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

Percentage Accurate: 99.9% → 99.9%

Time: 5.0s

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, 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 90.6%

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

   1. +-commutative 90.6%

      \[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot {x}^{2}\right)} \]

   2. mul-1-neg 90.6%

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

   3. unpow2 90.6%

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

   4. sub-neg 90.6%

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

   5. *-commutative 90.6%

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

   6. associate-*r* 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: 74.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -42000000000 \lor \neg \left(y \leq 7.3 \cdot 10^{-34}\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -42000000000.0) (not (<= y 7.3e-34))) (* x (* x (- y))) x))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -42000000000.0) || !(y <= 7.3e-34)) {
		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 <= (-42000000000.0d0)) .or. (.not. (y <= 7.3d-34))) 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 <= -42000000000.0) || !(y <= 7.3e-34)) {
		tmp = x * (x * -y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -42000000000.0) or not (y <= 7.3e-34):
		tmp = x * (x * -y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -42000000000.0) || !(y <= 7.3e-34))
		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 <= -42000000000.0) || ~((y <= 7.3e-34)))
		tmp = x * (x * -y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -42000000000.0], N[Not[LessEqual[y, 7.3e-34]], $MachinePrecision]], N[(x * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -4.2e10 or 7.29999999999999996e-34 < y

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf 71.0%

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

      1. mul-1-neg 71.0%

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

      2. unpow2 71.0%

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

      3. *-commutative 71.0%

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

      4. associate-*r* 75.4%

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

      5. distribute-rgt-neg-in 75.4%

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

      6. distribute-rgt-neg-in 75.4%

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

   4. Simplified 75.4%

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

   if -4.2e10 < y < 7.29999999999999996e-34

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 79.5%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -42000000000 \lor \neg \left(y \leq 7.3 \cdot 10^{-34}\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

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: 50.6% 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 52.5%

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

3. Final simplification 52.5%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023171 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))