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

Percentage Accurate: 99.9% → 99.9%

Time: 5.7s

Alternatives: 5

Speedup: 1.0×

• 24.3% 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. Step-by-step derivation

   1. flip-- 79.6%

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

   2. associate-*r/ 74.3%

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

   3. metadata-eval 74.3%

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

   4. pow2 74.3%

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

3. Applied egg-rr 74.3%

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

   1. *-commutative 74.3%

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

   2. associate-/l* 79.4%

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

5. Simplified 79.4%

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

6. Taylor expanded in x around 0 89.9%

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

   1. +-commutative 89.9%

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

   2. mul-1-neg 89.9%

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

   3. *-commutative 89.9%

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

   4. unpow2 89.9%

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

   5. associate-*r* 99.9%

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

   6. unsub-neg 99.9%

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

8. Simplified 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 71.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2800 \lor \neg \left(x \leq 9.2 \cdot 10^{-54}\right):\\ \;\;\;\;y \cdot \left(-x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2800.0) (not (<= x 9.2e-54))) (* y (- (* x x))) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2800.0) || !(x <= 9.2e-54)) {
		tmp = y * -(x * x);
	} 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 <= (-2800.0d0)) .or. (.not. (x <= 9.2d-54))) then
        tmp = y * -(x * x)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2800.0) || !(x <= 9.2e-54)) {
		tmp = y * -(x * x);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2800.0) or not (x <= 9.2e-54):
		tmp = y * -(x * x)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2800.0) || !(x <= 9.2e-54))
		tmp = Float64(y * Float64(-Float64(x * x)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2800.0) || ~((x <= 9.2e-54)))
		tmp = y * -(x * x);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2800.0], N[Not[LessEqual[x, 9.2e-54]], $MachinePrecision]], N[(y * (-N[(x * x), $MachinePrecision])), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -2800 or 9.1999999999999996e-54 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf 70.2%

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

      1. mul-1-neg 70.2%

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

      2. unpow2 70.2%

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

   4. Simplified 70.2%

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

   if -2800 < x < 9.1999999999999996e-54

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 80.5%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2800 \lor \neg \left(x \leq 9.2 \cdot 10^{-54}\right):\\ \;\;\;\;y \cdot \left(-x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 74.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2700 \lor \neg \left(x \leq 4.7 \cdot 10^{-52}\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2700.0) (not (<= x 4.7e-52))) (* x (* x (- y))) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2700.0) || !(x <= 4.7e-52)) {
		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 <= (-2700.0d0)) .or. (.not. (x <= 4.7d-52))) 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 <= -2700.0) || !(x <= 4.7e-52)) {
		tmp = x * (x * -y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2700.0) or not (x <= 4.7e-52):
		tmp = x * (x * -y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2700.0) || !(x <= 4.7e-52))
		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 <= -2700.0) || ~((x <= 4.7e-52)))
		tmp = x * (x * -y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2700.0], N[Not[LessEqual[x, 4.7e-52]], $MachinePrecision]], N[(x * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -2700 or 4.6999999999999997e-52 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf 70.2%

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

      1. mul-1-neg 70.2%

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

      2. *-commutative 70.2%

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

      3. unpow2 70.2%

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

      4. associate-*l* 79.6%

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

      5. distribute-rgt-neg-out 79.6%

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

      6. distribute-rgt-neg-in 79.6%

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

   4. Simplified 79.6%

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

   if -2700 < x < 4.6999999999999997e-52

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 80.5%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2700 \lor \neg \left(x \leq 4.7 \cdot 10^{-52}\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 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 5: 51.8% 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 48.1%

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

3. Final simplification 48.1%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023279 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))