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

Percentage Accurate: 100.0% → 100.0%

Time: 9.8s

Alternatives: 9

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: 61.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+219}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-101}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.8e+219) (* x y) (if (<= x -1e-101) x (if (<= x 1.0) y (* x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.8e+219) {
		tmp = x * y;
	} else if (x <= -1e-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 <= (-2.8d+219)) then
        tmp = x * y
    else if (x <= (-1d-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 <= -2.8e+219) {
		tmp = x * y;
	} else if (x <= -1e-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 <= -2.8e+219:
		tmp = x * y
	elif x <= -1e-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 <= -2.8e+219)
		tmp = Float64(x * y);
	elseif (x <= -1e-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 <= -2.8e+219)
		tmp = x * y;
	elseif (x <= -1e-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, -2.8e+219], N[(x * y), $MachinePrecision], If[LessEqual[x, -1e-101], x, If[LessEqual[x, 1.0], y, N[(x * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.80000000000000015e219 or 1 < x

   1. Initial program 100.0%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(y + x \cdot y\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\left(x + 1\right) \cdot \color{blue}{y}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(x + 1\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(x + 1\right)}\right)\right) \]

      8. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(1 + x\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(x + \color{blue}{1}\right)\right) \]

      3. +-lowering-+.f64 47.9%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right) \]

   7. Simplified 47.9%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 47.3%

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{x}\right) \]

   10. Simplified 47.3%

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

   if -2.80000000000000015e219 < x < -1.00000000000000005e-101

   1. Initial program 100.0%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(y + x \cdot y\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\left(x + 1\right) \cdot \color{blue}{y}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(x + 1\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(x + 1\right)}\right)\right) \]

      8. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x} \]
   6. Step-by-step derivation

   7. Simplified 45.0%

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

   if -1.00000000000000005e-101 < x < 1

   1. Initial program 100.0%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(y + x \cdot y\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\left(x + 1\right) \cdot \color{blue}{y}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(x + 1\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(x + 1\right)}\right)\right) \]

      8. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y} \]
   6. Step-by-step derivation

   7. Simplified 67.7%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+219}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-101}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-12}:\\ \;\;\;\;x + x \cdot y\\ \mathbf{elif}\;y \leq 0.0007:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7e-12) (+ x (* x y)) (if (<= y 0.0007) (+ x y) (+ y (* x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7e-12) {
		tmp = x + (x * y);
	} else if (y <= 0.0007) {
		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 (y <= (-7d-12)) then
        tmp = x + (x * y)
    else if (y <= 0.0007d0) 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 (y <= -7e-12) {
		tmp = x + (x * y);
	} else if (y <= 0.0007) {
		tmp = x + y;
	} else {
		tmp = y + (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7e-12:
		tmp = x + (x * y)
	elif y <= 0.0007:
		tmp = x + y
	else:
		tmp = y + (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7e-12)
		tmp = Float64(x + Float64(x * y));
	elseif (y <= 0.0007)
		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 (y <= -7e-12)
		tmp = x + (x * y);
	elseif (y <= 0.0007)
		tmp = x + y;
	else
		tmp = y + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7e-12], N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0007], N[(x + y), $MachinePrecision], N[(y + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.0000000000000001e-12

   1. Initial program 100.0%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(y + x \cdot y\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\left(x + 1\right) \cdot \color{blue}{y}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(x + 1\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(x + 1\right)}\right)\right) \]

      8. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(x \cdot y\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{x}\right)\right) \]

      2. *-lowering-*.f64 56.2%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{x}\right)\right) \]

   7. Simplified 56.2%

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

   if -7.0000000000000001e-12 < y < 6.99999999999999993e-4

   1. Initial program 100.0%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(y + x \cdot y\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\left(x + 1\right) \cdot \color{blue}{y}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(x + 1\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(x + 1\right)}\right)\right) \]

      8. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{y}\right) \]
   6. Step-by-step derivation

   7. Simplified 99.7%

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

   if 6.99999999999999993e-4 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(x \cdot y\right)}, y\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(y \cdot x\right), y\right) \]

      2. *-lowering-*.f64 99.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, x\right), y\right) \]

   5. Simplified 99.3%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-12}:\\ \;\;\;\;x + x \cdot y\\ \mathbf{elif}\;y \leq 0.0007:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-12}:\\ \;\;\;\;x + x \cdot y\\ \mathbf{elif}\;y \leq 0.0007:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7e-12) (+ x (* x y)) (if (<= y 0.0007) (+ x y) (* y (+ x 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7e-12) {
		tmp = x + (x * y);
	} else if (y <= 0.0007) {
		tmp = 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 <= (-7d-12)) then
        tmp = x + (x * y)
    else if (y <= 0.0007d0) then
        tmp = 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 <= -7e-12) {
		tmp = x + (x * y);
	} else if (y <= 0.0007) {
		tmp = x + y;
	} else {
		tmp = y * (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7e-12:
		tmp = x + (x * y)
	elif y <= 0.0007:
		tmp = x + y
	else:
		tmp = y * (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7e-12)
		tmp = Float64(x + Float64(x * y));
	elseif (y <= 0.0007)
		tmp = 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 <= -7e-12)
		tmp = x + (x * y);
	elseif (y <= 0.0007)
		tmp = x + y;
	else
		tmp = y * (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7e-12], N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0007], N[(x + y), $MachinePrecision], N[(y * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.0000000000000001e-12

   1. Initial program 100.0%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(y + x \cdot y\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\left(x + 1\right) \cdot \color{blue}{y}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(x + 1\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(x + 1\right)}\right)\right) \]

      8. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(x \cdot y\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{x}\right)\right) \]

      2. *-lowering-*.f64 56.2%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{x}\right)\right) \]

   7. Simplified 56.2%

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

   if -7.0000000000000001e-12 < y < 6.99999999999999993e-4

   1. Initial program 100.0%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(y + x \cdot y\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\left(x + 1\right) \cdot \color{blue}{y}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(x + 1\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(x + 1\right)}\right)\right) \]

      8. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{y}\right) \]
   6. Step-by-step derivation

   7. Simplified 99.7%

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

   if 6.99999999999999993e-4 < y

   1. Initial program 100.0%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(y + x \cdot y\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\left(x + 1\right) \cdot \color{blue}{y}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(x + 1\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(x + 1\right)}\right)\right) \]

      8. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(1 + x\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(x + \color{blue}{1}\right)\right) \]

      3. +-lowering-+.f64 99.2%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right) \]

   7. Simplified 99.2%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-12}:\\ \;\;\;\;x + x \cdot y\\ \mathbf{elif}\;y \leq 0.0007:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4500:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 0.0007:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4500.0) (* x y) (if (<= y 0.0007) (+ x y) (* y (+ x 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4500.0) {
		tmp = x * y;
	} else if (y <= 0.0007) {
		tmp = 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 <= (-4500.0d0)) then
        tmp = x * y
    else if (y <= 0.0007d0) then
        tmp = 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 <= -4500.0) {
		tmp = x * y;
	} else if (y <= 0.0007) {
		tmp = x + y;
	} else {
		tmp = y * (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4500.0:
		tmp = x * y
	elif y <= 0.0007:
		tmp = x + y
	else:
		tmp = y * (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4500.0)
		tmp = Float64(x * y);
	elseif (y <= 0.0007)
		tmp = 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 <= -4500.0)
		tmp = x * y;
	elseif (y <= 0.0007)
		tmp = x + y;
	else
		tmp = y * (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4500.0], N[(x * y), $MachinePrecision], If[LessEqual[y, 0.0007], N[(x + y), $MachinePrecision], N[(y * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4500

   1. Initial program 100.0%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(y + x \cdot y\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\left(x + 1\right) \cdot \color{blue}{y}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(x + 1\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(x + 1\right)}\right)\right) \]

      8. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(1 + x\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(x + \color{blue}{1}\right)\right) \]

      3. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 54.8%

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{x}\right) \]

   10. Simplified 54.8%

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

   if -4500 < y < 6.99999999999999993e-4

   1. Initial program 100.0%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(y + x \cdot y\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\left(x + 1\right) \cdot \color{blue}{y}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(x + 1\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(x + 1\right)}\right)\right) \]

      8. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{y}\right) \]
   6. Step-by-step derivation

   7. Simplified 98.7%

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

   if 6.99999999999999993e-4 < y

   1. Initial program 100.0%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(y + x \cdot y\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\left(x + 1\right) \cdot \color{blue}{y}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(x + 1\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(x + 1\right)}\right)\right) \]

      8. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(1 + x\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(x + \color{blue}{1}\right)\right) \]

      3. +-lowering-+.f64 99.2%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right) \]

   7. Simplified 99.2%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4500:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 0.0007:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4500:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+273}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4500.0) (* x y) (if (<= y 7.2e+273) (+ x y) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4500.0) {
		tmp = x * y;
	} else if (y <= 7.2e+273) {
		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 <= (-4500.0d0)) then
        tmp = x * y
    else if (y <= 7.2d+273) 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 <= -4500.0) {
		tmp = x * y;
	} else if (y <= 7.2e+273) {
		tmp = x + y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4500.0:
		tmp = x * y
	elif y <= 7.2e+273:
		tmp = x + y
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4500.0)
		tmp = Float64(x * y);
	elseif (y <= 7.2e+273)
		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 <= -4500.0)
		tmp = x * y;
	elseif (y <= 7.2e+273)
		tmp = x + y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4500.0], N[(x * y), $MachinePrecision], If[LessEqual[y, 7.2e+273], N[(x + y), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4500 or 7.20000000000000047e273 < y

   1. Initial program 100.0%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(y + x \cdot y\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\left(x + 1\right) \cdot \color{blue}{y}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(x + 1\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(x + 1\right)}\right)\right) \]

      8. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(1 + x\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(x + \color{blue}{1}\right)\right) \]

      3. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 57.4%

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{x}\right) \]

   10. Simplified 57.4%

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

   if -4500 < y < 7.20000000000000047e273

   1. Initial program 100.0%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(y + x \cdot y\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\left(x + 1\right) \cdot \color{blue}{y}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(x + 1\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(x + 1\right)}\right)\right) \]

      8. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{y}\right) \]
   6. Step-by-step derivation

   7. Simplified 91.1%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4500:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+273}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (+ x (* y (+ x 1.0))))

C

double code(double x, double y) {
	return x + (y * (x + 1.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + (y * (x + 1.0d0))
end function

Java

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

Python

def code(x, y):
	return x + (y * (x + 1.0))

Julia

function code(x, y)
	return Float64(x + Float64(y * Float64(x + 1.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = x + (y * (x + 1.0));
end

Wolfram

code[x_, y_] := N[(x + N[(y * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. associate-+l+ N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(y + x \cdot y\right)}\right) \]

   5. distribute-rgt1-in N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\left(x + 1\right) \cdot \color{blue}{y}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(x + 1\right)}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(x + 1\right)}\right)\right) \]

   8. +-lowering-+.f64 100.0%

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{x + y \cdot \left(x + 1\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 8: 50.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-103}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -7e-103) x y))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7e-103) {
		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 <= (-7d-103)) 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 <= -7e-103) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7e-103:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7e-103)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7e-103)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7e-103], x, y]

Derivation

1. Split input into 2 regimes

2. if x < -7.00000000000000032e-103

   1. Initial program 100.0%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(y + x \cdot y\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\left(x + 1\right) \cdot \color{blue}{y}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(x + 1\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(x + 1\right)}\right)\right) \]

      8. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x} \]
   6. Step-by-step derivation

   7. Simplified 45.8%

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

   if -7.00000000000000032e-103 < x

   1. Initial program 100.0%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(y + x \cdot y\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\left(x + 1\right) \cdot \color{blue}{y}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(x + 1\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(x + 1\right)}\right)\right) \]

      8. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y} \]
   6. Step-by-step derivation

   7. Simplified 47.1%

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

Alternative 9: 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. Step-by-step derivation

   1. associate-+l+ N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(y + x \cdot y\right)}\right) \]

   5. distribute-rgt1-in N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\left(x + 1\right) \cdot \color{blue}{y}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(x + 1\right)}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(x + 1\right)}\right)\right) \]

   8. +-lowering-+.f64 100.0%

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x} \]
6. Step-by-step derivation

7. Simplified 42.1%

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

Reproduce

herbie shell --seed 2024191 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))