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

Percentage Accurate: 100.0% → 100.0%

Time: 4.0s

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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. Add Preprocessing

3. Final simplification 100.0%

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

Alternative 2: 62.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{+166}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{+99}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-28}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-76}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-101}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.7e+166)
   x
   (if (<= x -2.8e+99)
     (* x y)
     (if (<= x -1.9e-28)
       x
       (if (<= x -3.4e-76)
         y
         (if (<= x -5.4e-101) x (if (<= x 1.0) y (* x y))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.7e+166) {
		tmp = x;
	} else if (x <= -2.8e+99) {
		tmp = x * y;
	} else if (x <= -1.9e-28) {
		tmp = x;
	} else if (x <= -3.4e-76) {
		tmp = y;
	} else if (x <= -5.4e-101) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.7d+166)) then
        tmp = x
    else if (x <= (-2.8d+99)) then
        tmp = x * y
    else if (x <= (-1.9d-28)) then
        tmp = x
    else if (x <= (-3.4d-76)) then
        tmp = y
    else if (x <= (-5.4d-101)) then
        tmp = x
    else if (x <= 1.0d0) then
        tmp = y
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.7e+166) {
		tmp = x;
	} else if (x <= -2.8e+99) {
		tmp = x * y;
	} else if (x <= -1.9e-28) {
		tmp = x;
	} else if (x <= -3.4e-76) {
		tmp = y;
	} else if (x <= -5.4e-101) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.7e+166:
		tmp = x
	elif x <= -2.8e+99:
		tmp = x * y
	elif x <= -1.9e-28:
		tmp = x
	elif x <= -3.4e-76:
		tmp = y
	elif x <= -5.4e-101:
		tmp = x
	elif x <= 1.0:
		tmp = y
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.7e+166)
		tmp = x;
	elseif (x <= -2.8e+99)
		tmp = Float64(x * y);
	elseif (x <= -1.9e-28)
		tmp = x;
	elseif (x <= -3.4e-76)
		tmp = y;
	elseif (x <= -5.4e-101)
		tmp = x;
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.7e+166)
		tmp = x;
	elseif (x <= -2.8e+99)
		tmp = x * y;
	elseif (x <= -1.9e-28)
		tmp = x;
	elseif (x <= -3.4e-76)
		tmp = y;
	elseif (x <= -5.4e-101)
		tmp = x;
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.7e+166], x, If[LessEqual[x, -2.8e+99], N[(x * y), $MachinePrecision], If[LessEqual[x, -1.9e-28], x, If[LessEqual[x, -3.4e-76], y, If[LessEqual[x, -5.4e-101], x, If[LessEqual[x, 1.0], y, N[(x * y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.69999999999999977e166 or -2.8e99 < x < -1.90000000000000005e-28 or -3.3999999999999999e-76 < x < -5.4000000000000003e-101

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 54.7%

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

   if -5.69999999999999977e166 < x < -2.8e99 or 1 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 99.4%

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

      1. +-commutative 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in y around inf 54.6%

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

   if -1.90000000000000005e-28 < x < -3.3999999999999999e-76 or -5.4000000000000003e-101 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 76.0%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{+166}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{+99}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-28}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-76}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-101}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-27}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-80}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-99}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.2e-27)
   (* x (+ y 1.0))
   (if (<= x -1.4e-80) y (if (<= x -2.1e-99) x (if (<= x 1.0) y (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.2e-27) {
		tmp = x * (y + 1.0);
	} else if (x <= -1.4e-80) {
		tmp = y;
	} else if (x <= -2.1e-99) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.2d-27)) then
        tmp = x * (y + 1.0d0)
    else if (x <= (-1.4d-80)) then
        tmp = y
    else if (x <= (-2.1d-99)) then
        tmp = x
    else if (x <= 1.0d0) then
        tmp = y
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.2e-27) {
		tmp = x * (y + 1.0);
	} else if (x <= -1.4e-80) {
		tmp = y;
	} else if (x <= -2.1e-99) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.2e-27:
		tmp = x * (y + 1.0)
	elif x <= -1.4e-80:
		tmp = y
	elif x <= -2.1e-99:
		tmp = x
	elif x <= 1.0:
		tmp = y
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.2e-27)
		tmp = Float64(x * Float64(y + 1.0));
	elseif (x <= -1.4e-80)
		tmp = y;
	elseif (x <= -2.1e-99)
		tmp = x;
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.2e-27)
		tmp = x * (y + 1.0);
	elseif (x <= -1.4e-80)
		tmp = y;
	elseif (x <= -2.1e-99)
		tmp = x;
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.2e-27], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.4e-80], y, If[LessEqual[x, -2.1e-99], x, If[LessEqual[x, 1.0], y, N[(x * y), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.20000000000000031e-27

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 89.1%

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

      1. +-commutative 89.1%

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

   5. Simplified 89.1%

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

   if -4.20000000000000031e-27 < x < -1.39999999999999995e-80 or -2.09999999999999984e-99 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 76.2%

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

   if -1.39999999999999995e-80 < x < -2.09999999999999984e-99

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 44.7%

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

   if 1 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 99.3%

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

      1. +-commutative 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in y around inf 49.6%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-27}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-80}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-99}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-142}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.1e-142) (* x (+ y 1.0)) (* y (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-142) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y * (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.1d-142) then
        tmp = x * (y + 1.0d0)
    else
        tmp = y * (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-142) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y * (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.1e-142:
		tmp = x * (y + 1.0)
	else:
		tmp = y * (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.1e-142)
		tmp = Float64(x * Float64(y + 1.0));
	else
		tmp = Float64(y * Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.1e-142)
		tmp = x * (y + 1.0);
	else
		tmp = y * (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.1e-142], 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 < 3.1e-142

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 67.5%

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

      1. +-commutative 67.5%

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

   5. Simplified 67.5%

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

   if 3.1e-142 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 80.7%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-142}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-139}:\\ \;\;\;\;x + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.1e-139) (+ x (* x y)) (* y (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.1e-139) {
		tmp = x + (x * y);
	} else {
		tmp = y * (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.1d-139) then
        tmp = x + (x * y)
    else
        tmp = y * (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.1e-139) {
		tmp = x + (x * y);
	} else {
		tmp = y * (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.1e-139:
		tmp = x + (x * y)
	else:
		tmp = y * (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.1e-139)
		tmp = Float64(x + Float64(x * y));
	else
		tmp = Float64(y * Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.1e-139)
		tmp = x + (x * y);
	else
		tmp = y * (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.1e-139], N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(y * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.10000000000000008e-139

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 67.7%

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

      1. +-commutative 67.7%

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

   5. Simplified 67.7%

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

      1. distribute-lft-in 67.7%

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

      2. *-rgt-identity 67.7%

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

   7. Applied egg-rr 67.7%

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

   if 2.10000000000000008e-139 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 81.4%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-139}:\\ \;\;\;\;x + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-142}:\\ \;\;\;\;x + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.1e-142) (+ x (* x y)) (+ y (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-142) {
		tmp = x + (x * y);
	} else {
		tmp = y + (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.1d-142) then
        tmp = x + (x * y)
    else
        tmp = y + (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-142) {
		tmp = x + (x * y);
	} else {
		tmp = y + (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.1e-142:
		tmp = x + (x * y)
	else:
		tmp = y + (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.1e-142)
		tmp = Float64(x + Float64(x * y));
	else
		tmp = Float64(y + Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.1e-142)
		tmp = x + (x * y);
	else
		tmp = y + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.1e-142], N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(y + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.1e-142

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 67.5%

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

      1. +-commutative 67.5%

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

   5. Simplified 67.5%

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

      1. distribute-lft-in 67.5%

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

      2. *-rgt-identity 67.5%

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

   7. Applied egg-rr 67.5%

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

   if 3.1e-142 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 80.7%

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

      1. +-commutative 80.7%

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

      2. distribute-rgt-in 80.7%

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

      3. *-un-lft-identity 80.7%

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

   5. Applied egg-rr 80.7%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-142}:\\ \;\;\;\;x + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 47.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-139}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.1e-139) x y))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.1e-139) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.1d-139) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.1e-139) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.1e-139:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.1e-139)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.1e-139)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.1e-139], x, y]

Derivation

1. Split input into 2 regimes

2. if y < 2.10000000000000008e-139

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 51.6%

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

   if 2.10000000000000008e-139 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 45.9%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-139}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 38.2% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

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 38.7%

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

4. Final simplification 38.7%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024044 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))