Numeric.Log:$clog1p from log-domain-0.10.2.1, A

Percentage Accurate: 99.9% → 99.9%

Time: 4.1s

Alternatives: 5

Speedup: 1.2×

• 42.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot 2 + x \cdot x\right) + y \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (+ (* x 2.0) (* x x)) (* y y)))

C

double code(double x, double y) {
	return ((x * 2.0) + (x * x)) + (y * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * 2.0d0) + (x * x)) + (y * y)
end function

Java

public static double code(double x, double y) {
	return ((x * 2.0) + (x * x)) + (y * y);
}

Python

def code(x, y):
	return ((x * 2.0) + (x * x)) + (y * y)

Julia

function code(x, y)
	return Float64(Float64(Float64(x * 2.0) + Float64(x * x)) + Float64(y * y))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * 2.0) + (x * x)) + (y * y);
end

Wolfram

code[x_, y_] := N[(N[(N[(x * 2.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot 2 + x \cdot x\right) + y \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (+ (* x 2.0) (* x x)) (* y y)))

C

double code(double x, double y) {
	return ((x * 2.0) + (x * x)) + (y * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * 2.0d0) + (x * x)) + (y * y)
end function

Java

public static double code(double x, double y) {
	return ((x * 2.0) + (x * x)) + (y * y);
}

Python

def code(x, y):
	return ((x * 2.0) + (x * x)) + (y * y)

Julia

function code(x, y)
	return Float64(Float64(Float64(x * 2.0) + Float64(x * x)) + Float64(y * y))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * 2.0) + (x * x)) + (y * y);
end

Wolfram

code[x_, y_] := N[(N[(N[(x * 2.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot 2 + x \cdot x\right) + y \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (+ (* x 2.0) (* x x)) (* y y)))

C

double code(double x, double y) {
	return ((x * 2.0) + (x * x)) + (y * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * 2.0d0) + (x * x)) + (y * y)
end function

Java

public static double code(double x, double y) {
	return ((x * 2.0) + (x * x)) + (y * y);
}

Python

def code(x, y):
	return ((x * 2.0) + (x * x)) + (y * y)

Julia

function code(x, y)
	return Float64(Float64(Float64(x * 2.0) + Float64(x * x)) + Float64(y * y))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * 2.0) + (x * x)) + (y * y);
end

Wolfram

code[x_, y_] := N[(N[(N[(x * 2.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

Alternative 2: 90.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(2 + x\right)\\ \mathbf{if}\;x \leq -33500000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+20}:\\ \;\;\;\;2 \cdot x + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (+ 2.0 x))))
   (if (<= x -33500000000000.0)
     t_0
     (if (<= x 7.5e+20) (+ (* 2.0 x) (* y y)) t_0))))

C

double code(double x, double y) {
	double t_0 = x * (2.0 + x);
	double tmp;
	if (x <= -33500000000000.0) {
		tmp = t_0;
	} else if (x <= 7.5e+20) {
		tmp = (2.0 * x) + (y * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (2.0d0 + x)
    if (x <= (-33500000000000.0d0)) then
        tmp = t_0
    else if (x <= 7.5d+20) then
        tmp = (2.0d0 * x) + (y * y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (2.0 + x);
	double tmp;
	if (x <= -33500000000000.0) {
		tmp = t_0;
	} else if (x <= 7.5e+20) {
		tmp = (2.0 * x) + (y * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (2.0 + x)
	tmp = 0
	if x <= -33500000000000.0:
		tmp = t_0
	elif x <= 7.5e+20:
		tmp = (2.0 * x) + (y * y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(2.0 + x))
	tmp = 0.0
	if (x <= -33500000000000.0)
		tmp = t_0;
	elseif (x <= 7.5e+20)
		tmp = Float64(Float64(2.0 * x) + Float64(y * y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (2.0 + x);
	tmp = 0.0;
	if (x <= -33500000000000.0)
		tmp = t_0;
	elseif (x <= 7.5e+20)
		tmp = (2.0 * x) + (y * y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(2.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -33500000000000.0], t$95$0, If[LessEqual[x, 7.5e+20], N[(N[(2.0 * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.35e13 or 7.5e20 < x

   1. Initial program 100.0%

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

      1. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{x \cdot \left(2 + x\right)} + y \cdot y \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(2 + x\right) + y \cdot y} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 83.6%

      \[\leadsto \color{blue}{x \cdot \left(2 + x\right)} \]

   if -3.35e13 < x < 7.5e20

   1. Initial program 100.0%

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

      1. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{x \cdot \left(2 + x\right)} + y \cdot y \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(2 + x\right) + y \cdot y} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 97.8%

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

Alternative 3: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(2 + x\right) + y \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (* x (+ 2.0 x)) (* y y)))

C

double code(double x, double y) {
	return (x * (2.0 + x)) + (y * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * (2.0d0 + x)) + (y * y)
end function

Java

public static double code(double x, double y) {
	return (x * (2.0 + x)) + (y * y);
}

Python

def code(x, y):
	return (x * (2.0 + x)) + (y * y)

Julia

function code(x, y)
	return Float64(Float64(x * Float64(2.0 + x)) + Float64(y * y))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * (2.0 + x)) + (y * y);
end

Wolfram

code[x_, y_] := N[(N[(x * N[(2.0 + x), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. distribute-lft-out 100.0%

      \[\leadsto \color{blue}{x \cdot \left(2 + x\right)} + y \cdot y \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \left(2 + x\right) + y \cdot y} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 4: 60.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(2 + x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (+ 2.0 x)))

C

double code(double x, double y) {
	return x * (2.0 + x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (2.0d0 + x)
end function

Java

public static double code(double x, double y) {
	return x * (2.0 + x);
}

Python

def code(x, y):
	return x * (2.0 + x)

Julia

function code(x, y)
	return Float64(x * Float64(2.0 + x))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (2.0 + x);
end

Wolfram

code[x_, y_] := N[(x * N[(2.0 + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. distribute-lft-out 100.0%

      \[\leadsto \color{blue}{x \cdot \left(2 + x\right)} + y \cdot y \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \left(2 + x\right) + y \cdot y} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 57.0%

   \[\leadsto \color{blue}{x \cdot \left(2 + x\right)} \]
6. Add Preprocessing

Alternative 5: 21.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ 2 \cdot x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* 2.0 x))

C

double code(double x, double y) {
	return 2.0 * x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 2.0d0 * x
end function

Java

public static double code(double x, double y) {
	return 2.0 * x;
}

Python

def code(x, y):
	return 2.0 * x

Julia

function code(x, y)
	return Float64(2.0 * x)
end

MATLAB

function tmp = code(x, y)
	tmp = 2.0 * x;
end

Wolfram

code[x_, y_] := N[(2.0 * x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. distribute-lft-out 100.0%

      \[\leadsto \color{blue}{x \cdot \left(2 + x\right)} + y \cdot y \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \left(2 + x\right) + y \cdot y} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 57.0%

   \[\leadsto \color{blue}{x \cdot \left(2 + x\right)} \]

6. Taylor expanded in x around 0 20.2%

   \[\leadsto \color{blue}{2 \cdot x} \]
7. Add Preprocessing

Developer target: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ y \cdot y + \left(2 \cdot x + x \cdot x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (* y y) (+ (* 2.0 x) (* x x))))

C

double code(double x, double y) {
	return (y * y) + ((2.0 * x) + (x * x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (y * y) + ((2.0d0 * x) + (x * x))
end function

Java

public static double code(double x, double y) {
	return (y * y) + ((2.0 * x) + (x * x));
}

Python

def code(x, y):
	return (y * y) + ((2.0 * x) + (x * x))

Julia

function code(x, y)
	return Float64(Float64(y * y) + Float64(Float64(2.0 * x) + Float64(x * x)))
end

MATLAB

function tmp = code(x, y)
	tmp = (y * y) + ((2.0 * x) + (x * x));
end

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] + N[(N[(2.0 * x), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024076 -o generate:simplify
(FPCore (x y)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, A"
  :precision binary64

  :alt
  (+ (* y y) (+ (* 2.0 x) (* x x)))

  (+ (+ (* x 2.0) (* x x)) (* y y)))