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

Percentage Accurate: 100.0% → 100.0%

Time: 6.6s

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: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6500:\\ \;\;\;\;x + x \cdot y\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-35}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6500.0) (+ x (* x y)) (if (<= x 3.7e-35) (+ x y) (+ y (* x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6500.0) {
		tmp = x + (x * y);
	} else if (x <= 3.7e-35) {
		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 <= (-6500.0d0)) then
        tmp = x + (x * y)
    else if (x <= 3.7d-35) 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 <= -6500.0) {
		tmp = x + (x * y);
	} else if (x <= 3.7e-35) {
		tmp = x + y;
	} else {
		tmp = y + (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6500.0:
		tmp = x + (x * y)
	elif x <= 3.7e-35:
		tmp = x + y
	else:
		tmp = y + (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6500.0)
		tmp = Float64(x + Float64(x * y));
	elseif (x <= 3.7e-35)
		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 <= -6500.0)
		tmp = x + (x * y);
	elseif (x <= 3.7e-35)
		tmp = x + y;
	else
		tmp = y + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6500.0], N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.7e-35], N[(x + y), $MachinePrecision], N[(y + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6500

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 99.9%

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

   5. Simplified 99.9%

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

      1. *-commutative N/A

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

      2. distribute-lft1-in N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 100.0%

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

   7. Applied egg-rr 100.0%

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

   if -6500 < x < 3.6999999999999999e-35

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 99.8%

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

   if 3.6999999999999999e-35 < x

   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 42.3%

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

   5. Simplified 42.3%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6500:\\ \;\;\;\;x + x \cdot y\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-35}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6500:\\ \;\;\;\;x + x \cdot y\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6500.0) (+ x (* x y)) (if (<= x 1.0) (+ x y) (* x (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6500.0) {
		tmp = x + (x * y);
	} else if (x <= 1.0) {
		tmp = x + y;
	} else {
		tmp = x * (y + 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 (x <= (-6500.0d0)) then
        tmp = x + (x * y)
    else if (x <= 1.0d0) then
        tmp = x + y
    else
        tmp = x * (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6500.0) {
		tmp = x + (x * y);
	} else if (x <= 1.0) {
		tmp = x + y;
	} else {
		tmp = x * (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6500.0:
		tmp = x + (x * y)
	elif x <= 1.0:
		tmp = x + y
	else:
		tmp = x * (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6500.0)
		tmp = Float64(x + Float64(x * y));
	elseif (x <= 1.0)
		tmp = Float64(x + y);
	else
		tmp = Float64(x * Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6500.0)
		tmp = x + (x * y);
	elseif (x <= 1.0)
		tmp = x + y;
	else
		tmp = x * (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6500.0], N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(x + y), $MachinePrecision], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6500

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 99.9%

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

   5. Simplified 99.9%

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

      1. *-commutative N/A

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

      2. distribute-lft1-in N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 100.0%

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

   7. Applied egg-rr 100.0%

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

   if -6500 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 99.3%

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

   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

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 99.3%

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

   5. Simplified 99.3%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6500:\\ \;\;\;\;x + x \cdot y\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y + 1\right)\\ \mathbf{if}\;x \leq -6500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (+ y 1.0))))
   (if (<= x -6500.0) t_0 (if (<= x 1.0) (+ x y) t_0))))

C

double code(double x, double y) {
	double t_0 = x * (y + 1.0);
	double tmp;
	if (x <= -6500.0) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = x + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (y + 1.0d0)
    if (x <= (-6500.0d0)) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = x + y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (y + 1.0);
	double tmp;
	if (x <= -6500.0) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = x + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (y + 1.0)
	tmp = 0
	if x <= -6500.0:
		tmp = t_0
	elif x <= 1.0:
		tmp = x + y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(y + 1.0))
	tmp = 0.0
	if (x <= -6500.0)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = Float64(x + y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (y + 1.0);
	tmp = 0.0;
	if (x <= -6500.0)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = x + y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6500.0], t$95$0, If[LessEqual[x, 1.0], N[(x + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6500 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

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 99.6%

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

   5. Simplified 99.6%

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

   if -6500 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 99.3%

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

Alternative 5: 61.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-105}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.8e-105) x (if (<= x 1.0) y (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.8e-105) {
		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 <= (-6.8d-105)) 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 <= -6.8e-105) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.8e-105:
		tmp = x
	elif x <= 1.0:
		tmp = y
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.8e-105)
		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 <= -6.8e-105)
		tmp = x;
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.8e-105], x, If[LessEqual[x, 1.0], y, N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.79999999999999984e-105

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 61.7%

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

   if -6.79999999999999984e-105 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 89.0%

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

   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

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 40.4%

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

   8. Simplified 40.4%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-105}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 10000000.0) (+ x y) (* x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 10000000.0) {
		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 <= 10000000.0d0) 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 <= 10000000.0) {
		tmp = x + y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 10000000.0:
		tmp = x + y
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 10000000.0)
		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 <= 10000000.0)
		tmp = x + y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 10000000.0], N[(x + y), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e7

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 84.4%

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

   if 1e7 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 40.4%

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

   8. Simplified 40.4%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-105}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -6.8e-105) x y))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.8e-105) {
		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 <= (-6.8d-105)) 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 <= -6.8e-105) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.8e-105:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.8e-105)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.8e-105)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.8e-105], x, y]

Derivation

1. Split input into 2 regimes

2. if x < -6.79999999999999984e-105

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 61.7%

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

   if -6.79999999999999984e-105 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 54.4%

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

Alternative 8: 38.4% 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

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

5. Simplified 43.7%

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

Reproduce

herbie shell --seed 2024170 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))