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

Percentage Accurate: 100.0% → 100.0%

Time: 3.1s

Alternatives: 8

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

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (+ y (fma x y x)))

C

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

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(y + fma(x, y, x))
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
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-define 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. Add Preprocessing
5. Add Preprocessing

Alternative 2: 70.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-128}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+262}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+276}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   (* y x)
   (if (<= y 2.6e-128)
     x
     (if (<= y 2.7e+262) y (if (<= y 1.15e+276) (* y x) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = y * x;
	} else if (y <= 2.6e-128) {
		tmp = x;
	} else if (y <= 2.7e+262) {
		tmp = y;
	} else if (y <= 1.15e+276) {
		tmp = y * x;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= 2.6d-128) then
        tmp = x
    else if (y <= 2.7d+262) then
        tmp = y
    else if (y <= 1.15d+276) then
        tmp = y * x
    else
        tmp = y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = y * x;
	} else if (y <= 2.6e-128) {
		tmp = x;
	} else if (y <= 2.7e+262) {
		tmp = y;
	} else if (y <= 1.15e+276) {
		tmp = y * x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = y * x
	elif y <= 2.6e-128:
		tmp = x
	elif y <= 2.7e+262:
		tmp = y
	elif y <= 1.15e+276:
		tmp = y * x
	else:
		tmp = y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(y * x);
	elseif (y <= 2.6e-128)
		tmp = x;
	elseif (y <= 2.7e+262)
		tmp = y;
	elseif (y <= 1.15e+276)
		tmp = Float64(y * x);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = y * x;
	elseif (y <= 2.6e-128)
		tmp = x;
	elseif (y <= 2.7e+262)
		tmp = y;
	elseif (y <= 1.15e+276)
		tmp = y * x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -1.0], N[(y * x), $MachinePrecision], If[LessEqual[y, 2.6e-128], x, If[LessEqual[y, 2.7e+262], y, If[LessEqual[y, 1.15e+276], N[(y * x), $MachinePrecision], y]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 2.7000000000000001e262 < y < 1.15000000000000002e276

   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-define 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. Add Preprocessing

   5. Taylor expanded in x around inf 47.6%

      \[\leadsto \color{blue}{x \cdot \left(1 + y\right)} \]
   6. Step-by-step derivation

      1. +-commutative 47.6%

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

   7. Simplified 47.6%

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

   8. Taylor expanded in y around inf 47.6%

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

   if -1 < y < 2.59999999999999981e-128

   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-define 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. Add Preprocessing

   5. Taylor expanded in y around 0 76.4%

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

   if 2.59999999999999981e-128 < y < 2.7000000000000001e262 or 1.15000000000000002e276 < y

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 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. Add Preprocessing

   5. Taylor expanded in x around 0 50.0%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-128}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+262}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+276}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-137} \lor \neg \left(x \leq 1.52 \cdot 10^{-27}\right):\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.3e-137) (not (<= x 1.52e-27))) (* x (+ y 1.0)) y))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if ((x <= -3.3e-137) || !(x <= 1.52e-27)) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-3.3d-137)) .or. (.not. (x <= 1.52d-27))) then
        tmp = x * (y + 1.0d0)
    else
        tmp = y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if ((x <= -3.3e-137) || !(x <= 1.52e-27)) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if (x <= -3.3e-137) or not (x <= 1.52e-27):
		tmp = x * (y + 1.0)
	else:
		tmp = y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if ((x <= -3.3e-137) || !(x <= 1.52e-27))
		tmp = Float64(x * Float64(y + 1.0));
	else
		tmp = y;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -3.3e-137) || ~((x <= 1.52e-27)))
		tmp = x * (y + 1.0);
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[Or[LessEqual[x, -3.3e-137], N[Not[LessEqual[x, 1.52e-27]], $MachinePrecision]], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], y]

Derivation

1. Split input into 2 regimes

2. if x < -3.3000000000000002e-137 or 1.52000000000000004e-27 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 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. Add Preprocessing

   5. Taylor expanded in x around inf 81.4%

      \[\leadsto \color{blue}{x \cdot \left(1 + y\right)} \]
   6. Step-by-step derivation

      1. +-commutative 81.4%

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

   7. Simplified 81.4%

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

   if -3.3000000000000002e-137 < x < 1.52000000000000004e-27

   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-define 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. Add Preprocessing

   5. Taylor expanded in x around 0 77.6%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-137} \lor \neg \left(x \leq 1.52 \cdot 10^{-27}\right):\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-137}:\\ \;\;\;\;x + y \cdot x\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-11}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.3e-137) (+ x (* y x)) (if (<= x 3.2e-11) y (* x (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.3e-137) {
		tmp = x + (y * x);
	} else if (x <= 3.2e-11) {
		tmp = y;
	} else {
		tmp = x * (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.3d-137)) then
        tmp = x + (y * x)
    else if (x <= 3.2d-11) then
        tmp = y
    else
        tmp = x * (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -3.3e-137) {
		tmp = x + (y * x);
	} else if (x <= 3.2e-11) {
		tmp = y;
	} else {
		tmp = x * (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.3e-137:
		tmp = x + (y * x)
	elif x <= 3.2e-11:
		tmp = y
	else:
		tmp = x * (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.3e-137)
		tmp = Float64(x + Float64(y * x));
	elseif (x <= 3.2e-11)
		tmp = y;
	else
		tmp = Float64(x * Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.3e-137)
		tmp = x + (y * x);
	elseif (x <= 3.2e-11)
		tmp = y;
	else
		tmp = x * (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -3.3e-137], N[(x + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.2e-11], y, N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.3000000000000002e-137

   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-define 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. Add Preprocessing

   5. Taylor expanded in x around inf 72.6%

      \[\leadsto \color{blue}{x \cdot \left(1 + y\right)} \]
   6. Step-by-step derivation

      1. +-commutative 72.6%

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

   7. Simplified 72.6%

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

      1. distribute-lft-in 72.6%

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

      2. *-rgt-identity 72.6%

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

   9. Applied egg-rr 72.6%

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

   if -3.3000000000000002e-137 < x < 3.19999999999999994e-11

   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-define 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. Add Preprocessing

   5. Taylor expanded in x around 0 75.4%

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

   if 3.19999999999999994e-11 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 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. Add Preprocessing

   5. Taylor expanded in x around inf 94.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + y\right)} \]
   6. Step-by-step derivation

      1. +-commutative 94.0%

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

   7. Simplified 94.0%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-137}:\\ \;\;\;\;x + y \cdot x\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-11}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.05 \cdot 10^{-128}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.05e-128) (* x (+ y 1.0)) (+ y (* y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.05e-128) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y + (y * x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.05d-128) then
        tmp = x * (y + 1.0d0)
    else
        tmp = y + (y * x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.05e-128) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y + (y * x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.05e-128:
		tmp = x * (y + 1.0)
	else:
		tmp = y + (y * x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.05e-128)
		tmp = Float64(x * Float64(y + 1.0));
	else
		tmp = Float64(y + Float64(y * x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.05e-128)
		tmp = x * (y + 1.0);
	else
		tmp = y + (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2.05e-128], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(y + N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.05e-128

   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-define 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. Add Preprocessing

   5. Taylor expanded in x around inf 67.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + y\right)} \]
   6. Step-by-step derivation

      1. +-commutative 67.0%

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

   7. Simplified 67.0%

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

   if 2.05e-128 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 84.1%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.05 \cdot 10^{-128}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 100.0% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (+ y (+ x (* y x))))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y + (x + (y * x))
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return y + (x + (y * x))

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(y + Float64(x + Float64(y * x)))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = y + (x + (y * x));
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
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. Add Preprocessing

3. Final simplification 100.0%

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

Alternative 7: 64.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{-128}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 2.6e-128) x y))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.6e-128) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.6d-128) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.6e-128) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.6e-128:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.6e-128)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.6e-128)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2.6e-128], x, y]

Derivation

1. Split input into 2 regimes

2. if y < 2.59999999999999981e-128

   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-define 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. Add Preprocessing

   5. Taylor expanded in y around 0 50.4%

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

   if 2.59999999999999981e-128 < y

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 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. Add Preprocessing

   5. Taylor expanded in x around 0 48.7%

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

Alternative 8: 39.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ x \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 x)

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return x

Julia

x, y = sort([x, y])
function code(x, y)
	return x
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
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%

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

   2. fma-define 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. Add Preprocessing

5. Taylor expanded in y around 0 41.0%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024086 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))