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

Percentage Accurate: 100.0% → 100.0%

Time: 2.2s

Alternatives: 6

Speedup: 1.2×

• 44.0% 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: 100.0% 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: 100.0% 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. Final simplification 100.0%

   \[\leadsto \left(x \cdot 2 + x \cdot x\right) + y \cdot y \]

Alternative 2: 49.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{-165} \lor \neg \left(y \leq 8.5 \cdot 10^{-151}\right) \land y \leq 9 \cdot 10^{-80}:\\ \;\;\;\;x + x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 5.4e-165) (and (not (<= y 8.5e-151)) (<= y 9e-80)))
   (+ x x)
   (+ (* x x) (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 5.4e-165) || (!(y <= 8.5e-151) && (y <= 9e-80))) {
		tmp = x + x;
	} else {
		tmp = (x * x) + (y * 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 <= 5.4d-165) .or. (.not. (y <= 8.5d-151)) .and. (y <= 9d-80)) then
        tmp = x + x
    else
        tmp = (x * x) + (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= 5.4e-165) || (!(y <= 8.5e-151) && (y <= 9e-80))) {
		tmp = x + x;
	} else {
		tmp = (x * x) + (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 5.4e-165) or (not (y <= 8.5e-151) and (y <= 9e-80)):
		tmp = x + x
	else:
		tmp = (x * x) + (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 5.4e-165) || (!(y <= 8.5e-151) && (y <= 9e-80)))
		tmp = Float64(x + x);
	else
		tmp = Float64(Float64(x * x) + Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 5.4e-165) || (~((y <= 8.5e-151)) && (y <= 9e-80)))
		tmp = x + x;
	else
		tmp = (x * x) + (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 5.4e-165], And[N[Not[LessEqual[y, 8.5e-151]], $MachinePrecision], LessEqual[y, 9e-80]]], N[(x + x), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.3999999999999995e-165 or 8.49999999999999999e-151 < y < 9.0000000000000006e-80

   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. Taylor expanded in x around 0 65.5%

      \[\leadsto \color{blue}{2 \cdot x} + y \cdot y \]
   5. Step-by-step derivation

      1. count-2 65.5%

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

   6. Simplified 65.5%

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

   7. Taylor expanded in x around inf 33.4%

      \[\leadsto \color{blue}{2 \cdot x} \]
   8. Step-by-step derivation

      1. count-2 33.4%

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

   9. Simplified 33.4%

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

   if 5.3999999999999995e-165 < y < 8.49999999999999999e-151 or 9.0000000000000006e-80 < y

   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. Taylor expanded in x around inf 94.0%

      \[\leadsto \color{blue}{{x}^{2}} + y \cdot y \]
   5. Step-by-step derivation

      1. unpow2 94.0%

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

   6. Simplified 94.0%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{-165} \lor \neg \left(y \leq 8.5 \cdot 10^{-151}\right) \land y \leq 9 \cdot 10^{-80}:\\ \;\;\;\;x + x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot y\\ \end{array} \]

Alternative 3: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;x \cdot x + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.0) (not (<= x 2.0)))
   (+ (* x x) (* y y))
   (+ (* y y) (+ x x))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.0) || !(x <= 2.0)) {
		tmp = (x * x) + (y * y);
	} else {
		tmp = (y * y) + (x + 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 <= (-2.0d0)) .or. (.not. (x <= 2.0d0))) then
        tmp = (x * x) + (y * y)
    else
        tmp = (y * y) + (x + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2.0) || !(x <= 2.0)) {
		tmp = (x * x) + (y * y);
	} else {
		tmp = (y * y) + (x + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.0) or not (x <= 2.0):
		tmp = (x * x) + (y * y)
	else:
		tmp = (y * y) + (x + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.0) || !(x <= 2.0))
		tmp = Float64(Float64(x * x) + Float64(y * y));
	else
		tmp = Float64(Float64(y * y) + Float64(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.0) || ~((x <= 2.0)))
		tmp = (x * x) + (y * y);
	else
		tmp = (y * y) + (x + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.0], N[Not[LessEqual[x, 2.0]], $MachinePrecision]], N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] + N[(x + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2 or 2 < 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. Taylor expanded in x around inf 99.1%

      \[\leadsto \color{blue}{{x}^{2}} + y \cdot y \]
   5. Step-by-step derivation

      1. unpow2 99.1%

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

   6. Simplified 99.1%

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

   if -2 < x < 2

   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. Taylor expanded in x around 0 98.7%

      \[\leadsto \color{blue}{2 \cdot x} + y \cdot y \]
   5. Step-by-step derivation

      1. count-2 98.7%

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

   6. Simplified 98.7%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;x \cdot x + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + \left(x + x\right)\\ \end{array} \]

Alternative 4: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] + N[(x * N[(x + 2.0), $MachinePrecision]), $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. Final simplification 100.0%

   \[\leadsto y \cdot y + x \cdot \left(x + 2\right) \]

Alternative 5: 40.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-31}:\\ \;\;\;\;x + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.6e-31) (+ x x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.6e-31) {
		tmp = x + x;
	} else {
		tmp = y * 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 <= 1.6d-31) then
        tmp = x + x
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.6e-31) {
		tmp = x + x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.6e-31:
		tmp = x + x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.6e-31)
		tmp = Float64(x + x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.6e-31)
		tmp = x + x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.6e-31], N[(x + x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.60000000000000009e-31

   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. Taylor expanded in x around 0 65.0%

      \[\leadsto \color{blue}{2 \cdot x} + y \cdot y \]
   5. Step-by-step derivation

      1. count-2 65.0%

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

   6. Simplified 65.0%

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

   7. Taylor expanded in x around inf 34.1%

      \[\leadsto \color{blue}{2 \cdot x} \]
   8. Step-by-step derivation

      1. count-2 34.1%

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

   9. Simplified 34.1%

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

   if 1.60000000000000009e-31 < y

   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. Taylor expanded in x around 0 82.4%

      \[\leadsto \color{blue}{2 \cdot x} + y \cdot y \]
   5. Step-by-step derivation

      1. count-2 82.4%

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

   6. Simplified 82.4%

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

   7. Taylor expanded in x around 0 82.5%

      \[\leadsto \color{blue}{{y}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 82.5%

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

   9. Simplified 82.5%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-31}:\\ \;\;\;\;x + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 6: 48.1% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(y * y), $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. Taylor expanded in x around 0 69.0%

   \[\leadsto \color{blue}{2 \cdot x} + y \cdot y \]
5. Step-by-step derivation

   1. count-2 69.0%

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

6. Simplified 69.0%

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

7. Taylor expanded in x around 0 44.9%

   \[\leadsto \color{blue}{{y}^{2}} \]
8. Step-by-step derivation

   1. unpow2 44.9%

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

9. Simplified 44.9%

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

10. Final simplification 44.9%

    \[\leadsto y \cdot y \]

Developer target: 100.0% 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 2023199 
(FPCore (x y)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, A"
  :precision binary64

  :herbie-target
  (+ (* y y) (+ (* 2.0 x) (* x x)))

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