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

Percentage Accurate: 100.0% → 100.0%

Time: 3.3s

Alternatives: 6

Speedup: 1.2×

• 43.3% 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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(y * y + 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. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative 100.0%

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

   2. fma-define 100.0%

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

   3. +-commutative 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 89.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{+71} \lor \neg \left(x \leq 8000000000\right) \land \left(x \leq 3.6 \cdot 10^{+41} \lor \neg \left(x \leq 8.5 \cdot 10^{+74}\right)\right):\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2 + y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.85e+71)
         (and (not (<= x 8000000000.0))
              (or (<= x 3.6e+41) (not (<= x 8.5e+74)))))
   (* x (+ x 2.0))
   (+ (* x 2.0) (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.85e+71) || (!(x <= 8000000000.0) && ((x <= 3.6e+41) || !(x <= 8.5e+74)))) {
		tmp = x * (x + 2.0);
	} else {
		tmp = (x * 2.0) + (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 ((x <= (-1.85d+71)) .or. (.not. (x <= 8000000000.0d0)) .and. (x <= 3.6d+41) .or. (.not. (x <= 8.5d+74))) then
        tmp = x * (x + 2.0d0)
    else
        tmp = (x * 2.0d0) + (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.85e+71) || (!(x <= 8000000000.0) && ((x <= 3.6e+41) || !(x <= 8.5e+74)))) {
		tmp = x * (x + 2.0);
	} else {
		tmp = (x * 2.0) + (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.85e+71) or (not (x <= 8000000000.0) and ((x <= 3.6e+41) or not (x <= 8.5e+74))):
		tmp = x * (x + 2.0)
	else:
		tmp = (x * 2.0) + (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.85e+71) || (!(x <= 8000000000.0) && ((x <= 3.6e+41) || !(x <= 8.5e+74))))
		tmp = Float64(x * Float64(x + 2.0));
	else
		tmp = Float64(Float64(x * 2.0) + Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.85e+71) || (~((x <= 8000000000.0)) && ((x <= 3.6e+41) || ~((x <= 8.5e+74)))))
		tmp = x * (x + 2.0);
	else
		tmp = (x * 2.0) + (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.85e+71], And[N[Not[LessEqual[x, 8000000000.0]], $MachinePrecision], Or[LessEqual[x, 3.6e+41], N[Not[LessEqual[x, 8.5e+74]], $MachinePrecision]]]], N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.85e71 or 8e9 < x < 3.60000000000000025e41 or 8.50000000000000028e74 < 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 89.4%

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

   if -1.85e71 < x < 8e9 or 3.60000000000000025e41 < x < 8.50000000000000028e74

   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 92.4%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{+71} \lor \neg \left(x \leq 8000000000\right) \land \left(x \leq 3.6 \cdot 10^{+41} \lor \neg \left(x \leq 8.5 \cdot 10^{+74}\right)\right):\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2 + y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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. Add Preprocessing

3. Final simplification 100.0%

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

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. Add Preprocessing

5. Final simplification 100.0%

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

Alternative 5: 60.4% 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. Add Preprocessing

5. Taylor expanded in y around 0 59.0%

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

6. Final simplification 59.0%

   \[\leadsto x \cdot \left(x + 2\right) \]
7. Add Preprocessing

Alternative 6: 20.5% 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. Add Preprocessing

5. Taylor expanded in x around 0 66.2%

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

6. Taylor expanded in x around inf 21.8%

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

7. Final simplification 21.8%

   \[\leadsto x \cdot 2 \]
8. Add Preprocessing

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 2024053 
(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)))