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

Percentage Accurate: 100.0% → 100.0%

Time: 1.6s

Alternatives: 5

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

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return y + ((y + 1.0) * 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 = y + ((y + 1.0d0) * x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   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%

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

Alternative 2: 79.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -4400000000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+175}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+210}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+225}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+247}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -4400000000.0)
   (* y x)
   (if (<= y 1.05e+175)
     (+ y x)
     (if (<= y 1.3e+210)
       (* y x)
       (if (<= y 2e+225) y (if (<= y 6.5e+247) (* y x) (+ y x)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -4400000000.0) {
		tmp = y * x;
	} else if (y <= 1.05e+175) {
		tmp = y + x;
	} else if (y <= 1.3e+210) {
		tmp = y * x;
	} else if (y <= 2e+225) {
		tmp = y;
	} else if (y <= 6.5e+247) {
		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 <= (-4400000000.0d0)) then
        tmp = y * x
    else if (y <= 1.05d+175) then
        tmp = y + x
    else if (y <= 1.3d+210) then
        tmp = y * x
    else if (y <= 2d+225) then
        tmp = y
    else if (y <= 6.5d+247) 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 <= -4400000000.0) {
		tmp = y * x;
	} else if (y <= 1.05e+175) {
		tmp = y + x;
	} else if (y <= 1.3e+210) {
		tmp = y * x;
	} else if (y <= 2e+225) {
		tmp = y;
	} else if (y <= 6.5e+247) {
		tmp = y * x;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -4400000000.0:
		tmp = y * x
	elif y <= 1.05e+175:
		tmp = y + x
	elif y <= 1.3e+210:
		tmp = y * x
	elif y <= 2e+225:
		tmp = y
	elif y <= 6.5e+247:
		tmp = y * x
	else:
		tmp = y + x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -4400000000.0)
		tmp = Float64(y * x);
	elseif (y <= 1.05e+175)
		tmp = Float64(y + x);
	elseif (y <= 1.3e+210)
		tmp = Float64(y * x);
	elseif (y <= 2e+225)
		tmp = y;
	elseif (y <= 6.5e+247)
		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 <= -4400000000.0)
		tmp = y * x;
	elseif (y <= 1.05e+175)
		tmp = y + x;
	elseif (y <= 1.3e+210)
		tmp = y * x;
	elseif (y <= 2e+225)
		tmp = y;
	elseif (y <= 6.5e+247)
		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, -4400000000.0], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.05e+175], N[(y + x), $MachinePrecision], If[LessEqual[y, 1.3e+210], N[(y * x), $MachinePrecision], If[LessEqual[y, 2e+225], y, If[LessEqual[y, 6.5e+247], N[(y * x), $MachinePrecision], N[(y + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.4e9 or 1.05e175 < y < 1.29999999999999995e210 or 1.99999999999999986e225 < y < 6.50000000000000023e247

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 99.5%

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

   3. Taylor expanded in x around inf 52.5%

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

   if -4.4e9 < y < 1.05e175 or 6.50000000000000023e247 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 90.1%

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

   if 1.29999999999999995e210 < y < 1.99999999999999986e225

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4400000000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+175}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+210}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+225}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+247}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 3: 97.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -4400000000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-32}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -4400000000.0) (* y x) (if (<= y 4.6e-32) (+ y x) (+ y (* y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -4400000000.0) {
		tmp = y * x;
	} else if (y <= 4.6e-32) {
		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 <= (-4400000000.0d0)) then
        tmp = y * x
    else if (y <= 4.6d-32) 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 <= -4400000000.0) {
		tmp = y * x;
	} else if (y <= 4.6e-32) {
		tmp = y + x;
	} else {
		tmp = y + (y * x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -4400000000.0:
		tmp = y * x
	elif y <= 4.6e-32:
		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 <= -4400000000.0)
		tmp = Float64(y * x);
	elseif (y <= 4.6e-32)
		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 <= -4400000000.0)
		tmp = y * x;
	elseif (y <= 4.6e-32)
		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, -4400000000.0], N[(y * x), $MachinePrecision], If[LessEqual[y, 4.6e-32], N[(y + x), $MachinePrecision], N[(y + N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.4e9

   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 45.2%

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

   if -4.4e9 < y < 4.6000000000000001e-32

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

   if 4.6000000000000001e-32 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 93.5%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4400000000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-32}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot x\\ \end{array} \]

Alternative 4: 61.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -95000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -95000.0) (* y x) (if (<= x 1.0) y (* y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -95000.0) {
		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 <= (-95000.0d0)) 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 <= -95000.0) {
		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 <= -95000.0:
		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 <= -95000.0)
		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 <= -95000.0)
		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, -95000.0], N[(y * x), $MachinePrecision], If[LessEqual[x, 1.0], y, N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -95000 or 1 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 47.5%

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

   3. Taylor expanded in x around inf 47.5%

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

   if -95000 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 70.7%

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

   3. Taylor expanded in x around 0 70.6%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -95000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5: 37.9% 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 40.9%

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

4. Final simplification 40.9%

   \[\leadsto y \]

Reproduce

herbie shell --seed 2023199 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))