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

Percentage Accurate: 100.0% → 100.0%

Time: 8.7s

Alternatives: 6

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

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. distribute-lft1-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: 86.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x + x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+295}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x (* x y)))))
   (if (<= t_0 -1e+295) (* x y) (if (<= t_0 5e+305) (+ x y) (* x y)))))

C

double code(double x, double y) {
	double t_0 = y + (x + (x * y));
	double tmp;
	if (t_0 <= -1e+295) {
		tmp = x * y;
	} else if (t_0 <= 5e+305) {
		tmp = x + y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + (x * y))
    if (t_0 <= (-1d+295)) then
        tmp = x * y
    else if (t_0 <= 5d+305) then
        tmp = x + y
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y + (x + (x * y));
	double tmp;
	if (t_0 <= -1e+295) {
		tmp = x * y;
	} else if (t_0 <= 5e+305) {
		tmp = x + y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y + (x + (x * y))
	tmp = 0
	if t_0 <= -1e+295:
		tmp = x * y
	elif t_0 <= 5e+305:
		tmp = x + y
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y + Float64(x + Float64(x * y)))
	tmp = 0.0
	if (t_0 <= -1e+295)
		tmp = Float64(x * y);
	elseif (t_0 <= 5e+305)
		tmp = Float64(x + y);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y + (x + (x * y));
	tmp = 0.0;
	if (t_0 <= -1e+295)
		tmp = x * y;
	elseif (t_0 <= 5e+305)
		tmp = x + y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+295], N[(x * y), $MachinePrecision], If[LessEqual[t$95$0, 5e+305], N[(x + y), $MachinePrecision], N[(x * y), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x y) x) y) < -9.9999999999999998e294 or 5.00000000000000009e305 < (+.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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 97.2

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

   8. Applied rewrites 97.2%

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

   if -9.9999999999999998e294 < (+.f64 (+.f64 (*.f64 x y) x) y) < 5.00000000000000009e305

   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

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft1-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 87.2%

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

      1. lower-+.f64 N/A

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

      2. *-lft-identity 87.2

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

   9. Applied rewrites 87.2%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x + x \cdot y\right) \leq -1 \cdot 10^{+295}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y + \left(x + x \cdot y\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 69.6

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

   7. Applied rewrites 69.6%

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

   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 4: 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 5: 87.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -59:\\ \;\;\;\;\mathsf{fma}\left(x, y, x\right)\\ \mathbf{elif}\;x \leq 950000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -59.0) (fma x y x) (if (<= x 950000.0) (+ x y) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -59.0) {
		tmp = fma(x, y, x);
	} else if (x <= 950000.0) {
		tmp = x + y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -59.0)
		tmp = fma(x, y, x);
	elseif (x <= 950000.0)
		tmp = Float64(x + y);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -59.0], N[(x * y + x), $MachinePrecision], If[LessEqual[x, 950000.0], N[(x + y), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -59

   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 -59 < x < 9.5e5

   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

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft1-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 98.2%

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

      1. lower-+.f64 N/A

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

      2. *-lft-identity 98.2

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

   9. Applied rewrites 98.2%

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

   if 9.5e5 < x

   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 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 46.6

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

   8. Applied rewrites 46.6%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -59:\\ \;\;\;\;\mathsf{fma}\left(x, y, x\right)\\ \mathbf{elif}\;x \leq 950000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.3% accurate, 3.0× speedup

Math

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

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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. distribute-lft1-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 76.0%

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

   1. lower-+.f64 N/A

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

   2. *-lft-identity 76.0

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

9. Applied rewrites 76.0%

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

10. Final simplification 76.0%

    \[\leadsto x + y \]
11. 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))