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

Percentage Accurate: 100.0% → 100.0%

Time: 3.6s

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: 61.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+276}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{+170}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{+101}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-33}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1e+276)
   (* x y)
   (if (<= x -1.15e+170)
     x
     (if (<= x -2.8e+101)
       (* x y)
       (if (<= x -6.2e-33) x (if (<= x 1.0) y (* x y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1e+276) {
		tmp = x * y;
	} else if (x <= -1.15e+170) {
		tmp = x;
	} else if (x <= -2.8e+101) {
		tmp = x * y;
	} else if (x <= -6.2e-33) {
		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 <= (-1d+276)) then
        tmp = x * y
    else if (x <= (-1.15d+170)) then
        tmp = x
    else if (x <= (-2.8d+101)) then
        tmp = x * y
    else if (x <= (-6.2d-33)) 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 <= -1e+276) {
		tmp = x * y;
	} else if (x <= -1.15e+170) {
		tmp = x;
	} else if (x <= -2.8e+101) {
		tmp = x * y;
	} else if (x <= -6.2e-33) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1e+276:
		tmp = x * y
	elif x <= -1.15e+170:
		tmp = x
	elif x <= -2.8e+101:
		tmp = x * y
	elif x <= -6.2e-33:
		tmp = x
	elif x <= 1.0:
		tmp = y
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1e+276)
		tmp = Float64(x * y);
	elseif (x <= -1.15e+170)
		tmp = x;
	elseif (x <= -2.8e+101)
		tmp = Float64(x * y);
	elseif (x <= -6.2e-33)
		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 <= -1e+276)
		tmp = x * y;
	elseif (x <= -1.15e+170)
		tmp = x;
	elseif (x <= -2.8e+101)
		tmp = x * y;
	elseif (x <= -6.2e-33)
		tmp = x;
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1e+276], N[(x * y), $MachinePrecision], If[LessEqual[x, -1.15e+170], x, If[LessEqual[x, -2.8e+101], N[(x * y), $MachinePrecision], If[LessEqual[x, -6.2e-33], x, If[LessEqual[x, 1.0], y, N[(x * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.0000000000000001e276 or -1.15e170 < x < -2.79999999999999981e101 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 98.9%

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

      1. +-commutative 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in y around inf 55.1%

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

   if -1.0000000000000001e276 < x < -1.15e170 or -2.79999999999999981e101 < x < -6.19999999999999994e-33

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 65.3%

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

   if -6.19999999999999994e-33 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 78.9%

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

Alternative 3: 82.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7500:\\ \;\;\;\;x + x \cdot y\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-161}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7500.0) (+ x (* x y)) (if (<= x 1.1e-161) (+ x y) (+ y (* x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7500.0) {
		tmp = x + (x * y);
	} else if (x <= 1.1e-161) {
		tmp = 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 (x <= (-7500.0d0)) then
        tmp = x + (x * y)
    else if (x <= 1.1d-161) then
        tmp = 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 (x <= -7500.0) {
		tmp = x + (x * y);
	} else if (x <= 1.1e-161) {
		tmp = x + y;
	} else {
		tmp = y + (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7500.0:
		tmp = x + (x * y)
	elif x <= 1.1e-161:
		tmp = x + y
	else:
		tmp = y + (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7500.0)
		tmp = Float64(x + Float64(x * y));
	elseif (x <= 1.1e-161)
		tmp = 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 (x <= -7500.0)
		tmp = x + (x * y);
	elseif (x <= 1.1e-161)
		tmp = x + y;
	else
		tmp = y + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7500.0], N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.1e-161], N[(x + y), $MachinePrecision], N[(y + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7500

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 99.7%

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

      1. +-commutative 99.7%

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

   5. Simplified 99.7%

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

      1. distribute-lft-in 99.7%

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

      2. *-rgt-identity 99.7%

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

   7. Applied egg-rr 99.7%

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

   if -7500 < x < 1.10000000000000001e-161

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 99.9%

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

      1. associate-+r+ 99.9%

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

      2. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 99.8%

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

   7. Taylor expanded in y around 0 99.8%

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

      1. +-commutative 99.8%

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

   9. Simplified 99.8%

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

   if 1.10000000000000001e-161 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 53.2%

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

      1. *-commutative 53.2%

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

   5. Simplified 53.2%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7500:\\ \;\;\;\;x + x \cdot y\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-161}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7500:\\ \;\;\;\;x + x \cdot y\\ \mathbf{elif}\;x \leq 820:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7500.0) (+ x (* x y)) (if (<= x 820.0) (+ x y) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7500.0) {
		tmp = x + (x * y);
	} else if (x <= 820.0) {
		tmp = x + 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 <= (-7500.0d0)) then
        tmp = x + (x * y)
    else if (x <= 820.0d0) then
        tmp = x + y
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -7500.0) {
		tmp = x + (x * y);
	} else if (x <= 820.0) {
		tmp = x + y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7500.0:
		tmp = x + (x * y)
	elif x <= 820.0:
		tmp = x + y
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7500.0)
		tmp = Float64(x + Float64(x * y));
	elseif (x <= 820.0)
		tmp = Float64(x + y);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7500.0)
		tmp = x + (x * y);
	elseif (x <= 820.0)
		tmp = x + y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7500.0], N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 820.0], N[(x + y), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7500

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 99.7%

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

      1. +-commutative 99.7%

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

   5. Simplified 99.7%

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

      1. distribute-lft-in 99.7%

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

      2. *-rgt-identity 99.7%

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

   7. Applied egg-rr 99.7%

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

   if -7500 < x < 820

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 99.9%

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

      1. associate-+r+ 99.9%

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

      2. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 98.8%

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

   7. Taylor expanded in y around 0 98.8%

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

      1. +-commutative 98.8%

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

   9. Simplified 98.8%

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

   if 820 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 98.4%

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

      1. +-commutative 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in y around inf 41.1%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7500:\\ \;\;\;\;x + x \cdot y\\ \mathbf{elif}\;x \leq 820:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7500:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;x \leq 820:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7500.0) (* x (+ y 1.0)) (if (<= x 820.0) (+ x y) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7500.0) {
		tmp = x * (y + 1.0);
	} else if (x <= 820.0) {
		tmp = x + 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 <= (-7500.0d0)) then
        tmp = x * (y + 1.0d0)
    else if (x <= 820.0d0) then
        tmp = x + y
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -7500.0) {
		tmp = x * (y + 1.0);
	} else if (x <= 820.0) {
		tmp = x + y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7500.0:
		tmp = x * (y + 1.0)
	elif x <= 820.0:
		tmp = x + y
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7500.0)
		tmp = Float64(x * Float64(y + 1.0));
	elseif (x <= 820.0)
		tmp = Float64(x + y);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7500.0)
		tmp = x * (y + 1.0);
	elseif (x <= 820.0)
		tmp = x + y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7500.0], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 820.0], N[(x + y), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7500

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 99.7%

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

      1. +-commutative 99.7%

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

   5. Simplified 99.7%

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

   if -7500 < x < 820

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 99.9%

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

      1. associate-+r+ 99.9%

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

      2. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 98.8%

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

   7. Taylor expanded in y around 0 98.8%

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

      1. +-commutative 98.8%

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

   9. Simplified 98.8%

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

   if 820 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 98.4%

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

      1. +-commutative 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in y around inf 41.1%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7500:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;x \leq 820:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+18}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -7.2e+18) (* x y) (+ x y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.2e+18) {
		tmp = x * 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 (y <= (-7.2d+18)) then
        tmp = x * y
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.2e+18) {
		tmp = x * y;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.2e+18:
		tmp = x * y
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.2e+18)
		tmp = Float64(x * y);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.2e+18)
		tmp = x * y;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.2e+18], N[(x * y), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.2e18

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 53.0%

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

      1. +-commutative 53.0%

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

   5. Simplified 53.0%

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

   6. Taylor expanded in y around inf 53.0%

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

   if -7.2e18 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 85.5%

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

      1. associate-+r+ 85.5%

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

      2. +-commutative 85.5%

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

   5. Simplified 85.5%

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

   6. Taylor expanded in x around 0 67.5%

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

   7. Taylor expanded in y around 0 82.0%

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

      1. +-commutative 82.0%

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

   9. Simplified 82.0%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+18}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{-33}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -2.35e-33) x y))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.35e-33) {
		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 (x <= (-2.35d-33)) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.35e-33) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.35e-33:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.35e-33)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.35e-33)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.35e-33], x, y]

Derivation

1. Split input into 2 regimes

2. if x < -2.3500000000000001e-33

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 49.4%

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

   if -2.3500000000000001e-33 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 55.3%

      \[\leadsto \color{blue}{y} \]
3. Recombined 2 regimes into one program.
4. 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 37.4%

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

Reproduce

herbie shell --seed 2024149 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))