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

Percentage Accurate: 100.0% → 100.0%

Time: 5.2s

Alternatives: 6

Speedup: 1.2×

• 43.4% 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: 84.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 6.5 \cdot 10^{-31} \lor \neg \left(y \cdot y \leq 9 \cdot 10^{+36}\right) \land y \cdot y \leq 7.2 \cdot 10^{+70}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* y y) 6.5e-31)
         (and (not (<= (* y y) 9e+36)) (<= (* y y) 7.2e+70)))
   (* x (+ x 2.0))
   (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (((y * y) <= 6.5e-31) || (!((y * y) <= 9e+36) && ((y * y) <= 7.2e+70))) {
		tmp = x * (x + 2.0);
	} 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 * y) <= 6.5d-31) .or. (.not. ((y * y) <= 9d+36)) .and. ((y * y) <= 7.2d+70)) then
        tmp = x * (x + 2.0d0)
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((y * y) <= 6.5e-31) || (!((y * y) <= 9e+36) && ((y * y) <= 7.2e+70))) {
		tmp = x * (x + 2.0);
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y * y) <= 6.5e-31) or (not ((y * y) <= 9e+36) and ((y * y) <= 7.2e+70)):
		tmp = x * (x + 2.0)
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(y * y) <= 6.5e-31) || (!(Float64(y * y) <= 9e+36) && (Float64(y * y) <= 7.2e+70)))
		tmp = Float64(x * Float64(x + 2.0));
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((y * y) <= 6.5e-31) || (~(((y * y) <= 9e+36)) && ((y * y) <= 7.2e+70)))
		tmp = x * (x + 2.0);
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(y * y), $MachinePrecision], 6.5e-31], And[N[Not[LessEqual[N[(y * y), $MachinePrecision], 9e+36]], $MachinePrecision], LessEqual[N[(y * y), $MachinePrecision], 7.2e+70]]], N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 6.49999999999999967e-31 or 8.99999999999999994e36 < (*.f64 y y) < 7.1999999999999999e70

   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 y around 0 90.1%

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

   if 6.49999999999999967e-31 < (*.f64 y y) < 8.99999999999999994e36 or 7.1999999999999999e70 < (*.f64 y 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 85.1%

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

      1. unpow2 85.1%

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

   6. Simplified 85.1%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 6.5 \cdot 10^{-31} \lor \neg \left(y \cdot y \leq 9 \cdot 10^{+36}\right) \land y \cdot y \leq 7.2 \cdot 10^{+70}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 3: 96.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 4 \cdot 10^{-190}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 4e-190) (* x (+ x 2.0)) (+ (* x x) (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 4e-190) {
		tmp = x * (x + 2.0);
	} 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 * y) <= 4d-190) then
        tmp = x * (x + 2.0d0)
    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 * y) <= 4e-190) {
		tmp = x * (x + 2.0);
	} else {
		tmp = (x * x) + (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 4e-190:
		tmp = x * (x + 2.0)
	else:
		tmp = (x * x) + (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 4e-190)
		tmp = Float64(x * Float64(x + 2.0));
	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 * y) <= 4e-190)
		tmp = x * (x + 2.0);
	else
		tmp = (x * x) + (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 4e-190], N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4.0000000000000001e-190

   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 y around 0 99.0%

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

   if 4.0000000000000001e-190 < (*.f64 y 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 98.0%

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

      1. unpow2 98.0%

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

   6. Simplified 98.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 4 \cdot 10^{-190}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot y\\ \end{array} \]

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: 63.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 1.45 \cdot 10^{-179}:\\ \;\;\;\;x + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 1.45e-179) (+ x x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1.45e-179) {
		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 * y) <= 1.45d-179) 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 * y) <= 1.45e-179) {
		tmp = x + x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 1.45e-179:
		tmp = x + x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 1.45e-179)
		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 * y) <= 1.45e-179)
		tmp = x + x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1.45e-179], N[(x + x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1.4499999999999999e-179

   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 0 53.7%

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

      1. count-2 53.7%

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

   6. Simplified 53.7%

      \[\leadsto \color{blue}{\left(x + x\right)} + y \cdot y \]
   7. Step-by-step derivation

      1. associate-+l+ 53.7%

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

      2. flip-+ 0.1%

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

   8. Applied egg-rr 0.1%

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

   9. Taylor expanded in x around inf 52.7%

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

       1. count-2 52.7%

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

   11. Simplified 52.7%

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

   if 1.4499999999999999e-179 < (*.f64 y 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 72.5%

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

      1. unpow2 72.5%

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

   6. Simplified 72.5%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 1.45 \cdot 10^{-179}:\\ \;\;\;\;x + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 6: 48.2% 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 46.7%

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

   1. unpow2 46.7%

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

6. Simplified 46.7%

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

7. Final simplification 46.7%

   \[\leadsto y \cdot y \]

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 2023215 
(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)))