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

Percentage Accurate: 100.0% → 100.0%

Time: 2.4s

Alternatives: 7

Speedup: 0.1×

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

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

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 2: 60.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-114}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+16}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+34}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+93}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+141}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{+200}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   (* x y)
   (if (<= y 3.8e-114)
     x
     (if (<= y 2.6e+16)
       y
       (if (<= y 2.2e+34)
         (* x y)
         (if (<= y 5.2e+93)
           y
           (if (<= y 1.8e+141) (* x y) (if (<= y 9.6e+200) y (* x y)))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x * y;
	} else if (y <= 3.8e-114) {
		tmp = x;
	} else if (y <= 2.6e+16) {
		tmp = y;
	} else if (y <= 2.2e+34) {
		tmp = x * y;
	} else if (y <= 5.2e+93) {
		tmp = y;
	} else if (y <= 1.8e+141) {
		tmp = x * y;
	} else if (y <= 9.6e+200) {
		tmp = 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) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x * y
    else if (y <= 3.8d-114) then
        tmp = x
    else if (y <= 2.6d+16) then
        tmp = y
    else if (y <= 2.2d+34) then
        tmp = x * y
    else if (y <= 5.2d+93) then
        tmp = y
    else if (y <= 1.8d+141) then
        tmp = x * y
    else if (y <= 9.6d+200) then
        tmp = y
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x * y;
	} else if (y <= 3.8e-114) {
		tmp = x;
	} else if (y <= 2.6e+16) {
		tmp = y;
	} else if (y <= 2.2e+34) {
		tmp = x * y;
	} else if (y <= 5.2e+93) {
		tmp = y;
	} else if (y <= 1.8e+141) {
		tmp = x * y;
	} else if (y <= 9.6e+200) {
		tmp = y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x * y
	elif y <= 3.8e-114:
		tmp = x
	elif y <= 2.6e+16:
		tmp = y
	elif y <= 2.2e+34:
		tmp = x * y
	elif y <= 5.2e+93:
		tmp = y
	elif y <= 1.8e+141:
		tmp = x * y
	elif y <= 9.6e+200:
		tmp = y
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(x * y);
	elseif (y <= 3.8e-114)
		tmp = x;
	elseif (y <= 2.6e+16)
		tmp = y;
	elseif (y <= 2.2e+34)
		tmp = Float64(x * y);
	elseif (y <= 5.2e+93)
		tmp = y;
	elseif (y <= 1.8e+141)
		tmp = Float64(x * y);
	elseif (y <= 9.6e+200)
		tmp = y;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x * y;
	elseif (y <= 3.8e-114)
		tmp = x;
	elseif (y <= 2.6e+16)
		tmp = y;
	elseif (y <= 2.2e+34)
		tmp = x * y;
	elseif (y <= 5.2e+93)
		tmp = y;
	elseif (y <= 1.8e+141)
		tmp = x * y;
	elseif (y <= 9.6e+200)
		tmp = y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], N[(x * y), $MachinePrecision], If[LessEqual[y, 3.8e-114], x, If[LessEqual[y, 2.6e+16], y, If[LessEqual[y, 2.2e+34], N[(x * y), $MachinePrecision], If[LessEqual[y, 5.2e+93], y, If[LessEqual[y, 1.8e+141], N[(x * y), $MachinePrecision], If[LessEqual[y, 9.6e+200], y, N[(x * y), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 2.6e16 < y < 2.2000000000000002e34 or 5.19999999999999999e93 < y < 1.8000000000000001e141 or 9.6000000000000003e200 < 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 x around inf 53.6%

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

   5. Taylor expanded in y around inf 52.0%

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

   if -1 < y < 3.7999999999999998e-114

   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 78.2%

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

   if 3.7999999999999998e-114 < y < 2.6e16 or 2.2000000000000002e34 < y < 5.19999999999999999e93 or 1.8000000000000001e141 < y < 9.6000000000000003e200

   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 55.4%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-114}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+16}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+34}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+93}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+141}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{+200}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3: 85.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-83} \lor \neg \left(x \leq 4 \cdot 10^{-8}\right):\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -6.2e-83) (not (<= x 4e-8))) (* x (+ y 1.0)) y))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -6.2e-83) || !(x <= 4e-8)) {
		tmp = x * (y + 1.0);
	} 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 <= (-6.2d-83)) .or. (.not. (x <= 4d-8))) then
        tmp = x * (y + 1.0d0)
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -6.2e-83) || !(x <= 4e-8)) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -6.2e-83) or not (x <= 4e-8):
		tmp = x * (y + 1.0)
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -6.2e-83) || !(x <= 4e-8))
		tmp = Float64(x * Float64(y + 1.0));
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -6.2e-83) || ~((x <= 4e-8)))
		tmp = x * (y + 1.0);
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -6.2e-83], N[Not[LessEqual[x, 4e-8]], $MachinePrecision]], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], y]

Derivation

1. Split input into 2 regimes

2. if x < -6.19999999999999985e-83 or 4.0000000000000001e-8 < 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 91.5%

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

   if -6.19999999999999985e-83 < x < 4.0000000000000001e-8

   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 78.7%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-83} \lor \neg \left(x \leq 4 \cdot 10^{-8}\right):\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 4: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-118}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.65e-118) (* x (+ y 1.0)) (+ y (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.65e-118) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y + (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.65d-118) then
        tmp = x * (y + 1.0d0)
    else
        tmp = y + (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.65e-118) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y + (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.65e-118:
		tmp = x * (y + 1.0)
	else:
		tmp = y + (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.65e-118)
		tmp = Float64(x * Float64(y + 1.0));
	else
		tmp = Float64(y + Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.65e-118)
		tmp = x * (y + 1.0);
	else
		tmp = y + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.65e-118], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(y + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.65e-118

   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 64.3%

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

   if 1.65e-118 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 87.6%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-118}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot y\\ \end{array} \]

Alternative 5: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + (y * (x + 1.0d0))
end function

Java

public static double code(double x, double y) {
	return x + (y * (x + 1.0));
}

Python

def code(x, y):
	return x + (y * (x + 1.0))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = x + (y * (x + 1.0));
end

Wolfram

code[x_, y_] := N[(x + N[(y * N[(x + 1.0), $MachinePrecision]), $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. Taylor expanded in y around 0 100.0%

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

5. Final simplification 100.0%

   \[\leadsto x + y \cdot \left(x + 1\right) \]

Alternative 6: 49.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-114}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 3.5e-114) x y))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.5e-114) {
		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 (y <= 3.5d-114) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.5e-114) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.5e-114:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.5e-114)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.5e-114)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.5e-114], x, y]

Derivation

1. Split input into 2 regimes

2. if y < 3.5e-114

   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 45.8%

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

   if 3.5e-114 < 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 x around 0 45.6%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-114}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 7: 39.1% 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 34.9%

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

5. Final simplification 34.9%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023257 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))