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

Percentage Accurate: 100.0% → 100.0%

Time: 2.7s

Alternatives: 6

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: 55.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-216}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+112}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+278}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   (* x y)
   (if (<= y 1.1e-216)
     x
     (if (<= y 5e+112) y (if (<= y 2.25e+278) (* x y) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x * y;
	} else if (y <= 1.1e-216) {
		tmp = x;
	} else if (y <= 5e+112) {
		tmp = y;
	} else if (y <= 2.25e+278) {
		tmp = x * y;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x * y
    else if (y <= 1.1d-216) then
        tmp = x
    else if (y <= 5d+112) then
        tmp = y
    else if (y <= 2.25d+278) then
        tmp = x * y
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x * y;
	} else if (y <= 1.1e-216) {
		tmp = x;
	} else if (y <= 5e+112) {
		tmp = y;
	} else if (y <= 2.25e+278) {
		tmp = x * y;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x * y
	elif y <= 1.1e-216:
		tmp = x
	elif y <= 5e+112:
		tmp = y
	elif y <= 2.25e+278:
		tmp = x * y
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(x * y);
	elseif (y <= 1.1e-216)
		tmp = x;
	elseif (y <= 5e+112)
		tmp = y;
	elseif (y <= 2.25e+278)
		tmp = Float64(x * y);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x * y;
	elseif (y <= 1.1e-216)
		tmp = x;
	elseif (y <= 5e+112)
		tmp = y;
	elseif (y <= 2.25e+278)
		tmp = x * y;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], N[(x * y), $MachinePrecision], If[LessEqual[y, 1.1e-216], x, If[LessEqual[y, 5e+112], y, If[LessEqual[y, 2.25e+278], N[(x * y), $MachinePrecision], y]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 5e112 < y < 2.25000000000000004e278

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 57.9%

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

      1. +-commutative 57.9%

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

   5. Simplified 57.9%

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

   6. Taylor expanded in y around inf 56.8%

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

   if -1 < y < 1.09999999999999995e-216

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 78.7%

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

   if 1.09999999999999995e-216 < y < 5e112 or 2.25000000000000004e278 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 53.4%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-216}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+112}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+278}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-156}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-178}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.4e-63)
   (* x (+ y 1.0))
   (if (<= x -8e-156) y (if (<= x -2.4e-178) x (if (<= x 1.0) y (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.4e-63) {
		tmp = x * (y + 1.0);
	} else if (x <= -8e-156) {
		tmp = y;
	} else if (x <= -2.4e-178) {
		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 <= (-1.4d-63)) then
        tmp = x * (y + 1.0d0)
    else if (x <= (-8d-156)) then
        tmp = y
    else if (x <= (-2.4d-178)) 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 <= -1.4e-63) {
		tmp = x * (y + 1.0);
	} else if (x <= -8e-156) {
		tmp = y;
	} else if (x <= -2.4e-178) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.4e-63:
		tmp = x * (y + 1.0)
	elif x <= -8e-156:
		tmp = y
	elif x <= -2.4e-178:
		tmp = x
	elif x <= 1.0:
		tmp = y
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.4e-63)
		tmp = Float64(x * Float64(y + 1.0));
	elseif (x <= -8e-156)
		tmp = y;
	elseif (x <= -2.4e-178)
		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 <= -1.4e-63)
		tmp = x * (y + 1.0);
	elseif (x <= -8e-156)
		tmp = y;
	elseif (x <= -2.4e-178)
		tmp = x;
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.4e-63], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8e-156], y, If[LessEqual[x, -2.4e-178], x, If[LessEqual[x, 1.0], y, N[(x * y), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.4000000000000001e-63

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 92.8%

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

      1. +-commutative 92.8%

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

   5. Simplified 92.8%

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

   if -1.4000000000000001e-63 < x < -8.00000000000000032e-156 or -2.40000000000000005e-178 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 81.6%

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

   if -8.00000000000000032e-156 < x < -2.40000000000000005e-178

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 67.3%

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

   if 1 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 99.2%

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

      1. +-commutative 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in y around inf 48.2%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-156}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-178}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-216}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.1e-216) (* x (+ y 1.0)) (* y (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.1e-216) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y * (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.1d-216) then
        tmp = x * (y + 1.0d0)
    else
        tmp = y * (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.1e-216) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y * (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.1e-216:
		tmp = x * (y + 1.0)
	else:
		tmp = y * (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.1e-216)
		tmp = Float64(x * Float64(y + 1.0));
	else
		tmp = Float64(y * Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.1e-216)
		tmp = x * (y + 1.0);
	else
		tmp = y * (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.1e-216], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.09999999999999995e-216

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 67.4%

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

      1. +-commutative 67.4%

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

   5. Simplified 67.4%

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

   if 1.09999999999999995e-216 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 78.2%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-216}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-216}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.1e-216) x y))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.1e-216) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.1d-216) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.1e-216) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.1e-216:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.1e-216)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.1e-216)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.1e-216], x, y]

Derivation

1. Split input into 2 regimes

2. if y < 1.09999999999999995e-216

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 44.0%

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

   if 1.09999999999999995e-216 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 48.0%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-216}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 37.4% 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 36.8%

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

4. Final simplification 36.8%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024054 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))