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

Percentage Accurate: 100.0% → 100.0%

Time: 3.0s

Alternatives: 7

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

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

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. Final simplification 100.0%

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

Alternative 2: 98.9% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.8) (not (<= x 1.0))) (* (+ y 1.0) x) (+ y x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.8) || !(x <= 1.0)) {
		tmp = (y + 1.0) * x;
	} else {
		tmp = y + 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 <= (-1.8d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (y + 1.0d0) * x
    else
        tmp = y + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.8) || !(x <= 1.0)) {
		tmp = (y + 1.0) * x;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.8) or not (x <= 1.0):
		tmp = (y + 1.0) * x
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.8) || !(x <= 1.0))
		tmp = Float64(Float64(y + 1.0) * x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.8) || ~((x <= 1.0)))
		tmp = (y + 1.0) * x;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.8], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(y + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.80000000000000004 or 1 < x

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. distribute-lft1-in 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 99.6%

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

      1. +-commutative 99.6%

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

   7. Simplified 99.6%

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

   if -1.80000000000000004 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.9%

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

4. Final simplification 99.7%

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

Alternative 3: 86.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{-15}:\\ \;\;\;\;\left(y + 1\right) \cdot x\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.7e-15)
   (* (+ y 1.0) x)
   (if (<= y 5.6e-11) (+ y x) (* y (+ 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.7e-15) {
		tmp = (y + 1.0) * x;
	} else if (y <= 5.6e-11) {
		tmp = y + 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 (y <= (-4.7d-15)) then
        tmp = (y + 1.0d0) * x
    else if (y <= 5.6d-11) then
        tmp = y + 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 (y <= -4.7e-15) {
		tmp = (y + 1.0) * x;
	} else if (y <= 5.6e-11) {
		tmp = y + x;
	} else {
		tmp = y * (1.0 + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.7e-15:
		tmp = (y + 1.0) * x
	elif y <= 5.6e-11:
		tmp = y + x
	else:
		tmp = y * (1.0 + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.7e-15)
		tmp = Float64(Float64(y + 1.0) * x);
	elseif (y <= 5.6e-11)
		tmp = Float64(y + x);
	else
		tmp = Float64(y * Float64(1.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.7e-15)
		tmp = (y + 1.0) * x;
	elseif (y <= 5.6e-11)
		tmp = y + x;
	else
		tmp = y * (1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.7e-15], N[(N[(y + 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 5.6e-11], N[(y + x), $MachinePrecision], N[(y * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.6999999999999999e-15

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft1-in 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 52.4%

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

      1. +-commutative 52.4%

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

   7. Simplified 52.4%

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

   if -4.6999999999999999e-15 < y < 5.6e-11

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.8%

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

   if 5.6e-11 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 98.2%

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

      1. +-commutative 98.2%

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

   5. Simplified 98.2%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{-15}:\\ \;\;\;\;\left(y + 1\right) \cdot x\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

Python

def code(x, y):
	return y + ((y + 1.0) * x)

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = y + ((y + 1.0) * x);
end

Wolfram

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. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative 100.0%

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

   2. distribute-lft1-in 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

   \[\leadsto y + \left(y + 1\right) \cdot x \]
6. Add Preprocessing

Alternative 5: 49.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-103}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -3.7e-103) x y))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.7e-103) {
		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 <= (-3.7d-103)) 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 <= -3.7e-103) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.7e-103:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.7e-103)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.7e-103)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.7e-103], x, y]

Derivation

1. Split input into 2 regimes

2. if x < -3.6999999999999999e-103

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. distribute-lft1-in 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 51.7%

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

   if -3.6999999999999999e-103 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 46.3%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-103}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ y + x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(y + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0 76.6%

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

4. Final simplification 76.6%

   \[\leadsto y + x \]
5. Add Preprocessing

Alternative 7: 39.3% 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. Step-by-step derivation

   1. *-commutative 100.0%

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

   2. distribute-lft1-in 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in y around 0 43.7%

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

6. Final simplification 43.7%

   \[\leadsto x \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024095 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))