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

Percentage Accurate: 100.0% → 100.0%

Time: 2.5s

Alternatives: 7

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, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \mathsf{fma}\left(x + 1, y, x\right) \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (fma (+ x 1.0) y x))

C

assert(x < y);
double code(double x, double y) {
	return fma((x + 1.0), y, x);
}

Julia

x, y = sort([x, y])
function code(x, y)
	return fma(Float64(x + 1.0), y, x)
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(x + 1.0), $MachinePrecision] * y + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. associate-+l+ 100.0%

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

   2. +-commutative 100.0%

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

   3. associate-+r+ 100.0%

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

   4. distribute-lft1-in 100.0%

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

   5. fma-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(x + 1, y, x\right) \]

Alternative 2: 68.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+240}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{+223}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{+200}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-168}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5.4e+240)
   (* x y)
   (if (<= x -2.2e+223)
     x
     (if (<= x -2.3e+200)
       (* x y)
       (if (<= x -7e-168) x (if (<= x 1.0) y (* x y)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5.4e+240) {
		tmp = x * y;
	} else if (x <= -2.2e+223) {
		tmp = x;
	} else if (x <= -2.3e+200) {
		tmp = x * y;
	} else if (x <= -7e-168) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.4d+240)) then
        tmp = x * y
    else if (x <= (-2.2d+223)) then
        tmp = x
    else if (x <= (-2.3d+200)) then
        tmp = x * y
    else if (x <= (-7d-168)) then
        tmp = x
    else if (x <= 1.0d0) then
        tmp = y
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -5.4e+240) {
		tmp = x * y;
	} else if (x <= -2.2e+223) {
		tmp = x;
	} else if (x <= -2.3e+200) {
		tmp = x * y;
	} else if (x <= -7e-168) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -5.4e+240:
		tmp = x * y
	elif x <= -2.2e+223:
		tmp = x
	elif x <= -2.3e+200:
		tmp = x * y
	elif x <= -7e-168:
		tmp = x
	elif x <= 1.0:
		tmp = y
	else:
		tmp = x * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5.4e+240)
		tmp = Float64(x * y);
	elseif (x <= -2.2e+223)
		tmp = x;
	elseif (x <= -2.3e+200)
		tmp = Float64(x * y);
	elseif (x <= -7e-168)
		tmp = x;
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.4e+240)
		tmp = x * y;
	elseif (x <= -2.2e+223)
		tmp = x;
	elseif (x <= -2.3e+200)
		tmp = x * y;
	elseif (x <= -7e-168)
		tmp = x;
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5.4e+240], N[(x * y), $MachinePrecision], If[LessEqual[x, -2.2e+223], x, If[LessEqual[x, -2.3e+200], N[(x * y), $MachinePrecision], If[LessEqual[x, -7e-168], x, If[LessEqual[x, 1.0], y, N[(x * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.3999999999999997e240 or -2.2e223 < x < -2.30000000000000003e200 or 1 < x

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]

   2. Taylor expanded in x around inf 100.0%

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

      1. +-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y around inf 58.2%

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

   if -5.3999999999999997e240 < x < -2.2e223 or -2.30000000000000003e200 < x < -6.99999999999999964e-168

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]

   2. Taylor expanded in y around 0 42.9%

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

   if -6.99999999999999964e-168 < x < 1

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]

   2. Taylor expanded in x around 0 84.5%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+240}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{+223}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{+200}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-168}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3: 87.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-168} \lor \neg \left(x \leq 0.026\right):\\ \;\;\;\;x \cdot \left(1 + y\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (or (<= x -7e-168) (not (<= x 0.026))) (* x (+ 1.0 y)) y))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if ((x <= -7e-168) || !(x <= 0.026)) {
		tmp = x * (1.0 + y);
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-7d-168)) .or. (.not. (x <= 0.026d0))) then
        tmp = x * (1.0d0 + y)
    else
        tmp = y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if ((x <= -7e-168) || !(x <= 0.026)) {
		tmp = x * (1.0 + y);
	} else {
		tmp = y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if (x <= -7e-168) or not (x <= 0.026):
		tmp = x * (1.0 + y)
	else:
		tmp = y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if ((x <= -7e-168) || !(x <= 0.026))
		tmp = Float64(x * Float64(1.0 + y));
	else
		tmp = y;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -7e-168) || ~((x <= 0.026)))
		tmp = x * (1.0 + y);
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[Or[LessEqual[x, -7e-168], N[Not[LessEqual[x, 0.026]], $MachinePrecision]], N[(x * N[(1.0 + y), $MachinePrecision]), $MachinePrecision], y]

Derivation

1. Split input into 2 regimes

2. if x < -6.99999999999999964e-168 or 0.0259999999999999988 < x

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]

   2. Taylor expanded in x around inf 86.1%

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

      1. +-commutative 86.1%

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

   4. Simplified 86.1%

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

   if -6.99999999999999964e-168 < x < 0.0259999999999999988

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]

   2. Taylor expanded in x around 0 84.5%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-168} \lor \neg \left(x \leq 0.026\right):\\ \;\;\;\;x \cdot \left(1 + y\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 4: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{-120}:\\ \;\;\;\;x \cdot \left(1 + y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.25e-120) (* x (+ 1.0 y)) (* (+ x 1.0) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.25e-120) {
		tmp = x * (1.0 + y);
	} else {
		tmp = (x + 1.0) * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.25d-120) then
        tmp = x * (1.0d0 + y)
    else
        tmp = (x + 1.0d0) * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.25e-120) {
		tmp = x * (1.0 + y);
	} else {
		tmp = (x + 1.0) * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.25e-120:
		tmp = x * (1.0 + y)
	else:
		tmp = (x + 1.0) * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.25e-120)
		tmp = Float64(x * Float64(1.0 + y));
	else
		tmp = Float64(Float64(x + 1.0) * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.25e-120)
		tmp = x * (1.0 + y);
	else
		tmp = (x + 1.0) * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2.25e-120], N[(x * N[(1.0 + y), $MachinePrecision]), $MachinePrecision], N[(N[(x + 1.0), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.25e-120

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]

   2. Taylor expanded in x around inf 66.3%

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

      1. +-commutative 66.3%

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

   4. Simplified 66.3%

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

   if 2.25e-120 < y

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]

   2. Taylor expanded in y around inf 84.4%

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

      1. +-commutative 84.4%

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

   4. Simplified 84.4%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{-120}:\\ \;\;\;\;x \cdot \left(1 + y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + 1\right) \cdot y\\ \end{array} \]

Alternative 5: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ y + \left(x + x \cdot y\right) \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (+ y (+ x (* x y))))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y + (x + (x * y))
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return y + (x + (x * y))

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(y + Float64(x + Float64(x * y)))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = y + (x + (x * y));
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
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. Final simplification 100.0%

   \[\leadsto y + \left(x + x \cdot y\right) \]

Alternative 6: 63.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{-120}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 2.25e-120) x y))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.25e-120) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.25d-120) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.25e-120) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.25e-120:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.25e-120)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.25e-120)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2.25e-120], x, y]

Derivation

1. Split input into 2 regimes

2. if y < 2.25e-120

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]

   2. Taylor expanded in y around 0 43.6%

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

   if 2.25e-120 < y

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]

   2. Taylor expanded in x around 0 44.9%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{-120}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 7: 38.2% accurate, 7.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ x \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 x)

C

assert(x < y);
double code(double x, double y) {
	return x;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x
end function

Java

assert x < y;
public static double code(double x, double y) {
	return x;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return x

Julia

x, y = sort([x, y])
function code(x, y)
	return x
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := x

Derivation

1. Initial program 100.0%

   \[\left(x \cdot y + x\right) + y \]

2. Taylor expanded in y around 0 33.7%

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

3. Final simplification 33.7%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023320 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))