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

Percentage Accurate: 100.0% → 100.0%

Time: 4.3s

Alternatives: 7

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

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (+ x (* y (+ x 1.0))))

C

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

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 + (y * (x + 1.0d0))
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return x + (y * (x + 1.0))

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x + Float64(y * Float64(x + 1.0)))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x + (y * (x + 1.0));
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
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. *-commutative 100.0%

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

   3. associate-+r+ 100.0%

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

   4. +-commutative 100.0%

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

   5. *-commutative 100.0%

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

   6. distribute-lft1-in 100.0%

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

   7. fma-def 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x + 1, y, x\right)} \]
4. Step-by-step derivation

   1. fma-udef 100.0%

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

   2. *-commutative 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 69.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{+272}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{+242}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{+169}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-110}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -4.6e+272)
   (* y x)
   (if (<= x -3.7e+242)
     x
     (if (<= x -3.2e+169)
       (* y x)
       (if (<= x -3.5e-110) x (if (<= x 1.0) y (* y x)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4.6e+272) {
		tmp = y * x;
	} else if (x <= -3.7e+242) {
		tmp = x;
	} else if (x <= -3.2e+169) {
		tmp = y * x;
	} else if (x <= -3.5e-110) {
		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 <= (-4.6d+272)) then
        tmp = y * x
    else if (x <= (-3.7d+242)) then
        tmp = x
    else if (x <= (-3.2d+169)) then
        tmp = y * x
    else if (x <= (-3.5d-110)) 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 <= -4.6e+272) {
		tmp = y * x;
	} else if (x <= -3.7e+242) {
		tmp = x;
	} else if (x <= -3.2e+169) {
		tmp = y * x;
	} else if (x <= -3.5e-110) {
		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 <= -4.6e+272:
		tmp = y * x
	elif x <= -3.7e+242:
		tmp = x
	elif x <= -3.2e+169:
		tmp = y * x
	elif x <= -3.5e-110:
		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 <= -4.6e+272)
		tmp = Float64(y * x);
	elseif (x <= -3.7e+242)
		tmp = x;
	elseif (x <= -3.2e+169)
		tmp = Float64(y * x);
	elseif (x <= -3.5e-110)
		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 <= -4.6e+272)
		tmp = y * x;
	elseif (x <= -3.7e+242)
		tmp = x;
	elseif (x <= -3.2e+169)
		tmp = y * x;
	elseif (x <= -3.5e-110)
		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, -4.6e+272], N[(y * x), $MachinePrecision], If[LessEqual[x, -3.7e+242], x, If[LessEqual[x, -3.2e+169], N[(y * x), $MachinePrecision], If[LessEqual[x, -3.5e-110], x, If[LessEqual[x, 1.0], y, N[(y * x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.5999999999999998e272 or -3.7e242 < x < -3.1999999999999998e169 or 1 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+r+ 100.0%

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

      4. +-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. fma-def 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x + 1, y, x\right)} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 99.4%

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

      1. *-commutative 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in y around inf 54.4%

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

   if -4.5999999999999998e272 < x < -3.7e242 or -3.1999999999999998e169 < x < -3.49999999999999974e-110

   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. *-commutative 100.0%

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

      3. associate-+r+ 100.0%

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

      4. +-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. fma-def 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x + 1, y, x\right)} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around 0 53.1%

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

   if -3.49999999999999974e-110 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 74.6%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{+272}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{+242}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{+169}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-110}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3: 79.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+272}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{+241}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{+152} \lor \neg \left(x \leq 600000\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.5e+272)
   (* y x)
   (if (<= x -2.3e+241)
     x
     (if (or (<= x -1.45e+152) (not (<= x 600000.0))) (* y x) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.5e+272) {
		tmp = y * x;
	} else if (x <= -2.3e+241) {
		tmp = x;
	} else if ((x <= -1.45e+152) || !(x <= 600000.0)) {
		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 (x <= (-3.5d+272)) then
        tmp = y * x
    else if (x <= (-2.3d+241)) then
        tmp = x
    else if ((x <= (-1.45d+152)) .or. (.not. (x <= 600000.0d0))) 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 (x <= -3.5e+272) {
		tmp = y * x;
	} else if (x <= -2.3e+241) {
		tmp = x;
	} else if ((x <= -1.45e+152) || !(x <= 600000.0)) {
		tmp = y * x;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.5e+272:
		tmp = y * x
	elif x <= -2.3e+241:
		tmp = x
	elif (x <= -1.45e+152) or not (x <= 600000.0):
		tmp = y * x
	else:
		tmp = y + x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.5e+272)
		tmp = Float64(y * x);
	elseif (x <= -2.3e+241)
		tmp = x;
	elseif ((x <= -1.45e+152) || !(x <= 600000.0))
		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 (x <= -3.5e+272)
		tmp = y * x;
	elseif (x <= -2.3e+241)
		tmp = x;
	elseif ((x <= -1.45e+152) || ~((x <= 600000.0)))
		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[LessEqual[x, -3.5e+272], N[(y * x), $MachinePrecision], If[LessEqual[x, -2.3e+241], x, If[Or[LessEqual[x, -1.45e+152], N[Not[LessEqual[x, 600000.0]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(y + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.50000000000000023e272 or -2.2999999999999999e241 < x < -1.4499999999999999e152 or 6e5 < 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. *-commutative 100.0%

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

      3. associate-+r+ 100.0%

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

      4. +-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. fma-def 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x + 1, y, x\right)} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 99.4%

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

      1. *-commutative 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in y around inf 55.7%

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

   if -3.50000000000000023e272 < x < -2.2999999999999999e241

   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. *-commutative 100.0%

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

      3. associate-+r+ 100.0%

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

      4. +-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. fma-def 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x + 1, y, x\right)} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around 0 50.1%

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

   if -1.4499999999999999e152 < x < 6e5

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 91.6%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+272}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{+241}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{+152} \lor \neg \left(x \leq 600000\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 4: 98.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -12000000000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.000155:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -12000000000.0) (* y x) (if (<= y 0.000155) (+ y x) (+ y (* y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -12000000000.0) {
		tmp = y * x;
	} else if (y <= 0.000155) {
		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 (y <= (-12000000000.0d0)) then
        tmp = y * x
    else if (y <= 0.000155d0) 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 (y <= -12000000000.0) {
		tmp = y * x;
	} else if (y <= 0.000155) {
		tmp = y + x;
	} else {
		tmp = y + (y * x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -12000000000.0:
		tmp = y * x
	elif y <= 0.000155:
		tmp = y + x
	else:
		tmp = y + (y * x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -12000000000.0)
		tmp = Float64(y * x);
	elseif (y <= 0.000155)
		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 (y <= -12000000000.0)
		tmp = y * x;
	elseif (y <= 0.000155)
		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[y, -12000000000.0], N[(y * x), $MachinePrecision], If[LessEqual[y, 0.000155], N[(y + x), $MachinePrecision], N[(y + N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2e10

   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. *-commutative 100.0%

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

      3. associate-+r+ 100.0%

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

      4. +-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. fma-def 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x + 1, y, x\right)} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 62.4%

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

      1. *-commutative 62.4%

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

   8. Simplified 62.4%

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

   9. Taylor expanded in y around inf 62.4%

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

   if -1.2e10 < y < 1.55e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.6%

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

   if 1.55e-4 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 98.7%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -12000000000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.000155:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot x\\ \end{array} \]

Alternative 5: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -38000:\\ \;\;\;\;x + y \cdot x\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-5}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -38000.0) (+ x (* y x)) (if (<= x 4.2e-5) (+ y x) (+ y (* y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -38000.0) {
		tmp = x + (y * x);
	} else if (x <= 4.2e-5) {
		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 <= (-38000.0d0)) then
        tmp = x + (y * x)
    else if (x <= 4.2d-5) 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 <= -38000.0) {
		tmp = x + (y * x);
	} else if (x <= 4.2e-5) {
		tmp = y + x;
	} else {
		tmp = y + (y * x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -38000.0:
		tmp = x + (y * x)
	elif x <= 4.2e-5:
		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 <= -38000.0)
		tmp = Float64(x + Float64(y * x));
	elseif (x <= 4.2e-5)
		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 <= -38000.0)
		tmp = x + (y * x);
	elseif (x <= 4.2e-5)
		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, -38000.0], N[(x + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.2e-5], N[(y + x), $MachinePrecision], N[(y + N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -38000

   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. *-commutative 100.0%

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

      3. associate-+r+ 100.0%

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

      4. +-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. fma-def 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x + 1, y, x\right)} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 99.6%

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

      1. *-commutative 99.6%

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

   8. Simplified 99.6%

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

   if -38000 < x < 4.19999999999999977e-5

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.8%

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

   if 4.19999999999999977e-5 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 49.5%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -38000:\\ \;\;\;\;x + y \cdot x\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-5}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot x\\ \end{array} \]

Alternative 6: 64.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-112}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -4.5e-112) x y))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4.5e-112) {
		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 <= (-4.5d-112)) 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 <= -4.5e-112) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -4.5e-112:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -4.5e-112)
		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 <= -4.5e-112)
		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, -4.5e-112], x, y]

Derivation

1. Split input into 2 regimes

2. if x < -4.50000000000000012e-112

   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. *-commutative 100.0%

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

      3. associate-+r+ 100.0%

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

      4. +-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. fma-def 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x + 1, y, x\right)} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around 0 46.6%

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

   if -4.50000000000000012e-112 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 45.8%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-112}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 7: 38.6% 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%

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

   2. *-commutative 100.0%

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

   3. associate-+r+ 100.0%

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

   4. +-commutative 100.0%

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

   5. *-commutative 100.0%

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

   6. distribute-lft1-in 100.0%

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

   7. fma-def 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x + 1, y, x\right)} \]
4. Step-by-step derivation

   1. fma-udef 100.0%

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

   2. *-commutative 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in y around 0 39.7%

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

7. Final simplification 39.7%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023336 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))