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

Percentage Accurate: 100.0% → 100.0%

Time: 2.9s

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, 0.1× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (fma (+ y 1.0) x y))

C

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

Julia

function code(x, y)
	return fma(Float64(y + 1.0), x, y)
end

Wolfram

code[x_, y_] := N[(N[(y + 1.0), $MachinePrecision] * x + y), $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. fma-define 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y + 1, x, y\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 85.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-95} \lor \neg \left(x \leq 8 \cdot 10^{-8}\right):\\ \;\;\;\;\left(y + 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -8e-95) (not (<= x 8e-8))) (* (+ y 1.0) x) y))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -8e-95) || !(x <= 8e-8)) {
		tmp = (y + 1.0) * 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 ((x <= (-8d-95)) .or. (.not. (x <= 8d-8))) then
        tmp = (y + 1.0d0) * x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -8e-95) || !(x <= 8e-8)) {
		tmp = (y + 1.0) * x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -8e-95) or not (x <= 8e-8):
		tmp = (y + 1.0) * x
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -8e-95) || !(x <= 8e-8))
		tmp = Float64(Float64(y + 1.0) * x);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -8e-95) || ~((x <= 8e-8)))
		tmp = (y + 1.0) * x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -8e-95], N[Not[LessEqual[x, 8e-8]], $MachinePrecision]], N[(N[(y + 1.0), $MachinePrecision] * x), $MachinePrecision], y]

Derivation

1. Split input into 2 regimes

2. if x < -7.99999999999999992e-95 or 8.0000000000000002e-8 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 90.0%

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

      1. +-commutative 90.0%

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

   5. Simplified 90.0%

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

   if -7.99999999999999992e-95 < x < 8.0000000000000002e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 83.0%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-95} \lor \neg \left(x \leq 8 \cdot 10^{-8}\right):\\ \;\;\;\;\left(y + 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-95}:\\ \;\;\;\;\left(y + 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -8e-95) (* (+ y 1.0) x) (* y (+ 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8e-95) {
		tmp = (y + 1.0) * x;
	} else {
		tmp = y * (1.0 + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-8d-95)) then
        tmp = (y + 1.0d0) * x
    else
        tmp = y * (1.0d0 + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -8e-95) {
		tmp = (y + 1.0) * x;
	} else {
		tmp = y * (1.0 + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -8e-95:
		tmp = (y + 1.0) * x
	else:
		tmp = y * (1.0 + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8e-95)
		tmp = Float64(Float64(y + 1.0) * x);
	else
		tmp = Float64(y * Float64(1.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8e-95)
		tmp = (y + 1.0) * x;
	else
		tmp = y * (1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8e-95], N[(N[(y + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(y * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.99999999999999992e-95

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 82.5%

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

      1. +-commutative 82.5%

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

   5. Simplified 82.5%

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

   if -7.99999999999999992e-95 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 71.3%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-95}:\\ \;\;\;\;\left(y + 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (+ x (+ y (* y x))))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + (y + (y * x))
end function

Java

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

Python

def code(x, y):
	return x + (y + (y * x))

Julia

function code(x, y)
	return Float64(x + Float64(y + Float64(y * x)))
end

MATLAB

function tmp = code(x, y)
	tmp = x + (y + (y * x));
end

Wolfram

code[x_, y_] := N[(x + N[(y + N[(y * x), $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. fma-define 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y + 1, x, y\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 100.0%

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

   2. distribute-lft1-in 100.0%

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

   3. *-commutative 100.0%

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

   4. +-commutative 100.0%

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

   5. associate-+r+ 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 5: 50.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-95}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -8e-95) x y))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8e-95) {
		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 (x <= (-8d-95)) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -8e-95) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -8e-95:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8e-95)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8e-95)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8e-95], x, y]

Derivation

1. Split input into 2 regimes

2. if x < -7.99999999999999992e-95

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 51.6%

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

   if -7.99999999999999992e-95 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 53.3%

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

Alternative 6: 38.2% 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 37.3%

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

Reproduce

herbie shell --seed 2024158 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))