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

Percentage Accurate: 100.0% → 100.0%

Time: 7.5s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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: 62.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ y (+ x (* y x))) -5e-268) (fma x y x) (fma x y y)))

C

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

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y + Float64(x + Float64(y * x))) <= -5e-268)
		tmp = fma(x, y, x);
	else
		tmp = fma(x, y, y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y + N[(x + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-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) < -4.9999999999999999e-268

   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 52.1

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

   5. Applied rewrites 52.1%

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

   if -4.9999999999999999e-268 < (+.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 60.2

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

   5. Applied rewrites 60.2%

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

4. Final simplification 55.9%

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

Alternative 3: 83.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -112000000000:\\ \;\;\;\;\mathsf{fma}\left(x, y, x\right)\\ \mathbf{elif}\;x \leq 3.75 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(1, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -112000000000.0)
   (fma x y x)
   (if (<= x 3.75e-121) (fma 1.0 x y) (fma x y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -112000000000.0) {
		tmp = fma(x, y, x);
	} else if (x <= 3.75e-121) {
		tmp = fma(1.0, x, y);
	} else {
		tmp = fma(x, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -112000000000.0)
		tmp = fma(x, y, x);
	elseif (x <= 3.75e-121)
		tmp = fma(1.0, x, y);
	else
		tmp = fma(x, y, y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -112000000000.0], N[(x * y + x), $MachinePrecision], If[LessEqual[x, 3.75e-121], N[(1.0 * x + y), $MachinePrecision], N[(x * y + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.12e11

   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 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.12e11 < x < 3.75000000000000013e-121

   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 0

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

   7. Applied rewrites 99.0%

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

   if 3.75000000000000013e-121 < 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 51.9

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

   5. Applied rewrites 51.9%

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

Alternative 4: 62.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 57.7

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

5. Applied rewrites 57.7%

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

Alternative 5: 27.4% 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. 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 64.7

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

5. Applied rewrites 64.7%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 23.3%

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

Reproduce

herbie shell --seed 2024221 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))