64(Float64(Float64(y * x) + x) + y) <= -5e-184)
		tmp = fma(y, x, x);
	else
		tmp = fma(y, x, y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[N[(N[(N[(y * x), $MachinePrecision] + x), $MachinePrecision] + y), $MachinePrecision], -5e-184], N[(y * x + x), $MachinePrecision], N[(y * x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x y) x) y) < -5.00000000000000003e-184

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 64.9

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

   5. Applied rewrites 64.9%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 58.9

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

   8. Applied rewrites 58.9%

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

   if -5.00000000000000003e-184 < (+.f64 (+.f64 (*.f64 x y) x) y)

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 64.2

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

   5. Applied rewrites 64.2%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x + x\right) + y \leq -5 \cdot 10^{-184}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{-9}:\\ \;\;\;\;\left(y + 1\right) \cdot x\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -2.65e-9)
   (* (+ y 1.0) x)
   (if (<= y 1.3e-7) (fma 1.0 y x) (fma y x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -2.65e-9) {
		tmp = (y + 1.0) * x;
	} else if (y <= 1.3e-7) {
		tmp = fma(1.0, y, x);
	} else {
		tmp = fma(y, x, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -2.65e-9)
		tmp = Float64(Float64(y + 1.0) * x);
	elseif (y <= 1.3e-7)
		tmp = fma(1.0, y, x);
	else
		tmp = fma(y, x, y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -2.65e-9], N[(N[(y + 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 1.3e-7], N[(1.0 * y + x), $MachinePrecision], N[(y * x + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.65000000000000015e-9

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 95.7

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

   5. Applied rewrites 95.7%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 44.2

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

   8. Applied rewrites 44.2%

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

   10. Applied rewrites 44.2%

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

   if -2.65000000000000015e-9 < y < 1.29999999999999999e-7

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-rgt1-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 100.0%

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

   if 1.29999999999999999e-7 < y

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{-9}:\\ \;\;\;\;\left(y + 1\right) \cdot x\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -2.65e-9) (fma y x x) (if (<= y 1.3e-7) (fma 1.0 y x) (fma y x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -2.65e-9) {
		tmp = fma(y, x, x);
	} else if (y <= 1.3e-7) {
		tmp = fma(1.0, y, x);
	} else {
		tmp = fma(y, x, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -2.65e-9)
		tmp = fma(y, x, x);
	elseif (y <= 1.3e-7)
		tmp = fma(1.0, y, x);
	else
		tmp = fma(y, x, y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -2.65e-9], N[(y * x + x), $MachinePrecision], If[LessEqual[y, 1.3e-7], N[(1.0 * y + x), $MachinePrecision], N[(y * x + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.65000000000000015e-9

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 95.7

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

   5. Applied rewrites 95.7%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 44.2

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

   8. Applied rewrites 44.2%

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

   if -2.65000000000000015e-9 < y < 1.29999999999999999e-7

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-rgt1-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 100.0%

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

   if 1.29999999999999999e-7 < y

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 99.6

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

   5. Applied rewrites 99.6%

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

Alternative 5: 62.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) (* y x) (if (<= y 1.0) (* 1.0 x) (* y x))))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = y * x
    else if (y <= 1.0d0) then
        tmp = 1.0d0 * x
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = y * x;
	elseif (y <= 1.0)
		tmp = 1.0 * x;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -1.0], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 * x), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 42.6%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 28.6

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

   5. Applied rewrites 28.6%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 73.1

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

   8. Applied rewrites 73.1%

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

   10. Applied rewrites 73.1%

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

   11. Taylor expanded in y around 0

       \[\leadsto 1 \cdot x \]
   12. Step-by-step derivation

   13. Applied rewrites 72.7%

       \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 63.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \mathsf{fma}\left(y, x, x\right) \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (fma y x x))

C

assert(x < y);
double code(double x, double y) {
	return fma(y, x, x);
}

Julia

x, y = sort([x, y])
function code(x, y)
	return fma(y, x, x)
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(y * x + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x \cdot y + x\right) + y \]
2. Add Preprocessing

3. Taylor expanded in y around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 64.5

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

5. Applied rewrites 64.5%

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

6. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. lower-fma.f64 58.0

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

8. Applied rewrites 58.0%

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

Alternative 7: 27.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ y \cdot x \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (* y x))

C

assert(x < y);
double code(double x, double y) {
	return y * x;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y * x
end function

Java

assert x < y;
public static double code(double x, double y) {
	return y * x;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return y * x

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(y * x)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = y * x;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(y * x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x \cdot y + x\right) + y \]
2. Add Preprocessing

3. Taylor expanded in y around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 64.5

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

5. Applied rewrites 64.5%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 23.7%

   \[\leadsto y \cdot \color{blue}{x} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024296 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))