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

Percentage Accurate: 100.0% → 100.0%

Time: 1.5s

Alternatives: 3

Speedup: 1.0×

• 24.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot \left(1 - x \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (- 1.0 (* x 0.5))))

C

double code(double x) {
	return x * (1.0 - (x * 0.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 - (x * 0.5d0))
end function

Java

public static double code(double x) {
	return x * (1.0 - (x * 0.5));
}

Python

def code(x):
	return x * (1.0 - (x * 0.5))

Julia

function code(x)
	return Float64(x * Float64(1.0 - Float64(x * 0.5)))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 - (x * 0.5));
end

Wolfram

code[x_] := N[(x * N[(1.0 - N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 - x \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (- 1.0 (* x 0.5))))

C

double code(double x) {
	return x * (1.0 - (x * 0.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 - (x * 0.5d0))
end function

Java

public static double code(double x) {
	return x * (1.0 - (x * 0.5));
}

Python

def code(x):
	return x * (1.0 - (x * 0.5))

Julia

function code(x)
	return Float64(x * Float64(1.0 - Float64(x * 0.5)))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 - (x * 0.5));
end

Wolfram

code[x_] := N[(x * N[(1.0 - N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 - x \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (- 1.0 (* x 0.5))))

C

double code(double x) {
	return x * (1.0 - (x * 0.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 - (x * 0.5d0))
end function

Java

public static double code(double x) {
	return x * (1.0 - (x * 0.5));
}

Python

def code(x):
	return x * (1.0 - (x * 0.5))

Julia

function code(x)
	return Float64(x * Float64(1.0 - Float64(x * 0.5)))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 - (x * 0.5));
end

Wolfram

code[x_] := N[(x * N[(1.0 - N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x \cdot \left(1 - x \cdot 0.5\right) \]

2. Final simplification 100.0%

   \[\leadsto x \cdot \left(1 - x \cdot 0.5\right) \]

Alternative 2: 97.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;x \cdot \left(x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -2.0) (not (<= x 2.0))) (* x (* x -0.5)) x))

C

double code(double x) {
	double tmp;
	if ((x <= -2.0) || !(x <= 2.0)) {
		tmp = x * (x * -0.5);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-2.0d0)) .or. (.not. (x <= 2.0d0))) then
        tmp = x * (x * (-0.5d0))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -2.0) || !(x <= 2.0)) {
		tmp = x * (x * -0.5);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -2.0) or not (x <= 2.0):
		tmp = x * (x * -0.5)
	else:
		tmp = x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -2.0) || !(x <= 2.0))
		tmp = Float64(x * Float64(x * -0.5));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -2.0) || ~((x <= 2.0)))
		tmp = x * (x * -0.5);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -2.0], N[Not[LessEqual[x, 2.0]], $MachinePrecision]], N[(x * N[(x * -0.5), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -2 or 2 < x

   1. Initial program 100.0%

      \[x \cdot \left(1 - x \cdot 0.5\right) \]

   2. Taylor expanded in x around inf 97.1%

      \[\leadsto \color{blue}{-0.5 \cdot {x}^{2}} \]
   3. Step-by-step derivation

      1. *-commutative 97.1%

         \[\leadsto \color{blue}{{x}^{2} \cdot -0.5} \]

      2. unpow2 97.1%

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot -0.5 \]

      3. associate-*r* 97.1%

         \[\leadsto \color{blue}{x \cdot \left(x \cdot -0.5\right)} \]

   4. Simplified 97.1%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot -0.5\right)} \]

   if -2 < x < 2

   1. Initial program 100.0%

      \[x \cdot \left(1 - x \cdot 0.5\right) \]

   2. Taylor expanded in x around 0 96.9%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;x \cdot \left(x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 51.6% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 100.0%

   \[x \cdot \left(1 - x \cdot 0.5\right) \]

2. Taylor expanded in x around 0 49.8%

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

3. Final simplification 49.8%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023196 
(FPCore (x)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, B"
  :precision binary64
  (* x (- 1.0 (* x 0.5))))