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

Percentage Accurate: 100.0% → 100.0%

Time: 4.6s

Alternatives: 6

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(x \cdot y + x\right) + y \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto y + \left(x + x \cdot y\right) \]
4. Add Preprocessing

Alternative 2: 60.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+273}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{+179}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1 \cdot 10^{+130}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{+103}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1.18 \cdot 10^{-110}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.2e+273)
   x
   (if (<= x -1.2e+179)
     (* x y)
     (if (<= x -1e+130)
       x
       (if (<= x -2.1e+103)
         (* x y)
         (if (<= x -1.18e-110) x (if (<= x 1.0) y (* x y))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.2e+273) {
		tmp = x;
	} else if (x <= -1.2e+179) {
		tmp = x * y;
	} else if (x <= -1e+130) {
		tmp = x;
	} else if (x <= -2.1e+103) {
		tmp = x * y;
	} else if (x <= -1.18e-110) {
		tmp = x;
	} else if (x <= 1.0) {
		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 (x <= (-2.2d+273)) then
        tmp = x
    else if (x <= (-1.2d+179)) then
        tmp = x * y
    else if (x <= (-1d+130)) then
        tmp = x
    else if (x <= (-2.1d+103)) then
        tmp = x * y
    else if (x <= (-1.18d-110)) then
        tmp = x
    else if (x <= 1.0d0) 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 (x <= -2.2e+273) {
		tmp = x;
	} else if (x <= -1.2e+179) {
		tmp = x * y;
	} else if (x <= -1e+130) {
		tmp = x;
	} else if (x <= -2.1e+103) {
		tmp = x * y;
	} else if (x <= -1.18e-110) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.2e+273:
		tmp = x
	elif x <= -1.2e+179:
		tmp = x * y
	elif x <= -1e+130:
		tmp = x
	elif x <= -2.1e+103:
		tmp = x * y
	elif x <= -1.18e-110:
		tmp = x
	elif x <= 1.0:
		tmp = y
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.2e+273)
		tmp = x;
	elseif (x <= -1.2e+179)
		tmp = Float64(x * y);
	elseif (x <= -1e+130)
		tmp = x;
	elseif (x <= -2.1e+103)
		tmp = Float64(x * y);
	elseif (x <= -1.18e-110)
		tmp = x;
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.2e+273)
		tmp = x;
	elseif (x <= -1.2e+179)
		tmp = x * y;
	elseif (x <= -1e+130)
		tmp = x;
	elseif (x <= -2.1e+103)
		tmp = x * y;
	elseif (x <= -1.18e-110)
		tmp = x;
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.2e+273], x, If[LessEqual[x, -1.2e+179], N[(x * y), $MachinePrecision], If[LessEqual[x, -1e+130], x, If[LessEqual[x, -2.1e+103], N[(x * y), $MachinePrecision], If[LessEqual[x, -1.18e-110], x, If[LessEqual[x, 1.0], y, N[(x * y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1999999999999999e273 or -1.20000000000000006e179 < x < -1.0000000000000001e130 or -2.1000000000000002e103 < x < -1.18e-110

   1. Initial program 100.0%

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

      1. add-cube-cbrt 98.3%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(x \cdot y + x\right) + y} \cdot \sqrt[3]{\left(x \cdot y + x\right) + y}\right) \cdot \sqrt[3]{\left(x \cdot y + x\right) + y}} \]

      2. pow3 98.3%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(x \cdot y + x\right) + y}\right)}^{3}} \]

      3. +-commutative 98.3%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(x + x \cdot y\right)} + y}\right)}^{3} \]

      4. *-un-lft-identity 98.3%

         \[\leadsto {\left(\sqrt[3]{\left(\color{blue}{1 \cdot x} + x \cdot y\right) + y}\right)}^{3} \]

      5. *-commutative 98.3%

         \[\leadsto {\left(\sqrt[3]{\left(1 \cdot x + \color{blue}{y \cdot x}\right) + y}\right)}^{3} \]

      6. distribute-rgt-out 98.2%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{x \cdot \left(1 + y\right)} + y}\right)}^{3} \]

      7. fma-define 98.2%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(x, 1 + y, y\right)}}\right)}^{3} \]

   4. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x, 1 + y, y\right)}\right)}^{3}} \]

   5. Taylor expanded in y around 0 55.1%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot x} \]
   6. Step-by-step derivation

      1. pow-base-1 55.1%

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

      2. *-lft-identity 55.1%

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

   7. Simplified 55.1%

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

   if -2.1999999999999999e273 < x < -1.20000000000000006e179 or -1.0000000000000001e130 < x < -2.1000000000000002e103 or 1 < x

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 62.0%

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

   4. Taylor expanded in x around inf 60.9%

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

      1. *-commutative 60.9%

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

   6. Simplified 60.9%

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

   if -1.18e-110 < x < 1

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 77.1%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+273}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{+179}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1 \cdot 10^{+130}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{+103}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1.18 \cdot 10^{-110}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+273}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -6 \cdot 10^{+178}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{+129}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{+97} \lor \neg \left(x \leq 3300\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.2e+273)
   x
   (if (<= x -6e+178)
     (* x y)
     (if (<= x -6.6e+129)
       x
       (if (or (<= x -1.4e+97) (not (<= x 3300.0))) (* x y) (+ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.2e+273) {
		tmp = x;
	} else if (x <= -6e+178) {
		tmp = x * y;
	} else if (x <= -6.6e+129) {
		tmp = x;
	} else if ((x <= -1.4e+97) || !(x <= 3300.0)) {
		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) :: tmp
    if (x <= (-1.2d+273)) then
        tmp = x
    else if (x <= (-6d+178)) then
        tmp = x * y
    else if (x <= (-6.6d+129)) then
        tmp = x
    else if ((x <= (-1.4d+97)) .or. (.not. (x <= 3300.0d0))) then
        tmp = x * y
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.2e+273) {
		tmp = x;
	} else if (x <= -6e+178) {
		tmp = x * y;
	} else if (x <= -6.6e+129) {
		tmp = x;
	} else if ((x <= -1.4e+97) || !(x <= 3300.0)) {
		tmp = x * y;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.2e+273:
		tmp = x
	elif x <= -6e+178:
		tmp = x * y
	elif x <= -6.6e+129:
		tmp = x
	elif (x <= -1.4e+97) or not (x <= 3300.0):
		tmp = x * y
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.2e+273)
		tmp = x;
	elseif (x <= -6e+178)
		tmp = Float64(x * y);
	elseif (x <= -6.6e+129)
		tmp = x;
	elseif ((x <= -1.4e+97) || !(x <= 3300.0))
		tmp = Float64(x * y);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.2e+273)
		tmp = x;
	elseif (x <= -6e+178)
		tmp = x * y;
	elseif (x <= -6.6e+129)
		tmp = x;
	elseif ((x <= -1.4e+97) || ~((x <= 3300.0)))
		tmp = x * y;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.2e+273], x, If[LessEqual[x, -6e+178], N[(x * y), $MachinePrecision], If[LessEqual[x, -6.6e+129], x, If[Or[LessEqual[x, -1.4e+97], N[Not[LessEqual[x, 3300.0]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(x + y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.2000000000000001e273 or -6.00000000000000031e178 < x < -6.5999999999999998e129

   1. Initial program 100.0%

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

      1. add-cube-cbrt 99.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(x \cdot y + x\right) + y} \cdot \sqrt[3]{\left(x \cdot y + x\right) + y}\right) \cdot \sqrt[3]{\left(x \cdot y + x\right) + y}} \]

      2. pow3 99.2%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(x \cdot y + x\right) + y}\right)}^{3}} \]

      3. +-commutative 99.2%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(x + x \cdot y\right)} + y}\right)}^{3} \]

      4. *-un-lft-identity 99.2%

         \[\leadsto {\left(\sqrt[3]{\left(\color{blue}{1 \cdot x} + x \cdot y\right) + y}\right)}^{3} \]

      5. *-commutative 99.2%

         \[\leadsto {\left(\sqrt[3]{\left(1 \cdot x + \color{blue}{y \cdot x}\right) + y}\right)}^{3} \]

      6. distribute-rgt-out 99.2%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{x \cdot \left(1 + y\right)} + y}\right)}^{3} \]

      7. fma-define 99.2%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(x, 1 + y, y\right)}}\right)}^{3} \]

   4. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x, 1 + y, y\right)}\right)}^{3}} \]

   5. Taylor expanded in y around 0 54.6%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot x} \]
   6. Step-by-step derivation

      1. pow-base-1 54.6%

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

      2. *-lft-identity 54.6%

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

   7. Simplified 54.6%

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

   if -1.2000000000000001e273 < x < -6.00000000000000031e178 or -6.5999999999999998e129 < x < -1.4e97 or 3300 < x

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 62.0%

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

   4. Taylor expanded in x around inf 60.9%

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

      1. *-commutative 60.9%

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

   6. Simplified 60.9%

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

   if -1.4e97 < x < 3300

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 95.5%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+273}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -6 \cdot 10^{+178}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{+129}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{+97} \lor \neg \left(x \leq 3300\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -44000000000:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-6}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -44000000000.0) (* x y) (if (<= y 2.3e-6) (+ x y) (+ y (* x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -44000000000.0) {
		tmp = x * y;
	} else if (y <= 2.3e-6) {
		tmp = x + y;
	} 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 <= (-44000000000.0d0)) then
        tmp = x * y
    else if (y <= 2.3d-6) then
        tmp = x + y
    else
        tmp = y + (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -44000000000.0) {
		tmp = x * y;
	} else if (y <= 2.3e-6) {
		tmp = x + y;
	} else {
		tmp = y + (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -44000000000.0:
		tmp = x * y
	elif y <= 2.3e-6:
		tmp = x + y
	else:
		tmp = y + (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -44000000000.0)
		tmp = Float64(x * y);
	elseif (y <= 2.3e-6)
		tmp = Float64(x + y);
	else
		tmp = Float64(y + Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -44000000000.0)
		tmp = x * y;
	elseif (y <= 2.3e-6)
		tmp = x + y;
	else
		tmp = y + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -44000000000.0], N[(x * y), $MachinePrecision], If[LessEqual[y, 2.3e-6], N[(x + y), $MachinePrecision], N[(y + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.4e10

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 100.0%

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

   4. Taylor expanded in x around inf 57.5%

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

      1. *-commutative 57.5%

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

   6. Simplified 57.5%

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

   if -4.4e10 < y < 2.3e-6

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 98.5%

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

   if 2.3e-6 < y

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 100.0%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -44000000000:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-6}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.18 \cdot 10^{-110}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -1.18e-110) x y))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.18e-110) {
		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 <= (-1.18d-110)) 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 <= -1.18e-110) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.18e-110:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.18e-110)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.18e-110)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.18e-110], x, y]

Derivation

1. Split input into 2 regimes

2. if x < -1.18e-110

   1. Initial program 100.0%

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

      1. add-cube-cbrt 98.5%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(x \cdot y + x\right) + y} \cdot \sqrt[3]{\left(x \cdot y + x\right) + y}\right) \cdot \sqrt[3]{\left(x \cdot y + x\right) + y}} \]

      2. pow3 98.5%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(x \cdot y + x\right) + y}\right)}^{3}} \]

      3. +-commutative 98.5%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(x + x \cdot y\right)} + y}\right)}^{3} \]

      4. *-un-lft-identity 98.5%

         \[\leadsto {\left(\sqrt[3]{\left(\color{blue}{1 \cdot x} + x \cdot y\right) + y}\right)}^{3} \]

      5. *-commutative 98.5%

         \[\leadsto {\left(\sqrt[3]{\left(1 \cdot x + \color{blue}{y \cdot x}\right) + y}\right)}^{3} \]

      6. distribute-rgt-out 98.5%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{x \cdot \left(1 + y\right)} + y}\right)}^{3} \]

      7. fma-define 98.5%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(x, 1 + y, y\right)}}\right)}^{3} \]

   4. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x, 1 + y, y\right)}\right)}^{3}} \]

   5. Taylor expanded in y around 0 48.6%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot x} \]
   6. Step-by-step derivation

      1. pow-base-1 48.6%

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

      2. *-lft-identity 48.6%

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

   7. Simplified 48.6%

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

   if -1.18e-110 < x

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 51.4%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.18 \cdot 10^{-110}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 38.8% 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. Add Preprocessing
3. Step-by-step derivation

   1. add-cube-cbrt 98.2%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\left(x \cdot y + x\right) + y} \cdot \sqrt[3]{\left(x \cdot y + x\right) + y}\right) \cdot \sqrt[3]{\left(x \cdot y + x\right) + y}} \]

   2. pow3 98.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(x \cdot y + x\right) + y}\right)}^{3}} \]

   3. +-commutative 98.2%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(x + x \cdot y\right)} + y}\right)}^{3} \]

   4. *-un-lft-identity 98.2%

      \[\leadsto {\left(\sqrt[3]{\left(\color{blue}{1 \cdot x} + x \cdot y\right) + y}\right)}^{3} \]

   5. *-commutative 98.2%

      \[\leadsto {\left(\sqrt[3]{\left(1 \cdot x + \color{blue}{y \cdot x}\right) + y}\right)}^{3} \]

   6. distribute-rgt-out 98.2%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{x \cdot \left(1 + y\right)} + y}\right)}^{3} \]

   7. fma-define 98.2%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(x, 1 + y, y\right)}}\right)}^{3} \]

4. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x, 1 + y, y\right)}\right)}^{3}} \]

5. Taylor expanded in y around 0 35.1%

   \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot x} \]
6. Step-by-step derivation

   1. pow-base-1 35.1%

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

   2. *-lft-identity 35.1%

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

7. Simplified 35.1%

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

8. Final simplification 35.1%

   \[\leadsto x \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))