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

Percentage Accurate: 100.0% → 100.0%

Time: 2.6s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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: 87.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -16.5:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\left(x + 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -16.5) (* x y) (if (<= y 1.0) (+ x y) (* (+ x 1.0) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -16.5) {
		tmp = x * y;
	} else if (y <= 1.0) {
		tmp = x + y;
	} else {
		tmp = (x + 1.0) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-16.5d0)) then
        tmp = x * y
    else if (y <= 1.0d0) then
        tmp = x + y
    else
        tmp = (x + 1.0d0) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -16.5) {
		tmp = x * y;
	} else if (y <= 1.0) {
		tmp = x + y;
	} else {
		tmp = (x + 1.0) * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -16.5:
		tmp = x * y
	elif y <= 1.0:
		tmp = x + y
	else:
		tmp = (x + 1.0) * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -16.5)
		tmp = Float64(x * y);
	elseif (y <= 1.0)
		tmp = Float64(x + y);
	else
		tmp = Float64(Float64(x + 1.0) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -16.5)
		tmp = x * y;
	elseif (y <= 1.0)
		tmp = x + y;
	else
		tmp = (x + 1.0) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -16.5], N[(x * y), $MachinePrecision], If[LessEqual[y, 1.0], N[(x + y), $MachinePrecision], N[(N[(x + 1.0), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -16.5

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 98.1%

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

      1. +-commutative 98.1%

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

   4. Simplified 98.1%

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

   5. Taylor expanded in x around inf 55.1%

      \[\leadsto \color{blue}{x \cdot y} \]
   6. Step-by-step derivation

      1. *-commutative 55.1%

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

   7. Simplified 55.1%

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

   if -16.5 < y < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 97.9%

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

   if 1 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -16.5:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\left(x + 1\right) \cdot y\\ \end{array} \]

Alternative 3: 60.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.5) (not (<= x 1.0))) (* x y) y))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.5) || !(x <= 1.0)) {
		tmp = x * y;
	} 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 <= (-3.5d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = x * y
    else
        tmp = y
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if (x <= -3.5) or not (x <= 1.0):
		tmp = x * y
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.5) || !(x <= 1.0))
		tmp = Float64(x * y);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -3.5) || ~((x <= 1.0)))
		tmp = x * y;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.5], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(x * y), $MachinePrecision], y]

Derivation

1. Split input into 2 regimes

2. if x < -3.5 or 1 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 53.6%

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

      1. +-commutative 53.6%

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

   4. Simplified 53.6%

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

   5. Taylor expanded in x around inf 52.6%

      \[\leadsto \color{blue}{x \cdot y} \]
   6. Step-by-step derivation

      1. *-commutative 52.6%

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

   7. Simplified 52.6%

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

   if -3.5 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 75.3%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 4: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+222} \lor \neg \left(x \leq 650000000000\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.9e+222) (not (<= x 650000000000.0))) (* x y) (+ x y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.9e+222) || !(x <= 650000000000.0)) {
		tmp = x * y;
	} else {
		tmp = x + 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 <= (-2.9d+222)) .or. (.not. (x <= 650000000000.0d0))) then
        tmp = x * y
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2.9e+222) || !(x <= 650000000000.0)) {
		tmp = x * y;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.9e+222) or not (x <= 650000000000.0):
		tmp = x * y
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.9e+222) || !(x <= 650000000000.0))
		tmp = Float64(x * y);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.9e+222) || ~((x <= 650000000000.0)))
		tmp = x * y;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.9e+222], N[Not[LessEqual[x, 650000000000.0]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.89999999999999981e222 or 6.5e11 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 56.1%

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

      1. +-commutative 56.1%

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

   4. Simplified 56.1%

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

   5. Taylor expanded in x around inf 56.1%

      \[\leadsto \color{blue}{x \cdot y} \]
   6. Step-by-step derivation

      1. *-commutative 56.1%

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

   7. Simplified 56.1%

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

   if -2.89999999999999981e222 < x < 6.5e11

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 88.4%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+222} \lor \neg \left(x \leq 650000000000\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 5: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y + (x * (1.0d0 + y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(y + N[(x * N[(1.0 + y), $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%

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

Alternative 6: 37.4% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 y)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

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

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0 40.1%

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

3. Final simplification 40.1%

   \[\leadsto y \]

Reproduce

herbie shell --seed 2023311 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))