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

Percentage Accurate: 100.0% → 100.0%

Time: 5.0s

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

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

FPCore

(FPCore (x y) :precision binary64 (fma (+ 1.0 x) y x))

C

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

Julia

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

Wolfram

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: 62.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(y \cdot x + x\right) + y \leq -4 \cdot 10^{-288}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (+ (* y x) x) y) -4e-288) (fma y x x) (fma y x y)))

C

double code(double x, double y) {
	double tmp;
	if ((((y * x) + x) + y) <= -4e-288) {
		tmp = fma(y, x, x);
	} else {
		tmp = fma(y, x, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(Float64(y * x) + x) + y) <= -4e-288)
		tmp = fma(y, x, x);
	else
		tmp = fma(y, x, y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[(y * x), $MachinePrecision] + x), $MachinePrecision] + y), $MachinePrecision], -4e-288], 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) < -4.00000000000000023e-288

   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 59.3

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

   5. Applied rewrites 59.3%

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

   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 60.5

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

   8. Applied rewrites 60.5%

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

   if -4.00000000000000023e-288 < (+.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 57.5

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

   5. Applied rewrites 57.5%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x + x\right) + y \leq -4 \cdot 10^{-288}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -108:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{elif}\;x \leq 5.5:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -108.0) (fma y x x) (if (<= x 5.5) (fma 1.0 y x) (fma y x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -108.0) {
		tmp = fma(y, x, x);
	} else if (x <= 5.5) {
		tmp = fma(1.0, y, x);
	} else {
		tmp = fma(y, x, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -108.0)
		tmp = fma(y, x, x);
	elseif (x <= 5.5)
		tmp = fma(1.0, y, x);
	else
		tmp = fma(y, x, y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -108.0], N[(y * x + x), $MachinePrecision], If[LessEqual[x, 5.5], N[(1.0 * y + x), $MachinePrecision], N[(y * x + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -108

   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 46.7

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

   5. Applied rewrites 46.7%

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

   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 100.0

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

   8. Applied rewrites 100.0%

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

   if -108 < x < 5.5

   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 98.7%

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

   if 5.5 < 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. +-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 36.5

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

   5. Applied rewrites 36.5%

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

Alternative 4: 63.1% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (fma y x x))

C

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

Julia

function code(x, y)
	return fma(y, x, x)
end

Wolfram

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 58.4

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

5. Applied rewrites 58.4%

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

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 62.5

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

8. Applied rewrites 62.5%

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

Alternative 5: 26.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ y \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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 58.4

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

5. Applied rewrites 58.4%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 22.2%

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

Reproduce

herbie shell --seed 2024331 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))