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

Percentage Accurate: 99.9% → 99.9%

Time: 3.7s

Alternatives: 3

Speedup: 1.0×

• 24.4% 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

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

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 2: 74.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+64} \lor \neg \left(y \leq -9.5 \cdot 10^{+46} \lor \neg \left(y \leq -1.25 \cdot 10^{-55}\right) \land y \leq 3.5 \cdot 10^{+14}\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.15e+64)
         (not
          (or (<= y -9.5e+46) (and (not (<= y -1.25e-55)) (<= y 3.5e+14)))))
   (* x (* x (- y)))
   x))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.15e+64) || !((y <= -9.5e+46) || (!(y <= -1.25e-55) && (y <= 3.5e+14)))) {
		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 ((y <= (-1.15d+64)) .or. (.not. (y <= (-9.5d+46)) .or. (.not. (y <= (-1.25d-55))) .and. (y <= 3.5d+14))) 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 ((y <= -1.15e+64) || !((y <= -9.5e+46) || (!(y <= -1.25e-55) && (y <= 3.5e+14)))) {
		tmp = x * (x * -y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.15e+64) or not ((y <= -9.5e+46) or (not (y <= -1.25e-55) and (y <= 3.5e+14))):
		tmp = x * (x * -y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.15e+64) || !((y <= -9.5e+46) || (!(y <= -1.25e-55) && (y <= 3.5e+14))))
		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 ((y <= -1.15e+64) || ~(((y <= -9.5e+46) || (~((y <= -1.25e-55)) && (y <= 3.5e+14)))))
		tmp = x * (x * -y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.15e+64], N[Not[Or[LessEqual[y, -9.5e+46], And[N[Not[LessEqual[y, -1.25e-55]], $MachinePrecision], LessEqual[y, 3.5e+14]]]], $MachinePrecision]], N[(x * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -1.15e64 or -9.5000000000000008e46 < y < -1.25e-55 or 3.5e14 < y

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf 73.9%

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

      1. mul-1-neg 73.9%

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

      2. unpow2 73.9%

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

      3. *-commutative 73.9%

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

      4. associate-*r* 78.9%

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

      5. distribute-rgt-neg-in 78.9%

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

      6. distribute-rgt-neg-in 78.9%

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

   4. Simplified 78.9%

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

   if -1.15e64 < y < -9.5000000000000008e46 or -1.25e-55 < y < 3.5e14

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 84.5%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+64} \lor \neg \left(y \leq -9.5 \cdot 10^{+46} \lor \neg \left(y \leq -1.25 \cdot 10^{-55}\right) \land y \leq 3.5 \cdot 10^{+14}\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 50.6% 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 50.3%

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

3. Final simplification 50.3%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023257 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))