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

Percentage Accurate: 100.0% → 100.0%

Time: 2.6s

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

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

FPCore

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

C

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

Julia

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

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x + N[(x * y + y), $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. associate-+r+ 100.0%

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

   3. fma-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 68.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-107}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+86}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+148}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+235}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   (* x y)
   (if (<= y 5.2e-107)
     x
     (if (<= y 1.1e+86)
       y
       (if (<= y 8.2e+148) (* x y) (if (<= y 2.4e+235) y (* x y)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x * y;
	} else if (y <= 5.2e-107) {
		tmp = x;
	} else if (y <= 1.1e+86) {
		tmp = y;
	} else if (y <= 8.2e+148) {
		tmp = x * y;
	} else if (y <= 2.4e+235) {
		tmp = y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x * y
    else if (y <= 5.2d-107) then
        tmp = x
    else if (y <= 1.1d+86) then
        tmp = y
    else if (y <= 8.2d+148) then
        tmp = x * y
    else if (y <= 2.4d+235) then
        tmp = y
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x * y;
	} else if (y <= 5.2e-107) {
		tmp = x;
	} else if (y <= 1.1e+86) {
		tmp = y;
	} else if (y <= 8.2e+148) {
		tmp = x * y;
	} else if (y <= 2.4e+235) {
		tmp = y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x * y
	elif y <= 5.2e-107:
		tmp = x
	elif y <= 1.1e+86:
		tmp = y
	elif y <= 8.2e+148:
		tmp = x * y
	elif y <= 2.4e+235:
		tmp = y
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(x * y);
	elseif (y <= 5.2e-107)
		tmp = x;
	elseif (y <= 1.1e+86)
		tmp = y;
	elseif (y <= 8.2e+148)
		tmp = Float64(x * y);
	elseif (y <= 2.4e+235)
		tmp = y;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x * y;
	elseif (y <= 5.2e-107)
		tmp = x;
	elseif (y <= 1.1e+86)
		tmp = y;
	elseif (y <= 8.2e+148)
		tmp = x * y;
	elseif (y <= 2.4e+235)
		tmp = y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -1.0], N[(x * y), $MachinePrecision], If[LessEqual[y, 5.2e-107], x, If[LessEqual[y, 1.1e+86], y, If[LessEqual[y, 8.2e+148], N[(x * y), $MachinePrecision], If[LessEqual[y, 2.4e+235], y, N[(x * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 1.10000000000000002e86 < y < 8.1999999999999996e148 or 2.3999999999999999e235 < y

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+r+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 52.8%

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

      1. +-commutative 52.8%

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

   6. Simplified 52.8%

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

   7. Taylor expanded in y around inf 52.2%

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

   if -1 < y < 5.2000000000000001e-107

   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. associate-+r+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 79.5%

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

   if 5.2000000000000001e-107 < y < 1.10000000000000002e86 or 8.1999999999999996e148 < y < 2.3999999999999999e235

   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. associate-+r+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 59.2%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-107}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+86}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+148}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+235}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3: 88.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-43}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-86}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-124}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5.5e-43)
   (* x (+ y 1.0))
   (if (<= x -1.45e-86) y (if (<= x -1.45e-124) x (if (<= x 1.0) y (* x y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5.5e-43) {
		tmp = x * (y + 1.0);
	} else if (x <= -1.45e-86) {
		tmp = y;
	} else if (x <= -1.45e-124) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = x * 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-43)) then
        tmp = x * (y + 1.0d0)
    else if (x <= (-1.45d-86)) then
        tmp = y
    else if (x <= (-1.45d-124)) then
        tmp = x
    else if (x <= 1.0d0) then
        tmp = y
    else
        tmp = x * 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-43) {
		tmp = x * (y + 1.0);
	} else if (x <= -1.45e-86) {
		tmp = y;
	} else if (x <= -1.45e-124) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -5.5e-43:
		tmp = x * (y + 1.0)
	elif x <= -1.45e-86:
		tmp = y
	elif x <= -1.45e-124:
		tmp = x
	elif x <= 1.0:
		tmp = y
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5.5e-43)
		tmp = Float64(x * Float64(y + 1.0));
	elseif (x <= -1.45e-86)
		tmp = y;
	elseif (x <= -1.45e-124)
		tmp = x;
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = Float64(x * 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-43)
		tmp = x * (y + 1.0);
	elseif (x <= -1.45e-86)
		tmp = y;
	elseif (x <= -1.45e-124)
		tmp = x;
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = x * 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-43], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.45e-86], y, If[LessEqual[x, -1.45e-124], x, If[LessEqual[x, 1.0], y, N[(x * y), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.50000000000000013e-43

   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. associate-+r+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 93.0%

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

      1. +-commutative 93.0%

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

   6. Simplified 93.0%

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

   if -5.50000000000000013e-43 < x < -1.45e-86 or -1.4500000000000001e-124 < x < 1

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+r+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 78.3%

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

   if -1.45e-86 < x < -1.4500000000000001e-124

   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. associate-+r+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 45.4%

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

   if 1 < x

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+r+ 99.9%

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

      3. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 99.7%

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

      1. +-commutative 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in y around inf 46.9%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-43}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-86}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-124}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 4: 88.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-107}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 5.2e-107) (* x (+ y 1.0)) (* y (+ x 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 5.2e-107) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y * (x + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.2d-107) then
        tmp = x * (y + 1.0d0)
    else
        tmp = y * (x + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 5.2e-107) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y * (x + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 5.2e-107:
		tmp = x * (y + 1.0)
	else:
		tmp = y * (x + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 5.2e-107)
		tmp = Float64(x * Float64(y + 1.0));
	else
		tmp = Float64(y * Float64(x + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.2e-107)
		tmp = x * (y + 1.0);
	else
		tmp = y * (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 5.2e-107], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.2000000000000001e-107

   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. associate-+r+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 68.0%

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

      1. +-commutative 68.0%

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

   6. Simplified 68.0%

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

   if 5.2000000000000001e-107 < y

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+r+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 90.7%

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

      1. +-commutative 90.7%

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

   6. Simplified 90.7%

      \[\leadsto \color{blue}{y \cdot \left(x + 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-107}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \]

Alternative 5: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 + (x * y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 6: 64.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-107}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 5.2e-107) x y))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 5.2e-107) {
		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 <= 5.2d-107) 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 <= 5.2e-107) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 5.2e-107:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 5.2e-107)
		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 <= 5.2e-107)
		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, 5.2e-107], x, y]

Derivation

1. Split input into 2 regimes

2. if y < 5.2000000000000001e-107

   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. associate-+r+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 50.5%

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

   if 5.2000000000000001e-107 < y

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+r+ 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 50.8%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-107}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 7: 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%

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

   2. associate-+r+ 100.0%

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

   3. fma-def 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in y around 0 39.1%

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

5. Final simplification 39.1%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023272 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))