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

Percentage Accurate: 99.9% → 99.9%

Time: 4.2s

Alternatives: 4

Speedup: 1.0×

• 24.8% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot \left(1 - x \cdot y\right) \]
2. Step-by-step derivation

   1. sub-neg 99.9%

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

   2. +-commutative 99.9%

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

   3. distribute-lft-neg-in 99.9%

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

   4. fma-def 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto x \cdot \mathsf{fma}\left(-x, y, 1\right) \]

Alternative 2: 75.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-54} \lor \neg \left(x \leq 4.8 \cdot 10^{-9}\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.25e-54) (not (<= x 4.8e-9))) (* x (* x (- y))) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.25e-54) || !(x <= 4.8e-9)) {
		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 <= (-1.25d-54)) .or. (.not. (x <= 4.8d-9))) 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 <= -1.25e-54) || !(x <= 4.8e-9)) {
		tmp = x * (x * -y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.25e-54) or not (x <= 4.8e-9):
		tmp = x * (x * -y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.25e-54) || !(x <= 4.8e-9))
		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 <= -1.25e-54) || ~((x <= 4.8e-9)))
		tmp = x * (x * -y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.25e-54], N[Not[LessEqual[x, 4.8e-9]], $MachinePrecision]], N[(x * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1.25000000000000004e-54 or 4.8e-9 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf 74.2%

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

      1. mul-1-neg 74.2%

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

      2. unpow2 74.2%

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

      3. *-commutative 74.2%

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

      4. associate-*r* 83.7%

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

      5. distribute-rgt-neg-in 83.7%

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

      6. distribute-rgt-neg-in 83.7%

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

   4. Simplified 83.7%

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

   if -1.25000000000000004e-54 < x < 4.8e-9

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 84.6%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-54} \lor \neg \left(x \leq 4.8 \cdot 10^{-9}\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 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 4: 50.8% 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 47.7%

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

3. Final simplification 47.7%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023249 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))