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

Percentage Accurate: 100.0% → 100.0%

Time: 8.1s

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(x + 1.0), $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. Add Preprocessing

Alternative 2: 62.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + x \cdot y\\ \mathbf{if}\;y + t\_0 \leq -1 \cdot 10^{-249}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (* x y)))) (if (<= (+ y t_0) -1e-249) t_0 (fma x y y))))

C

double code(double x, double y) {
	double t_0 = x + (x * y);
	double tmp;
	if ((y + t_0) <= -1e-249) {
		tmp = t_0;
	} else {
		tmp = fma(x, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x + Float64(x * y))
	tmp = 0.0
	if (Float64(y + t_0) <= -1e-249)
		tmp = t_0;
	else
		tmp = fma(x, y, y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y + t$95$0), $MachinePrecision], -1e-249], t$95$0, N[(x * y + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x y) x) y) < -1.00000000000000005e-249

   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 69.6

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

   5. Applied rewrites 69.6%

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

   7. Applied rewrites 69.6%

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

   if -1.00000000000000005e-249 < (+.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 62.6

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

   5. Applied rewrites 62.6%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x + x \cdot y\right) \leq -1 \cdot 10^{-249}:\\ \;\;\;\;x + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + \left(x + x \cdot y\right) \leq -1 \cdot 10^{-249}:\\ \;\;\;\;\mathsf{fma}\left(x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ y (+ x (* x y))) -1e-249) (fma x y x) (fma x y y)))

C

double code(double x, double y) {
	double tmp;
	if ((y + (x + (x * y))) <= -1e-249) {
		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(x * y))) <= -1e-249)
		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[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-249], 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) < -1.00000000000000005e-249

   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 69.6

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

   5. Applied rewrites 69.6%

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

   if -1.00000000000000005e-249 < (+.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 62.6

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

   5. Applied rewrites 62.6%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x + x \cdot y\right) \leq -1 \cdot 10^{-249}:\\ \;\;\;\;\mathsf{fma}\left(x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.6% 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 63.0

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

5. Applied rewrites 63.0%

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

Alternative 5: 26.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(x * y), $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 62.0

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

5. Applied rewrites 62.0%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 25.6%

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

9. Final simplification 25.6%

   \[\leadsto x \cdot y \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))