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

Percentage Accurate: 100.0% → 100.0%

Time: 3.3s

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, 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: 87.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + 1\right) \cdot x\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{-72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-84}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -2.85 \cdot 10^{-129}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-8}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -1.4e-72)
     t_0
     (if (<= x -4.5e-84) y (if (<= x -2.85e-129) x (if (<= x 7e-8) y t_0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + 1.0) * x;
	double tmp;
	if (x <= -1.4e-72) {
		tmp = t_0;
	} else if (x <= -4.5e-84) {
		tmp = y;
	} else if (x <= -2.85e-129) {
		tmp = x;
	} else if (x <= 7e-8) {
		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 <= (-1.4d-72)) then
        tmp = t_0
    else if (x <= (-4.5d-84)) then
        tmp = y
    else if (x <= (-2.85d-129)) then
        tmp = x
    else if (x <= 7d-8) 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 <= -1.4e-72) {
		tmp = t_0;
	} else if (x <= -4.5e-84) {
		tmp = y;
	} else if (x <= -2.85e-129) {
		tmp = x;
	} else if (x <= 7e-8) {
		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 <= -1.4e-72:
		tmp = t_0
	elif x <= -4.5e-84:
		tmp = y
	elif x <= -2.85e-129:
		tmp = x
	elif x <= 7e-8:
		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 <= -1.4e-72)
		tmp = t_0;
	elseif (x <= -4.5e-84)
		tmp = y;
	elseif (x <= -2.85e-129)
		tmp = x;
	elseif (x <= 7e-8)
		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 <= -1.4e-72)
		tmp = t_0;
	elseif (x <= -4.5e-84)
		tmp = y;
	elseif (x <= -2.85e-129)
		tmp = x;
	elseif (x <= 7e-8)
		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, -1.4e-72], t$95$0, If[LessEqual[x, -4.5e-84], y, If[LessEqual[x, -2.85e-129], x, If[LessEqual[x, 7e-8], y, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3999999999999999e-72 or 7.00000000000000048e-8 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 90.8%

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

      1. +-commutative 90.8%

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

   5. Simplified 90.8%

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

   if -1.3999999999999999e-72 < x < -4.50000000000000016e-84 or -2.85e-129 < x < 7.00000000000000048e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 81.2%

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

   if -4.50000000000000016e-84 < x < -2.85e-129

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 72.3%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-72}:\\ \;\;\;\;\left(y + 1\right) \cdot x\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-84}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -2.85 \cdot 10^{-129}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-8}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\left(y + 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+116}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{+82}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-129}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -6e+116)
   x
   (if (<= x -9.2e+82)
     (* y x)
     (if (<= x -3e-129) x (if (<= x 1.0) y (* y x))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -6e+116) {
		tmp = x;
	} else if (x <= -9.2e+82) {
		tmp = y * x;
	} else if (x <= -3e-129) {
		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 <= (-6d+116)) then
        tmp = x
    else if (x <= (-9.2d+82)) then
        tmp = y * x
    else if (x <= (-3d-129)) 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 <= -6e+116) {
		tmp = x;
	} else if (x <= -9.2e+82) {
		tmp = y * x;
	} else if (x <= -3e-129) {
		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 <= -6e+116:
		tmp = x
	elif x <= -9.2e+82:
		tmp = y * x
	elif x <= -3e-129:
		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 <= -6e+116)
		tmp = x;
	elseif (x <= -9.2e+82)
		tmp = Float64(y * x);
	elseif (x <= -3e-129)
		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 <= -6e+116)
		tmp = x;
	elseif (x <= -9.2e+82)
		tmp = y * x;
	elseif (x <= -3e-129)
		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, -6e+116], x, If[LessEqual[x, -9.2e+82], N[(y * x), $MachinePrecision], If[LessEqual[x, -3e-129], x, If[LessEqual[x, 1.0], y, N[(y * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.9999999999999997e116 or -9.19999999999999953e82 < x < -2.9999999999999998e-129

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 53.3%

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

   if -5.9999999999999997e116 < x < -9.19999999999999953e82 or 1 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 51.3%

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

      1. *-commutative 51.3%

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

   5. Simplified 51.3%

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

   6. Taylor expanded in x around inf 50.6%

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

      1. *-commutative 50.6%

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

   8. Simplified 50.6%

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

   if -2.9999999999999998e-129 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 81.2%

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

Alternative 4: 88.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-72}:\\ \;\;\;\;\left(y + 1\right) \cdot x\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-84}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-129}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.4e-72)
   (* (+ y 1.0) x)
   (if (<= x -4.5e-84) y (if (<= x -3e-129) x (* y (+ 1.0 x))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.4e-72) {
		tmp = (y + 1.0) * x;
	} else if (x <= -4.5e-84) {
		tmp = y;
	} else if (x <= -3e-129) {
		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 <= (-1.4d-72)) then
        tmp = (y + 1.0d0) * x
    else if (x <= (-4.5d-84)) then
        tmp = y
    else if (x <= (-3d-129)) 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 <= -1.4e-72) {
		tmp = (y + 1.0) * x;
	} else if (x <= -4.5e-84) {
		tmp = y;
	} else if (x <= -3e-129) {
		tmp = x;
	} else {
		tmp = y * (1.0 + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.4e-72:
		tmp = (y + 1.0) * x
	elif x <= -4.5e-84:
		tmp = y
	elif x <= -3e-129:
		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 <= -1.4e-72)
		tmp = Float64(Float64(y + 1.0) * x);
	elseif (x <= -4.5e-84)
		tmp = y;
	elseif (x <= -3e-129)
		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 <= -1.4e-72)
		tmp = (y + 1.0) * x;
	elseif (x <= -4.5e-84)
		tmp = y;
	elseif (x <= -3e-129)
		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, -1.4e-72], N[(N[(y + 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, -4.5e-84], y, If[LessEqual[x, -3e-129], x, N[(y * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.3999999999999999e-72

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 83.3%

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

      1. +-commutative 83.3%

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

   5. Simplified 83.3%

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

   if -1.3999999999999999e-72 < x < -4.50000000000000016e-84

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 83.8%

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

   if -4.50000000000000016e-84 < x < -2.9999999999999998e-129

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 72.3%

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

   if -2.9999999999999998e-129 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 69.0%

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

      1. +-commutative 69.0%

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

   5. Simplified 69.0%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-72}:\\ \;\;\;\;\left(y + 1\right) \cdot x\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-84}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-129}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-194}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-73}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-39}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 6.5e-194) x (if (<= y 2.6e-73) y (if (<= y 5e-39) x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 6.5e-194) {
		tmp = x;
	} else if (y <= 2.6e-73) {
		tmp = y;
	} else if (y <= 5e-39) {
		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 <= 6.5d-194) then
        tmp = x
    else if (y <= 2.6d-73) then
        tmp = y
    else if (y <= 5d-39) 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 <= 6.5e-194) {
		tmp = x;
	} else if (y <= 2.6e-73) {
		tmp = y;
	} else if (y <= 5e-39) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 6.5e-194:
		tmp = x
	elif y <= 2.6e-73:
		tmp = y
	elif y <= 5e-39:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 6.5e-194)
		tmp = x;
	elseif (y <= 2.6e-73)
		tmp = y;
	elseif (y <= 5e-39)
		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 <= 6.5e-194)
		tmp = x;
	elseif (y <= 2.6e-73)
		tmp = y;
	elseif (y <= 5e-39)
		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, 6.5e-194], x, If[LessEqual[y, 2.6e-73], y, If[LessEqual[y, 5e-39], x, y]]]

Derivation

1. Split input into 2 regimes

2. if y < 6.50000000000000019e-194 or 2.6000000000000001e-73 < y < 4.9999999999999998e-39

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 53.8%

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

   if 6.50000000000000019e-194 < y < 2.6000000000000001e-73 or 4.9999999999999998e-39 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 56.4%

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

Alternative 6: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return y + ((y + 1.0) * 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 + ((y + 1.0d0) * x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. distribute-lft1-in 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

   \[\leadsto y + \left(y + 1\right) \cdot x \]
6. Add Preprocessing

Alternative 7: 38.5% 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. Add Preprocessing

3. Taylor expanded in y around 0 39.6%

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

Reproduce

herbie shell --seed 2024093 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))