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

Percentage Accurate: 100.0% → 100.0%

Time: 3.9s

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: 67.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+120}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-94}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.8e+120)
   (* y x)
   (if (<= x -1.1e-94) x (if (<= x 1.0) y (* y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.8e+120) {
		tmp = y * x;
	} else if (x <= -1.1e-94) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = 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 (x <= (-2.8d+120)) then
        tmp = y * x
    else if (x <= (-1.1d-94)) then
        tmp = x
    else if (x <= 1.0d0) then
        tmp = y
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.8e+120) {
		tmp = y * x;
	} else if (x <= -1.1e-94) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.8e+120:
		tmp = y * x
	elif x <= -1.1e-94:
		tmp = x
	elif x <= 1.0:
		tmp = y
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.8e+120)
		tmp = Float64(y * x);
	elseif (x <= -1.1e-94)
		tmp = x;
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.8e+120)
		tmp = y * x;
	elseif (x <= -1.1e-94)
		tmp = x;
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = 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[x, -2.8e+120], N[(y * x), $MachinePrecision], If[LessEqual[x, -1.1e-94], x, If[LessEqual[x, 1.0], y, N[(y * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.8000000000000001e120 or 1 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      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 99.0%

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

      1. +-commutative 99.0%

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

      2. distribute-rgt-in 99.0%

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

      3. *-un-lft-identity 99.0%

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

   7. Applied egg-rr 99.0%

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

   8. Taylor expanded in y around inf 61.0%

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

   if -2.8000000000000001e120 < x < -1.10000000000000001e-94

   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 42.7%

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

   if -1.10000000000000001e-94 < x < 1

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      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 79.8%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+120}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-94}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1150000000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (or (<= x -1150000000.0) (not (<= x 1.0))) (* x (+ y 1.0)) (+ y x)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if ((x <= -1150000000.0) || !(x <= 1.0)) {
		tmp = x * (y + 1.0);
	} else {
		tmp = 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 ((x <= (-1150000000.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = x * (y + 1.0d0)
    else
        tmp = y + x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if ((x <= -1150000000.0) || !(x <= 1.0)) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if (x <= -1150000000.0) or not (x <= 1.0):
		tmp = x * (y + 1.0)
	else:
		tmp = y + x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if ((x <= -1150000000.0) || !(x <= 1.0))
		tmp = Float64(x * Float64(y + 1.0));
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1150000000.0) || ~((x <= 1.0)))
		tmp = x * (y + 1.0);
	else
		tmp = 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[Or[LessEqual[x, -1150000000.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.15e9 or 1 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      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 99.1%

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

   if -1.15e9 < x < 1

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      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 inf 100.0%

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

   6. Taylor expanded in y around 0 98.6%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1150000000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1150000000:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-146}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1150000000.0)
   (* x (+ y 1.0))
   (if (<= x 5e-146) (+ y x) (+ y (* y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1150000000.0) {
		tmp = x * (y + 1.0);
	} else if (x <= 5e-146) {
		tmp = y + x;
	} 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 (x <= (-1150000000.0d0)) then
        tmp = x * (y + 1.0d0)
    else if (x <= 5d-146) then
        tmp = y + x
    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 (x <= -1150000000.0) {
		tmp = x * (y + 1.0);
	} else if (x <= 5e-146) {
		tmp = y + x;
	} else {
		tmp = y + (y * x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1150000000.0:
		tmp = x * (y + 1.0)
	elif x <= 5e-146:
		tmp = y + x
	else:
		tmp = y + (y * x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1150000000.0)
		tmp = Float64(x * Float64(y + 1.0));
	elseif (x <= 5e-146)
		tmp = Float64(y + x);
	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 (x <= -1150000000.0)
		tmp = x * (y + 1.0);
	elseif (x <= 5e-146)
		tmp = y + x;
	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[x, -1150000000.0], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e-146], N[(y + x), $MachinePrecision], N[(y + N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.15e9

   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 99.7%

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

   if -1.15e9 < x < 4.99999999999999957e-146

   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 inf 100.0%

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

   6. Taylor expanded in y around 0 98.5%

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

   if 4.99999999999999957e-146 < 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 y around inf 63.5%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1150000000:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-146}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -49000000 \lor \neg \left(y \leq 9 \cdot 10^{+139}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (or (<= y -49000000.0) (not (<= y 9e+139))) (* y x) (+ y x)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if ((y <= -49000000.0) || !(y <= 9e+139)) {
		tmp = y * x;
	} else {
		tmp = 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 <= (-49000000.0d0)) .or. (.not. (y <= 9d+139))) then
        tmp = y * x
    else
        tmp = y + x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if ((y <= -49000000.0) || !(y <= 9e+139)) {
		tmp = y * x;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if (y <= -49000000.0) or not (y <= 9e+139):
		tmp = y * x
	else:
		tmp = y + x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if ((y <= -49000000.0) || !(y <= 9e+139))
		tmp = Float64(y * x);
	else
		tmp = 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 <= -49000000.0) || ~((y <= 9e+139)))
		tmp = y * x;
	else
		tmp = 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[Or[LessEqual[y, -49000000.0], N[Not[LessEqual[y, 9e+139]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.9e7 or 8.9999999999999999e139 < 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 inf 55.7%

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

      1. +-commutative 55.7%

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

      2. distribute-rgt-in 55.7%

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

      3. *-un-lft-identity 55.7%

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

   7. Applied egg-rr 55.7%

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

   8. Taylor expanded in y around inf 55.7%

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

   if -4.9e7 < y < 8.9999999999999999e139

   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 inf 84.8%

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

   6. Taylor expanded in y around 0 88.9%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -49000000 \lor \neg \left(y \leq 9 \cdot 10^{+139}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
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: 64.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.42 \cdot 10^{-95}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -1.42e-95) x y))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.42e-95) {
		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 (x <= (-1.42d-95)) 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 (x <= -1.42e-95) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.42e-95:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.42e-95)
		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 (x <= -1.42e-95)
		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[x, -1.42e-95], x, y]

Derivation

1. Split input into 2 regimes

2. if x < -1.42000000000000007e-95

   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 39.4%

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

   if -1.42000000000000007e-95 < 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 0 53.3%

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

Alternative 8: 39.7% 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 32.5%

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

Reproduce

herbie shell --seed 2024131 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))