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

Percentage Accurate: 100.0% → 100.0%

Time: 2.5s

Alternatives: 6

Speedup: 1.2×

• 43.5% 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: 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. fma-def 100.0%

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

   3. +-commutative 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(x, x + 2, y \cdot y\right) \]

Alternative 2: 90.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+48} \lor \neg \left(x \leq -2.7 \cdot 10^{+27} \lor \neg \left(x \leq -6 \cdot 10^{-8}\right) \land x \leq 7.5 \cdot 10^{+32}\right):\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -9.2e+48)
         (not (or (<= x -2.7e+27) (and (not (<= x -6e-8)) (<= x 7.5e+32)))))
   (* x (+ x 2.0))
   (+ (* y y) (* x 2.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -9.2e+48) || !((x <= -2.7e+27) || (!(x <= -6e-8) && (x <= 7.5e+32)))) {
		tmp = x * (x + 2.0);
	} else {
		tmp = (y * y) + (x * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-9.2d+48)) .or. (.not. (x <= (-2.7d+27)) .or. (.not. (x <= (-6d-8))) .and. (x <= 7.5d+32))) then
        tmp = x * (x + 2.0d0)
    else
        tmp = (y * y) + (x * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -9.2e+48) || !((x <= -2.7e+27) || (!(x <= -6e-8) && (x <= 7.5e+32)))) {
		tmp = x * (x + 2.0);
	} else {
		tmp = (y * y) + (x * 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -9.2e+48) or not ((x <= -2.7e+27) or (not (x <= -6e-8) and (x <= 7.5e+32))):
		tmp = x * (x + 2.0)
	else:
		tmp = (y * y) + (x * 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -9.2e+48) || !((x <= -2.7e+27) || (!(x <= -6e-8) && (x <= 7.5e+32))))
		tmp = Float64(x * Float64(x + 2.0));
	else
		tmp = Float64(Float64(y * y) + Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -9.2e+48) || ~(((x <= -2.7e+27) || (~((x <= -6e-8)) && (x <= 7.5e+32)))))
		tmp = x * (x + 2.0);
	else
		tmp = (y * y) + (x * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -9.2e+48], N[Not[Or[LessEqual[x, -2.7e+27], And[N[Not[LessEqual[x, -6e-8]], $MachinePrecision], LessEqual[x, 7.5e+32]]]], $MachinePrecision]], N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.2000000000000001e48 or -2.6999999999999997e27 < x < -5.99999999999999946e-8 or 7.49999999999999959e32 < 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 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 85.0%

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

      1. unpow2 85.0%

         \[\leadsto 2 \cdot x + \color{blue}{x \cdot x} \]

      2. distribute-rgt-in 85.0%

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

   6. Simplified 85.0%

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

   if -9.2000000000000001e48 < x < -2.6999999999999997e27 or -5.99999999999999946e-8 < x < 7.49999999999999959e32

   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 99.0%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+48} \lor \neg \left(x \leq -2.7 \cdot 10^{+27} \lor \neg \left(x \leq -6 \cdot 10^{-8}\right) \land x \leq 7.5 \cdot 10^{+32}\right):\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot 2\\ \end{array} \]

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return (y * y) + ((x * 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) + (x * 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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: 60.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(x * N[(x + 2.0), $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. Taylor expanded in x around inf 60.1%

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

   1. unpow2 60.1%

      \[\leadsto 2 \cdot x + \color{blue}{x \cdot x} \]

   2. distribute-rgt-in 60.1%

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

6. Simplified 60.1%

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

7. Final simplification 60.1%

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

Alternative 6: 20.1% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(x * 2.0), $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 66.2%

   \[\leadsto \color{blue}{2 \cdot x} + y \cdot y \]

5. Taylor expanded in x around inf 20.8%

   \[\leadsto \color{blue}{2 \cdot x} \]

6. Final simplification 20.8%

   \[\leadsto x \cdot 2 \]

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 2023322 
(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)))