Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A

Percentage Accurate: 99.9% → 99.9%

Time: 3.5s

Alternatives: 5

Speedup: 1.0×

• 24.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot \left(1 - x \cdot y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (- 1.0 (* x y))))

C

double code(double x, double y) {
	return x * (1.0 - (x * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 - (x * y))
end function

Java

public static double code(double x, double y) {
	return x * (1.0 - (x * y));
}

Python

def code(x, y):
	return x * (1.0 - (x * y))

Julia

function code(x, y)
	return Float64(x * Float64(1.0 - Float64(x * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 - (x * y));
end

Wolfram

code[x_, y_] := N[(x * N[(1.0 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 - x \cdot y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (- 1.0 (* x y))))

C

double code(double x, double y) {
	return x * (1.0 - (x * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 - (x * y))
end function

Java

public static double code(double x, double y) {
	return x * (1.0 - (x * y));
}

Python

def code(x, y):
	return x * (1.0 - (x * y))

Julia

function code(x, y)
	return Float64(x * Float64(1.0 - Float64(x * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 - (x * y));
end

Wolfram

code[x_, y_] := N[(x * N[(1.0 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot \left(1 - x \cdot y\right) \]

2. Taylor expanded in x around 0 91.3%

   \[\leadsto \color{blue}{x + -1 \cdot \left({x}^{2} \cdot y\right)} \]
3. Step-by-step derivation

   1. mul-1-neg 91.3%

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

   2. unsub-neg 91.3%

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

   3. unpow2 91.3%

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

   4. associate-*l* 99.9%

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

4. Simplified 99.9%

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

5. Final simplification 99.9%

   \[\leadsto x - x \cdot \left(x \cdot y\right) \]

Alternative 2: 70.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-11} \lor \neg \left(x \leq 1.22 \cdot 10^{-61}\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.9e-11) (not (<= x 1.22e-61))) (* y (* x (- x))) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.9e-11) || !(x <= 1.22e-61)) {
		tmp = y * (x * -x);
	} else {
		tmp = 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 <= (-1.9d-11)) .or. (.not. (x <= 1.22d-61))) then
        tmp = y * (x * -x)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.9e-11) || !(x <= 1.22e-61)) {
		tmp = y * (x * -x);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.9e-11) or not (x <= 1.22e-61):
		tmp = y * (x * -x)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.9e-11) || !(x <= 1.22e-61))
		tmp = Float64(y * Float64(x * Float64(-x)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.9e-11) || ~((x <= 1.22e-61)))
		tmp = y * (x * -x);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.9e-11], N[Not[LessEqual[x, 1.22e-61]], $MachinePrecision]], N[(y * N[(x * (-x)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1.8999999999999999e-11 or 1.22e-61 < x

   1. Initial program 99.9%

      \[x \cdot \left(1 - x \cdot y\right) \]

   2. Taylor expanded in x around inf 72.1%

      \[\leadsto \color{blue}{-1 \cdot \left({x}^{2} \cdot y\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 72.1%

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

      2. unpow2 72.1%

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

   4. Simplified 72.1%

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

   if -1.8999999999999999e-11 < x < 1.22e-61

   1. Initial program 99.9%

      \[x \cdot \left(1 - x \cdot y\right) \]

   2. Taylor expanded in x around 0 79.8%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-11} \lor \neg \left(x \leq 1.22 \cdot 10^{-61}\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 73.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-11} \lor \neg \left(x \leq 1.5 \cdot 10^{-65}\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.85e-11) (not (<= x 1.5e-65))) (* x (* x (- y))) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.85e-11) || !(x <= 1.5e-65)) {
		tmp = x * (x * -y);
	} else {
		tmp = 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 <= (-1.85d-11)) .or. (.not. (x <= 1.5d-65))) then
        tmp = x * (x * -y)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.85e-11) || !(x <= 1.5e-65)) {
		tmp = x * (x * -y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.85e-11) or not (x <= 1.5e-65):
		tmp = x * (x * -y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.85e-11) || !(x <= 1.5e-65))
		tmp = Float64(x * Float64(x * Float64(-y)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.85e-11) || ~((x <= 1.5e-65)))
		tmp = x * (x * -y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.85e-11], N[Not[LessEqual[x, 1.5e-65]], $MachinePrecision]], N[(x * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1.8500000000000001e-11 or 1.49999999999999999e-65 < x

   1. Initial program 99.9%

      \[x \cdot \left(1 - x \cdot y\right) \]

   2. Taylor expanded in x around inf 72.1%

      \[\leadsto \color{blue}{-1 \cdot \left({x}^{2} \cdot y\right)} \]
   3. Step-by-step derivation

      1. unpow2 72.1%

         \[\leadsto -1 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right) \]

      2. associate-*l* 76.4%

         \[\leadsto -1 \cdot \color{blue}{\left(x \cdot \left(x \cdot y\right)\right)} \]

      3. associate-*r* 76.4%

         \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \left(x \cdot y\right)} \]

      4. neg-mul-1 76.4%

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

   4. Simplified 76.4%

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

   if -1.8500000000000001e-11 < x < 1.49999999999999999e-65

   1. Initial program 99.9%

      \[x \cdot \left(1 - x \cdot y\right) \]

   2. Taylor expanded in x around 0 79.8%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-11} \lor \neg \left(x \leq 1.5 \cdot 10^{-65}\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 - x \cdot y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (- 1.0 (* x y))))

C

double code(double x, double y) {
	return x * (1.0 - (x * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 - (x * y))
end function

Java

public static double code(double x, double y) {
	return x * (1.0 - (x * y));
}

Python

def code(x, y):
	return x * (1.0 - (x * y))

Julia

function code(x, y)
	return Float64(x * Float64(1.0 - Float64(x * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 - (x * y));
end

Wolfram

code[x_, y_] := N[(x * N[(1.0 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot \left(1 - x \cdot y\right) \]

2. Final simplification 99.9%

   \[\leadsto x \cdot \left(1 - x \cdot y\right) \]

Alternative 5: 50.1% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.9%

   \[x \cdot \left(1 - x \cdot y\right) \]

2. Taylor expanded in x around 0 52.3%

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

3. Final simplification 52.3%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023293 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))