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

Percentage Accurate: 100.0% → 100.0%

Time: 2.1s

Alternatives: 6

Speedup: 0.1×

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. fma-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 80.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -330000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+157} \lor \neg \left(y \leq 1.06 \cdot 10^{+251}\right) \land y \leq 3.5 \cdot 10^{+304}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -330000.0)
   (* y x)
   (if (or (<= y 1.95e+157) (and (not (<= y 1.06e+251)) (<= y 3.5e+304)))
     (+ y x)
     (* y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -330000.0) {
		tmp = y * x;
	} else if ((y <= 1.95e+157) || (!(y <= 1.06e+251) && (y <= 3.5e+304))) {
		tmp = y + x;
	} else {
		tmp = y * x;
	}
	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 <= (-330000.0d0)) then
        tmp = y * x
    else if ((y <= 1.95d+157) .or. (.not. (y <= 1.06d+251)) .and. (y <= 3.5d+304)) then
        tmp = y + x
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -330000.0) {
		tmp = y * x;
	} else if ((y <= 1.95e+157) || (!(y <= 1.06e+251) && (y <= 3.5e+304))) {
		tmp = y + x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -330000.0:
		tmp = y * x
	elif (y <= 1.95e+157) or (not (y <= 1.06e+251) and (y <= 3.5e+304)):
		tmp = y + x
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -330000.0)
		tmp = Float64(y * x);
	elseif ((y <= 1.95e+157) || (!(y <= 1.06e+251) && (y <= 3.5e+304)))
		tmp = Float64(y + x);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -330000.0)
		tmp = y * x;
	elseif ((y <= 1.95e+157) || (~((y <= 1.06e+251)) && (y <= 3.5e+304)))
		tmp = y + x;
	else
		tmp = y * x;
	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, -330000.0], N[(y * x), $MachinePrecision], If[Or[LessEqual[y, 1.95e+157], And[N[Not[LessEqual[y, 1.06e+251]], $MachinePrecision], LessEqual[y, 3.5e+304]]], N[(y + x), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.3e5 or 1.94999999999999985e157 < y < 1.05999999999999998e251 or 3.4999999999999998e304 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 99.4%

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

   3. Taylor expanded in x around inf 57.8%

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

   if -3.3e5 < y < 1.94999999999999985e157 or 1.05999999999999998e251 < y < 3.4999999999999998e304

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 88.2%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -330000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+157} \lor \neg \left(y \leq 1.06 \cdot 10^{+251}\right) \land y \leq 3.5 \cdot 10^{+304}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3: 98.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -490000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -490000.0) (* y x) (if (<= y 1.0) (+ y x) (+ y (* y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -490000.0) {
		tmp = y * x;
	} else if (y <= 1.0) {
		tmp = y + x;
	} else {
		tmp = y + (y * x);
	}
	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 <= (-490000.0d0)) then
        tmp = y * x
    else if (y <= 1.0d0) then
        tmp = y + x
    else
        tmp = y + (y * x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -490000.0) {
		tmp = y * x;
	} else if (y <= 1.0) {
		tmp = y + x;
	} else {
		tmp = y + (y * x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -490000.0:
		tmp = y * x
	elif y <= 1.0:
		tmp = y + x
	else:
		tmp = y + (y * x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -490000.0)
		tmp = Float64(y * x);
	elseif (y <= 1.0)
		tmp = Float64(y + x);
	else
		tmp = Float64(y + Float64(y * x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -490000.0)
		tmp = y * x;
	elseif (y <= 1.0)
		tmp = y + x;
	else
		tmp = y + (y * x);
	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, -490000.0], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.0], N[(y + x), $MachinePrecision], N[(y + N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.9e5

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 99.1%

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

   3. Taylor expanded in x around inf 47.7%

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

   if -4.9e5 < y < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.3%

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

   if 1 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 99.0%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -490000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot x\\ \end{array} \]

Alternative 4: 61.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-10}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.5e-10) (* y x) (if (<= x 1.0) y (* y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.5e-10) {
		tmp = y * x;
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = y * x;
	}
	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 <= (-3.5d-10)) then
        tmp = y * x
    else if (x <= 1.0d0) then
        tmp = y
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -3.5e-10) {
		tmp = y * x;
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.5e-10:
		tmp = y * x
	elif x <= 1.0:
		tmp = y
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.5e-10)
		tmp = Float64(y * x);
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.5e-10)
		tmp = y * x;
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = y * x;
	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, -3.5e-10], N[(y * x), $MachinePrecision], If[LessEqual[x, 1.0], y, N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.4999999999999998e-10 or 1 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 50.1%

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

   3. Taylor expanded in x around inf 48.5%

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

   if -3.4999999999999998e-10 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 73.5%

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

   3. Taylor expanded in x around 0 72.8%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-10}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * (y + 1.0d0))
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return y + (x * (y + 1.0))

Julia

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

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = y + (x * (y + 1.0));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. distribute-lft1-in 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto y + x \cdot \left(y + 1\right) \]

Alternative 6: 38.4% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in y around inf 60.5%

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

3. Taylor expanded in x around 0 34.2%

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

4. Final simplification 34.2%

   \[\leadsto y \]

Reproduce

herbie shell --seed 2023268 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))