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

Percentage Accurate: 99.9% → 99.9%

Time: 4.3s

Alternatives: 5

Speedup: 1.0×

• 25.2% 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 91.5%

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

   1. +-commutative 91.5%

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

   2. mul-1-neg 91.5%

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

   3. unpow2 91.5%

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

   4. sub-neg 91.5%

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

   5. *-commutative 91.5%

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

   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: 75.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.4 \cdot 10^{+61} \lor \neg \left(y \leq -1.35 \cdot 10^{+31}\right) \land \left(y \leq -1.85 \cdot 10^{-28} \lor \neg \left(y \leq 4.2 \cdot 10^{-38}\right)\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -9.4e+61)
         (and (not (<= y -1.35e+31))
              (or (<= y -1.85e-28) (not (<= y 4.2e-38)))))
   (* x (* x (- y)))
   x))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -9.4e+61) || (!(y <= -1.35e+31) && ((y <= -1.85e-28) || !(y <= 4.2e-38)))) {
		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 <= (-9.4d+61)) .or. (.not. (y <= (-1.35d+31))) .and. (y <= (-1.85d-28)) .or. (.not. (y <= 4.2d-38))) 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 <= -9.4e+61) || (!(y <= -1.35e+31) && ((y <= -1.85e-28) || !(y <= 4.2e-38)))) {
		tmp = x * (x * -y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -9.4e+61) or (not (y <= -1.35e+31) and ((y <= -1.85e-28) or not (y <= 4.2e-38))):
		tmp = x * (x * -y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -9.4e+61) || (!(y <= -1.35e+31) && ((y <= -1.85e-28) || !(y <= 4.2e-38))))
		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 <= -9.4e+61) || (~((y <= -1.35e+31)) && ((y <= -1.85e-28) || ~((y <= 4.2e-38)))))
		tmp = x * (x * -y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -9.4e+61], And[N[Not[LessEqual[y, -1.35e+31]], $MachinePrecision], Or[LessEqual[y, -1.85e-28], N[Not[LessEqual[y, 4.2e-38]], $MachinePrecision]]]], N[(x * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -9.3999999999999997e61 or -1.34999999999999993e31 < y < -1.8500000000000001e-28 or 4.20000000000000026e-38 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 72.6%

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

      1. mul-1-neg 72.6%

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

      2. unpow2 72.6%

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

      3. *-commutative 72.6%

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

      4. associate-*r* 78.0%

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

      5. distribute-rgt-neg-in 78.0%

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

      6. distribute-rgt-neg-in 78.0%

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

   4. Simplified 78.0%

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

   if -9.3999999999999997e61 < y < -1.34999999999999993e31 or -1.8500000000000001e-28 < y < 4.20000000000000026e-38

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 81.8%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.4 \cdot 10^{+61} \lor \neg \left(y \leq -1.35 \cdot 10^{+31}\right) \land \left(y \leq -1.85 \cdot 10^{-28} \lor \neg \left(y \leq 4.2 \cdot 10^{-38}\right)\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 75.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{if}\;y \leq -1.25 \cdot 10^{+62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -4.9 \cdot 10^{+31}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-28}:\\ \;\;\;\;\left(-y\right) \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-38}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* x (- y)))))
   (if (<= y -1.25e+62)
     t_0
     (if (<= y -4.9e+31)
       x
       (if (<= y -1.7e-28) (* (- y) (* x x)) (if (<= y 7.5e-38) x t_0))))))

C

double code(double x, double y) {
	double t_0 = x * (x * -y);
	double tmp;
	if (y <= -1.25e+62) {
		tmp = t_0;
	} else if (y <= -4.9e+31) {
		tmp = x;
	} else if (y <= -1.7e-28) {
		tmp = -y * (x * x);
	} else if (y <= 7.5e-38) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * -y)
    if (y <= (-1.25d+62)) then
        tmp = t_0
    else if (y <= (-4.9d+31)) then
        tmp = x
    else if (y <= (-1.7d-28)) then
        tmp = -y * (x * x)
    else if (y <= 7.5d-38) then
        tmp = x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (x * -y);
	double tmp;
	if (y <= -1.25e+62) {
		tmp = t_0;
	} else if (y <= -4.9e+31) {
		tmp = x;
	} else if (y <= -1.7e-28) {
		tmp = -y * (x * x);
	} else if (y <= 7.5e-38) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (x * -y)
	tmp = 0
	if y <= -1.25e+62:
		tmp = t_0
	elif y <= -4.9e+31:
		tmp = x
	elif y <= -1.7e-28:
		tmp = -y * (x * x)
	elif y <= 7.5e-38:
		tmp = x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x * Float64(-y)))
	tmp = 0.0
	if (y <= -1.25e+62)
		tmp = t_0;
	elseif (y <= -4.9e+31)
		tmp = x;
	elseif (y <= -1.7e-28)
		tmp = Float64(Float64(-y) * Float64(x * x));
	elseif (y <= 7.5e-38)
		tmp = x;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (x * -y);
	tmp = 0.0;
	if (y <= -1.25e+62)
		tmp = t_0;
	elseif (y <= -4.9e+31)
		tmp = x;
	elseif (y <= -1.7e-28)
		tmp = -y * (x * x);
	elseif (y <= 7.5e-38)
		tmp = x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x * (-y)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.25e+62], t$95$0, If[LessEqual[y, -4.9e+31], x, If[LessEqual[y, -1.7e-28], N[((-y) * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.5e-38], x, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.25000000000000007e62 or 7.5e-38 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 72.6%

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

      1. mul-1-neg 72.6%

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

      2. unpow2 72.6%

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

      3. *-commutative 72.6%

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

      4. associate-*r* 78.6%

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

      5. distribute-rgt-neg-in 78.6%

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

      6. distribute-rgt-neg-in 78.6%

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

   4. Simplified 78.6%

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

   if -1.25000000000000007e62 < y < -4.89999999999999996e31 or -1.7e-28 < y < 7.5e-38

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 81.8%

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

   if -4.89999999999999996e31 < y < -1.7e-28

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. mul-1-neg 99.9%

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

      3. unpow2 99.9%

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

      4. sub-neg 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*r* 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in x around inf 72.7%

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

      1. mul-1-neg 72.7%

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

      2. unpow2 72.7%

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

      3. *-commutative 72.7%

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

      4. distribute-rgt-neg-in 72.7%

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

   7. Simplified 72.7%

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(-y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+62}:\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq -4.9 \cdot 10^{+31}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-28}:\\ \;\;\;\;\left(-y\right) \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-38}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \end{array} \]

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: 50.0% 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 50.5%

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

3. Final simplification 50.5%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023240 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))