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

Percentage Accurate: 100.0% → 100.0%

Time: 2.3s

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: 73.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-70}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.8e-70) (* x (+ y 1.0)) (if (<= x 1.0) y (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.8e-70) {
		tmp = x * (y + 1.0);
	} 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 <= (-1.8d-70)) then
        tmp = x * (y + 1.0d0)
    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 <= -1.8e-70) {
		tmp = x * (y + 1.0);
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.8e-70:
		tmp = x * (y + 1.0)
	elif x <= 1.0:
		tmp = y
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.8e-70)
		tmp = Float64(x * Float64(y + 1.0));
	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 <= -1.8e-70)
		tmp = x * (y + 1.0);
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.8e-70], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], y, N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.8000000000000001e-70

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 95.1%

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

      1. +-commutative 95.1%

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

   5. Simplified 95.1%

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

   if -1.8000000000000001e-70 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 79.4%

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

   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 100.0%

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

      1. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf 33.5%

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

      1. *-commutative 33.5%

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

   8. Simplified 33.5%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-70}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-55}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) (* x y) (if (<= y 2.45e-55) x y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x * y;
	} else if (y <= 2.45e-55) {
		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 <= (-1.0d0)) then
        tmp = x * y
    else if (y <= 2.45d-55) 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 <= -1.0) {
		tmp = x * y;
	} else if (y <= 2.45e-55) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x * y
	elif y <= 2.45e-55:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(x * y);
	elseif (y <= 2.45e-55)
		tmp = x;
	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 <= 2.45e-55)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], N[(x * y), $MachinePrecision], If[LessEqual[y, 2.45e-55], x, y]]

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 42.1%

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

      1. +-commutative 42.1%

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

   5. Simplified 42.1%

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

   6. Taylor expanded in y around inf 42.1%

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

      1. *-commutative 42.1%

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

   8. Simplified 42.1%

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

   if -1 < y < 2.45000000000000018e-55

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 73.3%

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

   if 2.45000000000000018e-55 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 58.7%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-55}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-57}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3e-57) (* x (+ y 1.0)) (+ y (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3e-57) {
		tmp = x * (y + 1.0);
	} 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 <= 3d-57) then
        tmp = x * (y + 1.0d0)
    else
        tmp = y + (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3e-57) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y + (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3e-57:
		tmp = x * (y + 1.0)
	else:
		tmp = y + (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3e-57)
		tmp = Float64(x * Float64(y + 1.0));
	else
		tmp = Float64(y + Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3e-57)
		tmp = x * (y + 1.0);
	else
		tmp = y + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3e-57], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(y + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.00000000000000001e-57

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 64.1%

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

      1. +-commutative 64.1%

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

   5. Simplified 64.1%

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

   if 3.00000000000000001e-57 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 86.8%

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

      1. *-commutative 86.8%

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

   5. Simplified 86.8%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-57}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-55}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.2e-55) x y))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.2e-55) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.2d-55) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.2e-55) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.2e-55:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.2e-55)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.2e-55)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.2e-55], x, y]

Derivation

1. Split input into 2 regimes

2. if y < 2.2e-55

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 49.8%

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

   if 2.2e-55 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 58.7%

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

Alternative 6: 38.3% 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 40.5%

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

Reproduce

herbie shell --seed 2024180 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))