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

Percentage Accurate: 100.0% → 100.0%

Time: 3.0s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x y) :precision binary64 (+ (+ (* x y) x) y))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * y) + x) + y
end function

Java

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

Python

def code(x, y):
	return ((x * y) + x) + y

Julia

function code(x, y)
	return Float64(Float64(Float64(x * y) + x) + y)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * y) + x) + y;
end

Wolfram

code[x_, y_] := N[(N[(N[(x * y), $MachinePrecision] + x), $MachinePrecision] + y), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (+ (+ (* x y) x) y))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * y) + x) + y
end function

Java

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

Python

def code(x, y):
	return ((x * y) + x) + y

Julia

function code(x, y)
	return Float64(Float64(Float64(x * y) + x) + y)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * y) + x) + y;
end

Wolfram

code[x_, y_] := N[(N[(N[(x * y), $MachinePrecision] + x), $MachinePrecision] + y), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. fma-def 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 68.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+240}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{+164}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -8.8 \cdot 10^{+130}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-62}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-118}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-153}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.7e+240)
   (* y x)
   (if (<= x -7.8e+164)
     x
     (if (<= x -8.8e+130)
       (* y x)
       (if (<= x -3.2e-62)
         x
         (if (<= x -7.2e-118)
           y
           (if (<= x -1.25e-153) x (if (<= x 1.0) y (* y x)))))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.7e+240) {
		tmp = y * x;
	} else if (x <= -7.8e+164) {
		tmp = x;
	} else if (x <= -8.8e+130) {
		tmp = y * x;
	} else if (x <= -3.2e-62) {
		tmp = x;
	} else if (x <= -7.2e-118) {
		tmp = y;
	} else if (x <= -1.25e-153) {
		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 <= (-1.7d+240)) then
        tmp = y * x
    else if (x <= (-7.8d+164)) then
        tmp = x
    else if (x <= (-8.8d+130)) then
        tmp = y * x
    else if (x <= (-3.2d-62)) then
        tmp = x
    else if (x <= (-7.2d-118)) then
        tmp = y
    else if (x <= (-1.25d-153)) 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 <= -1.7e+240) {
		tmp = y * x;
	} else if (x <= -7.8e+164) {
		tmp = x;
	} else if (x <= -8.8e+130) {
		tmp = y * x;
	} else if (x <= -3.2e-62) {
		tmp = x;
	} else if (x <= -7.2e-118) {
		tmp = y;
	} else if (x <= -1.25e-153) {
		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 <= -1.7e+240:
		tmp = y * x
	elif x <= -7.8e+164:
		tmp = x
	elif x <= -8.8e+130:
		tmp = y * x
	elif x <= -3.2e-62:
		tmp = x
	elif x <= -7.2e-118:
		tmp = y
	elif x <= -1.25e-153:
		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 <= -1.7e+240)
		tmp = Float64(y * x);
	elseif (x <= -7.8e+164)
		tmp = x;
	elseif (x <= -8.8e+130)
		tmp = Float64(y * x);
	elseif (x <= -3.2e-62)
		tmp = x;
	elseif (x <= -7.2e-118)
		tmp = y;
	elseif (x <= -1.25e-153)
		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 <= -1.7e+240)
		tmp = y * x;
	elseif (x <= -7.8e+164)
		tmp = x;
	elseif (x <= -8.8e+130)
		tmp = y * x;
	elseif (x <= -3.2e-62)
		tmp = x;
	elseif (x <= -7.2e-118)
		tmp = y;
	elseif (x <= -1.25e-153)
		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, -1.7e+240], N[(y * x), $MachinePrecision], If[LessEqual[x, -7.8e+164], x, If[LessEqual[x, -8.8e+130], N[(y * x), $MachinePrecision], If[LessEqual[x, -3.2e-62], x, If[LessEqual[x, -7.2e-118], y, If[LessEqual[x, -1.25e-153], x, If[LessEqual[x, 1.0], y, N[(y * x), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.70000000000000004e240 or -7.79999999999999971e164 < x < -8.79999999999999974e130 or 1 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around inf 57.9%

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

   if -1.70000000000000004e240 < x < -7.79999999999999971e164 or -8.79999999999999974e130 < x < -3.20000000000000021e-62 or -7.2000000000000004e-118 < x < -1.25000000000000008e-153

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 61.9%

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

   if -3.20000000000000021e-62 < x < -7.2000000000000004e-118 or -1.25000000000000008e-153 < x < 1

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 88.3%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+240}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{+164}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -8.8 \cdot 10^{+130}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-62}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-118}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-153}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.45 \cdot 10^{-62}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-118}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-153}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.45e-62)
   (* x (+ y 1.0))
   (if (<= x -7.2e-118) y (if (<= x -1.25e-153) x (if (<= x 1.0) y (* y x))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.45e-62) {
		tmp = x * (y + 1.0);
	} else if (x <= -7.2e-118) {
		tmp = y;
	} else if (x <= -1.25e-153) {
		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 <= (-3.45d-62)) then
        tmp = x * (y + 1.0d0)
    else if (x <= (-7.2d-118)) then
        tmp = y
    else if (x <= (-1.25d-153)) 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 <= -3.45e-62) {
		tmp = x * (y + 1.0);
	} else if (x <= -7.2e-118) {
		tmp = y;
	} else if (x <= -1.25e-153) {
		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 <= -3.45e-62:
		tmp = x * (y + 1.0)
	elif x <= -7.2e-118:
		tmp = y
	elif x <= -1.25e-153:
		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 <= -3.45e-62)
		tmp = Float64(x * Float64(y + 1.0));
	elseif (x <= -7.2e-118)
		tmp = y;
	elseif (x <= -1.25e-153)
		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 <= -3.45e-62)
		tmp = x * (y + 1.0);
	elseif (x <= -7.2e-118)
		tmp = y;
	elseif (x <= -1.25e-153)
		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, -3.45e-62], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7.2e-118], y, If[LessEqual[x, -1.25e-153], x, If[LessEqual[x, 1.0], y, N[(y * x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.44999999999999979e-62

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 96.0%

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

      1. +-commutative 96.0%

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

   7. Simplified 96.0%

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

   if -3.44999999999999979e-62 < x < -7.2000000000000004e-118 or -1.25000000000000008e-153 < x < 1

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 88.3%

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

   if -7.2000000000000004e-118 < x < -1.25000000000000008e-153

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 51.4%

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

   if 1 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around inf 53.8%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.45 \cdot 10^{-62}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-118}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-153}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-62}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-118}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-153}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.3e-62) x (if (<= x -9e-118) y (if (<= x -1.25e-153) x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.3e-62) {
		tmp = x;
	} else if (x <= -9e-118) {
		tmp = y;
	} else if (x <= -1.25e-153) {
		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 <= (-2.3d-62)) then
        tmp = x
    else if (x <= (-9d-118)) then
        tmp = y
    else if (x <= (-1.25d-153)) 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 <= -2.3e-62) {
		tmp = x;
	} else if (x <= -9e-118) {
		tmp = y;
	} else if (x <= -1.25e-153) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.3e-62:
		tmp = x
	elif x <= -9e-118:
		tmp = y
	elif x <= -1.25e-153:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.3e-62)
		tmp = x;
	elseif (x <= -9e-118)
		tmp = y;
	elseif (x <= -1.25e-153)
		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 <= -2.3e-62)
		tmp = x;
	elseif (x <= -9e-118)
		tmp = y;
	elseif (x <= -1.25e-153)
		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, -2.3e-62], x, If[LessEqual[x, -9e-118], y, If[LessEqual[x, -1.25e-153], x, y]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.3e-62 or -9.0000000000000001e-118 < x < -1.25000000000000008e-153

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 54.1%

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

   if -2.3e-62 < x < -9.0000000000000001e-118 or -1.25000000000000008e-153 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 52.2%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-62}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-118}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-153}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.4e-12) (* x (+ y 1.0)) (* y (+ x 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.4e-12) {
		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 <= 1.4d-12) 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 <= 1.4e-12) {
		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 <= 1.4e-12:
		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 <= 1.4e-12)
		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 <= 1.4e-12)
		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, 1.4e-12], 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 < 1.4000000000000001e-12

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 70.1%

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

      1. +-commutative 70.1%

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

   7. Simplified 70.1%

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

   if 1.4000000000000001e-12 < y

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 98.1%

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

      1. +-commutative 98.1%

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

   7. Simplified 98.1%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-12}:\\ \;\;\;\;x + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.4e-12) (+ x (* y x)) (* y (+ x 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.4e-12) {
		tmp = x + (y * x);
	} 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 <= 2.4d-12) then
        tmp = x + (y * x)
    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 <= 2.4e-12) {
		tmp = x + (y * x);
	} else {
		tmp = y * (x + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.4e-12:
		tmp = x + (y * x)
	else:
		tmp = y * (x + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.4e-12)
		tmp = Float64(x + Float64(y * x));
	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 <= 2.4e-12)
		tmp = x + (y * x);
	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, 2.4e-12], N[(x + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(y * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.39999999999999987e-12

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 70.1%

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

      1. +-commutative 70.1%

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

   7. Simplified 70.1%

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

      1. distribute-lft-in 70.1%

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

      2. *-rgt-identity 70.1%

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

   9. Applied egg-rr 70.1%

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

   if 2.39999999999999987e-12 < y

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 98.1%

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

      1. +-commutative 98.1%

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

   7. Simplified 98.1%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-12}:\\ \;\;\;\;x + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \]
5. 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: 38.9% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := x

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. fma-def 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in y around 0 37.9%

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

6. Final simplification 37.9%

   \[\leadsto x \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))