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

Percentage Accurate: 100.0% → 100.0%

Time: 3.2s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. distribute-lft1-in 100.0%

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

   3. fma-define 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y + 1, x, y\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 68.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-84}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{-71}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-50}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+121} \lor \neg \left(y \leq 3 \cdot 10^{+143}\right) \land y \leq 1.4 \cdot 10^{+243}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   (* y x)
   (if (<= y 5.2e-84)
     x
     (if (<= y 5.3e-71)
       y
       (if (<= y 1.8e-50)
         x
         (if (or (<= y 1.02e+121) (and (not (<= y 3e+143)) (<= y 1.4e+243)))
           y
           (* y x)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = y * x;
	} else if (y <= 5.2e-84) {
		tmp = x;
	} else if (y <= 5.3e-71) {
		tmp = y;
	} else if (y <= 1.8e-50) {
		tmp = x;
	} else if ((y <= 1.02e+121) || (!(y <= 3e+143) && (y <= 1.4e+243))) {
		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 (y <= (-1.0d0)) then
        tmp = y * x
    else if (y <= 5.2d-84) then
        tmp = x
    else if (y <= 5.3d-71) then
        tmp = y
    else if (y <= 1.8d-50) then
        tmp = x
    else if ((y <= 1.02d+121) .or. (.not. (y <= 3d+143)) .and. (y <= 1.4d+243)) 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 (y <= -1.0) {
		tmp = y * x;
	} else if (y <= 5.2e-84) {
		tmp = x;
	} else if (y <= 5.3e-71) {
		tmp = y;
	} else if (y <= 1.8e-50) {
		tmp = x;
	} else if ((y <= 1.02e+121) || (!(y <= 3e+143) && (y <= 1.4e+243))) {
		tmp = y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = y * x
	elif y <= 5.2e-84:
		tmp = x
	elif y <= 5.3e-71:
		tmp = y
	elif y <= 1.8e-50:
		tmp = x
	elif (y <= 1.02e+121) or (not (y <= 3e+143) and (y <= 1.4e+243)):
		tmp = y
	else:
		tmp = y * x
	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 <= 5.2e-84)
		tmp = x;
	elseif (y <= 5.3e-71)
		tmp = y;
	elseif (y <= 1.8e-50)
		tmp = x;
	elseif ((y <= 1.02e+121) || (!(y <= 3e+143) && (y <= 1.4e+243)))
		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 (y <= -1.0)
		tmp = y * x;
	elseif (y <= 5.2e-84)
		tmp = x;
	elseif (y <= 5.3e-71)
		tmp = y;
	elseif (y <= 1.8e-50)
		tmp = x;
	elseif ((y <= 1.02e+121) || (~((y <= 3e+143)) && (y <= 1.4e+243)))
		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[y, -1.0], N[(y * x), $MachinePrecision], If[LessEqual[y, 5.2e-84], x, If[LessEqual[y, 5.3e-71], y, If[LessEqual[y, 1.8e-50], x, If[Or[LessEqual[y, 1.02e+121], And[N[Not[LessEqual[y, 3e+143]], $MachinePrecision], LessEqual[y, 1.4e+243]]], y, N[(y * x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 1.02000000000000005e121 < y < 3.0000000000000001e143 or 1.4e243 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 58.6%

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

      1. +-commutative 58.6%

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

   5. Simplified 58.6%

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

   6. Taylor expanded in y around inf 58.5%

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

   if -1 < y < 5.2e-84 or 5.29999999999999999e-71 < y < 1.7999999999999999e-50

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 74.0%

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

   if 5.2e-84 < y < 5.29999999999999999e-71 or 1.7999999999999999e-50 < y < 1.02000000000000005e121 or 3.0000000000000001e143 < y < 1.4e243

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 50.5%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-84}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{-71}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-50}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+121} \lor \neg \left(y \leq 3 \cdot 10^{+143}\right) \land y \leq 1.4 \cdot 10^{+243}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + 1\right) \cdot x\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{-28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-85}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-155}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-11}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (+ y 1.0) x)))
   (if (<= x -4.2e-28)
     t_0
     (if (<= x -1.85e-85)
       y
       (if (<= x -3.9e-155) x (if (<= x 2.85e-11) y t_0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + 1.0) * x;
	double tmp;
	if (x <= -4.2e-28) {
		tmp = t_0;
	} else if (x <= -1.85e-85) {
		tmp = y;
	} else if (x <= -3.9e-155) {
		tmp = x;
	} else if (x <= 2.85e-11) {
		tmp = y;
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (y + 1.0d0) * x
    if (x <= (-4.2d-28)) then
        tmp = t_0
    else if (x <= (-1.85d-85)) then
        tmp = y
    else if (x <= (-3.9d-155)) then
        tmp = x
    else if (x <= 2.85d-11) then
        tmp = y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y + 1.0) * x;
	double tmp;
	if (x <= -4.2e-28) {
		tmp = t_0;
	} else if (x <= -1.85e-85) {
		tmp = y;
	} else if (x <= -3.9e-155) {
		tmp = x;
	} else if (x <= 2.85e-11) {
		tmp = y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y + 1.0) * x
	tmp = 0
	if x <= -4.2e-28:
		tmp = t_0
	elif x <= -1.85e-85:
		tmp = y
	elif x <= -3.9e-155:
		tmp = x
	elif x <= 2.85e-11:
		tmp = y
	else:
		tmp = t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y + 1.0) * x)
	tmp = 0.0
	if (x <= -4.2e-28)
		tmp = t_0;
	elseif (x <= -1.85e-85)
		tmp = y;
	elseif (x <= -3.9e-155)
		tmp = x;
	elseif (x <= 2.85e-11)
		tmp = y;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (y + 1.0) * x;
	tmp = 0.0;
	if (x <= -4.2e-28)
		tmp = t_0;
	elseif (x <= -1.85e-85)
		tmp = y;
	elseif (x <= -3.9e-155)
		tmp = x;
	elseif (x <= 2.85e-11)
		tmp = y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(y + 1.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -4.2e-28], t$95$0, If[LessEqual[x, -1.85e-85], y, If[LessEqual[x, -3.9e-155], x, If[LessEqual[x, 2.85e-11], y, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.20000000000000013e-28 or 2.8499999999999999e-11 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 96.4%

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

      1. +-commutative 96.4%

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

   5. Simplified 96.4%

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

   if -4.20000000000000013e-28 < x < -1.84999999999999992e-85 or -3.9000000000000003e-155 < x < 2.8499999999999999e-11

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 79.6%

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

   if -1.84999999999999992e-85 < x < -3.9000000000000003e-155

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 36.0%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-28}:\\ \;\;\;\;\left(y + 1\right) \cdot x\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-85}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-155}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-11}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\left(y + 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-28}:\\ \;\;\;\;x + y \cdot x\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-85}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-155}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.85e-28)
   (+ x (* y x))
   (if (<= x -1.85e-85) y (if (<= x -3.9e-155) x (* y (+ 1.0 x))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.85e-28) {
		tmp = x + (y * x);
	} else if (x <= -1.85e-85) {
		tmp = y;
	} else if (x <= -3.9e-155) {
		tmp = x;
	} else {
		tmp = y * (1.0 + 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 <= (-1.85d-28)) then
        tmp = x + (y * x)
    else if (x <= (-1.85d-85)) then
        tmp = y
    else if (x <= (-3.9d-155)) then
        tmp = x
    else
        tmp = y * (1.0d0 + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.85e-28) {
		tmp = x + (y * x);
	} else if (x <= -1.85e-85) {
		tmp = y;
	} else if (x <= -3.9e-155) {
		tmp = x;
	} else {
		tmp = y * (1.0 + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.85e-28:
		tmp = x + (y * x)
	elif x <= -1.85e-85:
		tmp = y
	elif x <= -3.9e-155:
		tmp = x
	else:
		tmp = y * (1.0 + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.85e-28)
		tmp = Float64(x + Float64(y * x));
	elseif (x <= -1.85e-85)
		tmp = y;
	elseif (x <= -3.9e-155)
		tmp = x;
	else
		tmp = Float64(y * Float64(1.0 + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.85e-28)
		tmp = x + (y * x);
	elseif (x <= -1.85e-85)
		tmp = y;
	elseif (x <= -3.9e-155)
		tmp = x;
	else
		tmp = y * (1.0 + 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, -1.85e-28], N[(x + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.85e-85], y, If[LessEqual[x, -3.9e-155], x, N[(y * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.8500000000000001e-28

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 95.3%

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

      1. +-commutative 95.3%

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

   5. Simplified 95.3%

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

      1. distribute-lft-in 95.3%

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

      2. *-rgt-identity 95.3%

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

   7. Applied egg-rr 95.3%

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

   if -1.8500000000000001e-28 < x < -1.84999999999999992e-85

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 83.5%

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

   if -1.84999999999999992e-85 < x < -3.9000000000000003e-155

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 36.0%

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

   if -3.9000000000000003e-155 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 67.3%

      \[\leadsto \color{blue}{y \cdot \left(1 + x\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-28}:\\ \;\;\;\;x + y \cdot x\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-85}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-155}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-29}:\\ \;\;\;\;\left(y + 1\right) \cdot x\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-85}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-155}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.25e-29)
   (* (+ y 1.0) x)
   (if (<= x -1.85e-85) y (if (<= x -3.9e-155) x (* y (+ 1.0 x))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.25e-29) {
		tmp = (y + 1.0) * x;
	} else if (x <= -1.85e-85) {
		tmp = y;
	} else if (x <= -3.9e-155) {
		tmp = x;
	} else {
		tmp = y * (1.0 + 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 <= (-1.25d-29)) then
        tmp = (y + 1.0d0) * x
    else if (x <= (-1.85d-85)) then
        tmp = y
    else if (x <= (-3.9d-155)) then
        tmp = x
    else
        tmp = y * (1.0d0 + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.25e-29) {
		tmp = (y + 1.0) * x;
	} else if (x <= -1.85e-85) {
		tmp = y;
	} else if (x <= -3.9e-155) {
		tmp = x;
	} else {
		tmp = y * (1.0 + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.25e-29:
		tmp = (y + 1.0) * x
	elif x <= -1.85e-85:
		tmp = y
	elif x <= -3.9e-155:
		tmp = x
	else:
		tmp = y * (1.0 + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.25e-29)
		tmp = Float64(Float64(y + 1.0) * x);
	elseif (x <= -1.85e-85)
		tmp = y;
	elseif (x <= -3.9e-155)
		tmp = x;
	else
		tmp = Float64(y * Float64(1.0 + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.25e-29)
		tmp = (y + 1.0) * x;
	elseif (x <= -1.85e-85)
		tmp = y;
	elseif (x <= -3.9e-155)
		tmp = x;
	else
		tmp = y * (1.0 + 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, -1.25e-29], N[(N[(y + 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, -1.85e-85], y, If[LessEqual[x, -3.9e-155], x, N[(y * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.24999999999999996e-29

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 95.3%

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

      1. +-commutative 95.3%

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

   5. Simplified 95.3%

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

   if -1.24999999999999996e-29 < x < -1.84999999999999992e-85

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 83.5%

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

   if -1.84999999999999992e-85 < x < -3.9000000000000003e-155

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 36.0%

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

   if -3.9000000000000003e-155 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 67.3%

      \[\leadsto \color{blue}{y \cdot \left(1 + x\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-29}:\\ \;\;\;\;\left(y + 1\right) \cdot x\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-85}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-155}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-28}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-85}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-155}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5.5e-28) x (if (<= x -1.85e-85) y (if (<= x -3.9e-155) x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5.5e-28) {
		tmp = x;
	} else if (x <= -1.85e-85) {
		tmp = y;
	} else if (x <= -3.9e-155) {
		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 <= (-5.5d-28)) then
        tmp = x
    else if (x <= (-1.85d-85)) then
        tmp = y
    else if (x <= (-3.9d-155)) 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 <= -5.5e-28) {
		tmp = x;
	} else if (x <= -1.85e-85) {
		tmp = y;
	} else if (x <= -3.9e-155) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -5.5e-28:
		tmp = x
	elif x <= -1.85e-85:
		tmp = y
	elif x <= -3.9e-155:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5.5e-28)
		tmp = x;
	elseif (x <= -1.85e-85)
		tmp = y;
	elseif (x <= -3.9e-155)
		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 <= -5.5e-28)
		tmp = x;
	elseif (x <= -1.85e-85)
		tmp = y;
	elseif (x <= -3.9e-155)
		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, -5.5e-28], x, If[LessEqual[x, -1.85e-85], y, If[LessEqual[x, -3.9e-155], x, y]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.49999999999999967e-28 or -1.84999999999999992e-85 < x < -3.9000000000000003e-155

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 40.9%

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

   if -5.49999999999999967e-28 < x < -1.84999999999999992e-85 or -3.9000000000000003e-155 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 49.2%

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

Alternative 7: 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 8: 39.1% 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. Add Preprocessing

3. Taylor expanded in y around 0 35.6%

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

Reproduce

herbie shell --seed 2024089 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))