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

Percentage Accurate: 100.0% → 100.0%

Time: 2.2s

Alternatives: 5

Speedup: N/A×

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

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

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. 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: 87.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -30500000000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-13}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -30500000000.0) (* y x) (if (<= y 8.5e-13) (+ y x) (+ y (* y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -30500000000.0) {
		tmp = y * x;
	} else if (y <= 8.5e-13) {
		tmp = y + x;
	} else {
		tmp = y + (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 (y <= (-30500000000.0d0)) then
        tmp = y * x
    else if (y <= 8.5d-13) then
        tmp = y + x
    else
        tmp = y + (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -30500000000.0) {
		tmp = y * x;
	} else if (y <= 8.5e-13) {
		tmp = y + x;
	} else {
		tmp = y + (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -30500000000.0:
		tmp = y * x
	elif y <= 8.5e-13:
		tmp = y + x
	else:
		tmp = y + (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -30500000000.0)
		tmp = Float64(y * x);
	elseif (y <= 8.5e-13)
		tmp = Float64(y + x);
	else
		tmp = Float64(y + Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -30500000000.0)
		tmp = y * x;
	elseif (y <= 8.5e-13)
		tmp = y + x;
	else
		tmp = y + (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -30500000000.0], N[(y * x), $MachinePrecision], If[LessEqual[y, 8.5e-13], N[(y + x), $MachinePrecision], N[(y + N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.05e10

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 99.7%

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

   3. Taylor expanded in x around inf 53.7%

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

      1. *-commutative 53.7%

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

   5. Simplified 53.7%

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

   if -3.05e10 < y < 8.5000000000000001e-13

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.9%

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

   if 8.5000000000000001e-13 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 99.7%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -30500000000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-13}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot x\\ \end{array} \]

Alternative 3: 62.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.1) (not (<= x 1.0))) (* y x) y))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.1) || !(x <= 1.0)) {
		tmp = y * 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 <= (-3.1d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = y * x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -3.1) || !(x <= 1.0)) {
		tmp = y * x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.1) or not (x <= 1.0):
		tmp = y * x
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.1) || !(x <= 1.0))
		tmp = Float64(y * x);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -3.1) || ~((x <= 1.0)))
		tmp = y * x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.1], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(y * x), $MachinePrecision], y]

Derivation

1. Split input into 2 regimes

2. if x < -3.10000000000000009 or 1 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 55.0%

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

   3. Taylor expanded in x around inf 54.8%

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

      1. *-commutative 54.8%

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

   5. Simplified 54.8%

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

   if -3.10000000000000009 < x < 1

   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. Taylor expanded in x around 0 74.2%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 4: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -720000 \lor \neg \left(y \leq 2.7 \cdot 10^{+249}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -720000.0) (not (<= y 2.7e+249))) (* y x) (+ y x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -720000.0) || !(y <= 2.7e+249)) {
		tmp = y * 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 ((y <= (-720000.0d0)) .or. (.not. (y <= 2.7d+249))) then
        tmp = y * x
    else
        tmp = y + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -720000.0) || !(y <= 2.7e+249)) {
		tmp = y * x;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -720000.0) or not (y <= 2.7e+249):
		tmp = y * x
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -720000.0) || !(y <= 2.7e+249))
		tmp = Float64(y * x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -720000.0) || ~((y <= 2.7e+249)))
		tmp = y * x;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -720000.0], N[Not[LessEqual[y, 2.7e+249]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.2e5 or 2.70000000000000018e249 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 99.7%

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

   3. Taylor expanded in x around inf 54.6%

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

      1. *-commutative 54.6%

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

   5. Simplified 54.6%

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

   if -7.2e5 < y < 2.70000000000000018e249

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 85.1%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -720000 \lor \neg \left(y \leq 2.7 \cdot 10^{+249}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 5: 38.5% 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. 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. Taylor expanded in x around 0 38.2%

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

5. Final simplification 38.2%

   \[\leadsto y \]

Reproduce

herbie shell --seed 2023310 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))