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

Percentage Accurate: 100.0% → 100.0%

Time: 6.0s

Alternatives: 5

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

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

FPCore

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

C

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

Julia

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

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(y + 1.0), $MachinePrecision] * x + y), $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. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. distribute-lft1-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 64.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + y \cdot x\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-268}:\\ \;\;\;\;\mathsf{fma}\left(x, y, x\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+176}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x (* y x)))))
   (if (<= t_0 -1e-268)
     (fma x y x)
     (if (<= t_0 5e+176) (* y 2.0) (fma x y x)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + (y * x));
	double tmp;
	if (t_0 <= -1e-268) {
		tmp = fma(x, y, x);
	} else if (t_0 <= 5e+176) {
		tmp = y * 2.0;
	} else {
		tmp = fma(x, y, x);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + Float64(y * x)))
	tmp = 0.0
	if (t_0 <= -1e-268)
		tmp = fma(x, y, x);
	elseif (t_0 <= 5e+176)
		tmp = Float64(y * 2.0);
	else
		tmp = fma(x, y, x);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-268], N[(x * y + x), $MachinePrecision], If[LessEqual[t$95$0, 5e+176], N[(y * 2.0), $MachinePrecision], N[(x * y + x), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x y) x) y) < -9.99999999999999958e-269 or 5e176 < (+.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 x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 74.2

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

   5. Applied rewrites 74.2%

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

   if -9.99999999999999958e-269 < (+.f64 (+.f64 (*.f64 x y) x) y) < 5e176

   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. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around -inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot 2} \]

      2. lower-*.f64 11.9

         \[\leadsto \color{blue}{y \cdot 2} \]

   7. Applied rewrites 11.9%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x + y \cdot x\right) \leq -1 \cdot 10^{-268}:\\ \;\;\;\;\mathsf{fma}\left(x, y, x\right)\\ \mathbf{elif}\;y + \left(x + y \cdot x\right) \leq 5 \cdot 10^{+176}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y + \left(x + y \cdot x\right) \leq -1 \cdot 10^{-268}:\\ \;\;\;\;\mathsf{fma}\left(x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= (+ y (+ x (* y x))) -1e-268) (fma x y x) (fma x y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if ((y + (x + (y * x))) <= -1e-268) {
		tmp = fma(x, y, x);
	} else {
		tmp = fma(x, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (Float64(y + Float64(x + Float64(y * x))) <= -1e-268)
		tmp = fma(x, y, x);
	else
		tmp = fma(x, y, 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[(y + N[(x + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-268], N[(x * y + x), $MachinePrecision], N[(x * y + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x y) x) y) < -9.99999999999999958e-269

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 72.6

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

   5. Applied rewrites 72.6%

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

   if -9.99999999999999958e-269 < (+.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. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 64.5

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

   5. Applied rewrites 64.5%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x + y \cdot x\right) \leq -1 \cdot 10^{-268}:\\ \;\;\;\;\mathsf{fma}\left(x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 32.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -12000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -12000.0) (* y x) (if (<= x 2.0) (* y 2.0) (* y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -12000.0) {
		tmp = y * x;
	} else if (x <= 2.0) {
		tmp = y * 2.0;
	} 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 (x <= (-12000.0d0)) then
        tmp = y * x
    else if (x <= 2.0d0) then
        tmp = y * 2.0d0
    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 (x <= -12000.0) {
		tmp = y * x;
	} else if (x <= 2.0) {
		tmp = y * 2.0;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -12000.0:
		tmp = y * x
	elif x <= 2.0:
		tmp = y * 2.0
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -12000.0)
		tmp = Float64(y * x);
	elseif (x <= 2.0)
		tmp = Float64(y * 2.0);
	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 (x <= -12000.0)
		tmp = y * x;
	elseif (x <= 2.0)
		tmp = y * 2.0;
	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[x, -12000.0], N[(y * x), $MachinePrecision], If[LessEqual[x, 2.0], N[(y * 2.0), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -12000 or 2 < x

   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. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 48.9

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

   5. Applied rewrites 48.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 47.7%

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

   if -12000 < x < 2

   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. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around -inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot 2} \]

      2. lower-*.f64 15.0

         \[\leadsto \color{blue}{y \cdot 2} \]

   7. Applied rewrites 15.0%

      \[\leadsto \color{blue}{y \cdot 2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 8.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ y \cdot 2 \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (* y 2.0))

C

assert(x < y);
double code(double x, double y) {
	return y * 2.0;
}

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 * 2.0d0
end function

Java

assert x < y;
public static double code(double x, double y) {
	return y * 2.0;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return y * 2.0

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(y * 2.0)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = y * 2.0;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(y * 2.0), $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. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. distribute-lft1-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in y around -inf

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{y \cdot 2} \]

   2. lower-*.f64 8.8

      \[\leadsto \color{blue}{y \cdot 2} \]

7. Applied rewrites 8.8%

   \[\leadsto \color{blue}{y \cdot 2} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))