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

Percentage Accurate: 100.0% → 100.0%

Time: 2.1s

Alternatives: 9

Speedup: 1.0×

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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(x \cdot y + x\right) + y \]
2. Step-by-step derivation

   1. +-commutative 100.0%

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

   2. fma-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(x \cdot y + x\right) + y \]
2. Step-by-step derivation

   1. +-commutative 100.0%

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

   2. associate-+l+ 100.0%

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

   3. fma-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 58.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+258}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-179}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.8e+258)
   (* y x)
   (if (<= x -9.2e-179) x (if (<= x 1.0) y (* y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.8e+258) {
		tmp = y * x;
	} else if (x <= -9.2e-179) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.8d+258)) then
        tmp = y * x
    else if (x <= (-9.2d-179)) then
        tmp = x
    else if (x <= 1.0d0) then
        tmp = y
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.8e+258) {
		tmp = y * x;
	} else if (x <= -9.2e-179) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.8e+258:
		tmp = y * x
	elif x <= -9.2e-179:
		tmp = x
	elif x <= 1.0:
		tmp = y
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.8e+258)
		tmp = Float64(y * x);
	elseif (x <= -9.2e-179)
		tmp = x;
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.8e+258)
		tmp = y * x;
	elseif (x <= -9.2e-179)
		tmp = x;
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.8e+258], N[(y * x), $MachinePrecision], If[LessEqual[x, -9.2e-179], x, If[LessEqual[x, 1.0], y, N[(y * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.8000000000000002e258 or 1 < x

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. associate-+l+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

   5. Taylor expanded in y around inf 63.9%

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

   if -5.8000000000000002e258 < x < -9.1999999999999995e-179

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. associate-+l+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 50.3%

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

   if -9.1999999999999995e-179 < x < 1

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. associate-+l+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 82.9%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+258}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-179}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 4: 72.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-113}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) (* y x) (if (<= y 2.7e-113) x (* y (+ x 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = y * x;
	} else if (y <= 2.7e-113) {
		tmp = x;
	} else {
		tmp = y * (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = y * x
    else if (y <= 2.7d-113) then
        tmp = x
    else
        tmp = y * (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = y * x;
	} else if (y <= 2.7e-113) {
		tmp = x;
	} else {
		tmp = y * (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = y * x
	elif y <= 2.7e-113:
		tmp = x
	else:
		tmp = y * (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(y * x);
	elseif (y <= 2.7e-113)
		tmp = x;
	else
		tmp = Float64(y * Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = y * x;
	elseif (y <= 2.7e-113)
		tmp = x;
	else
		tmp = y * (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], N[(y * x), $MachinePrecision], If[LessEqual[y, 2.7e-113], x, N[(y * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 99.9%

      \[\left(x \cdot y + x\right) + y \]
   2. Step-by-step derivation

      1. +-commutative 99.9%

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

      2. associate-+l+ 99.9%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 57.8%

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

   5. Taylor expanded in y around inf 56.1%

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

   if -1 < y < 2.69999999999999996e-113

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. associate-+l+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 74.7%

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

   if 2.69999999999999996e-113 < y

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. associate-+l+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 86.0%

      \[\leadsto \color{blue}{\left(1 + x\right) \cdot y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-113}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \]

Alternative 5: 70.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-179}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9.2e-179) (* x (+ y 1.0)) (* y (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9.2e-179) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y * (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-9.2d-179)) then
        tmp = x * (y + 1.0d0)
    else
        tmp = y * (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -9.2e-179) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y * (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9.2e-179:
		tmp = x * (y + 1.0)
	else:
		tmp = y * (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9.2e-179)
		tmp = Float64(x * Float64(y + 1.0));
	else
		tmp = Float64(y * Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9.2e-179)
		tmp = x * (y + 1.0);
	else
		tmp = y * (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9.2e-179], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.1999999999999995e-179

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. associate-+l+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 77.2%

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

   if -9.1999999999999995e-179 < x

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. associate-+l+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 74.9%

      \[\leadsto \color{blue}{\left(1 + x\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-179}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \]

Alternative 6: 70.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-179}:\\ \;\;\;\;x + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -8.8e-179) (+ x (* y x)) (* y (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8.8e-179) {
		tmp = x + (y * x);
	} else {
		tmp = y * (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-8.8d-179)) then
        tmp = x + (y * x)
    else
        tmp = y * (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -8.8e-179) {
		tmp = x + (y * x);
	} else {
		tmp = y * (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -8.8e-179:
		tmp = x + (y * x)
	else:
		tmp = y * (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8.8e-179)
		tmp = Float64(x + Float64(y * x));
	else
		tmp = Float64(y * Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8.8e-179)
		tmp = x + (y * x);
	else
		tmp = y * (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8.8e-179], N[(x + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(y * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.80000000000000018e-179

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. associate-+l+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 77.2%

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

   5. Taylor expanded in y around 0 77.2%

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

   if -8.80000000000000018e-179 < x

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. associate-+l+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 74.9%

      \[\leadsto \color{blue}{\left(1 + x\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-179}:\\ \;\;\;\;x + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \]

Alternative 7: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ y + \left(x + y \cdot x\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(x \cdot y + x\right) + y \]

2. Final simplification 100.0%

   \[\leadsto y + \left(x + y \cdot x\right) \]

Alternative 8: 47.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-179}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -9.2e-179) x y))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9.2e-179) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-9.2d-179)) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -9.2e-179) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9.2e-179:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9.2e-179)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9.2e-179)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9.2e-179], x, y]

Derivation

1. Split input into 2 regimes

2. if x < -9.1999999999999995e-179

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. associate-+l+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 49.0%

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

   if -9.1999999999999995e-179 < x

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. associate-+l+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 52.2%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-179}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 9: 38.4% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

   \[\left(x \cdot y + x\right) + y \]
2. Step-by-step derivation

   1. +-commutative 100.0%

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

   2. associate-+l+ 100.0%

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

   3. fma-def 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in y around 0 35.5%

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

5. Final simplification 35.5%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023200 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))