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

Percentage Accurate: 99.9% → 99.9%

Time: 4.3s

Alternatives: 4

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

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

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 91.9%

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

   1. +-commutative 91.9%

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

   2. mul-1-neg 91.9%

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

   3. unsub-neg 91.9%

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

   4. *-commutative 91.9%

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

   5. unpow2 91.9%

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

   6. 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: 74.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-14} \lor \neg \left(x \leq 5.6 \cdot 10^{-80} \lor \neg \left(x \leq 2.35 \cdot 10^{-36}\right) \land x \leq 2.1 \cdot 10^{+29}\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -7.2e-14)
         (not (or (<= x 5.6e-80) (and (not (<= x 2.35e-36)) (<= x 2.1e+29)))))
   (* x (* x (- y)))
   x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -7.2e-14) || !((x <= 5.6e-80) || (!(x <= 2.35e-36) && (x <= 2.1e+29)))) {
		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 <= (-7.2d-14)) .or. (.not. (x <= 5.6d-80) .or. (.not. (x <= 2.35d-36)) .and. (x <= 2.1d+29))) 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 <= -7.2e-14) || !((x <= 5.6e-80) || (!(x <= 2.35e-36) && (x <= 2.1e+29)))) {
		tmp = x * (x * -y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -7.2e-14) or not ((x <= 5.6e-80) or (not (x <= 2.35e-36) and (x <= 2.1e+29))):
		tmp = x * (x * -y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -7.2e-14) || !((x <= 5.6e-80) || (!(x <= 2.35e-36) && (x <= 2.1e+29))))
		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 <= -7.2e-14) || ~(((x <= 5.6e-80) || (~((x <= 2.35e-36)) && (x <= 2.1e+29)))))
		tmp = x * (x * -y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -7.2e-14], N[Not[Or[LessEqual[x, 5.6e-80], And[N[Not[LessEqual[x, 2.35e-36]], $MachinePrecision], LessEqual[x, 2.1e+29]]]], $MachinePrecision]], N[(x * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -7.1999999999999996e-14 or 5.59999999999999978e-80 < x < 2.3500000000000001e-36 or 2.1000000000000002e29 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 76.2%

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

      1. associate-*r* 76.2%

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

      2. *-commutative 76.2%

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

      3. unpow2 76.2%

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

      4. associate-*r* 81.6%

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

      5. mul-1-neg 81.6%

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

   4. Simplified 81.6%

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

   if -7.1999999999999996e-14 < x < 5.59999999999999978e-80 or 2.3500000000000001e-36 < x < 2.1000000000000002e29

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 78.6%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-14} \lor \neg \left(x \leq 5.6 \cdot 10^{-80} \lor \neg \left(x \leq 2.35 \cdot 10^{-36}\right) \land x \leq 2.1 \cdot 10^{+29}\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

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 46.8%

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

3. Final simplification 46.8%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023274 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))