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

Percentage Accurate: 99.9% → 99.9%

Time: 6.1s

Alternatives: 5

Speedup: 1.0×

• 25.1% 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. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg 99.9%

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

   2. distribute-rgt-in 99.9%

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

   3. *-un-lft-identity 99.9%

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

   4. distribute-rgt-neg-in 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

   \[\leadsto x - x \cdot \left(x \cdot y\right) \]
6. Add Preprocessing

Alternative 2: 73.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+115} \lor \neg \left(x \leq -5.6 \cdot 10^{+60}\right) \land \left(x \leq -1.3 \cdot 10^{-50} \lor \neg \left(x \leq 9 \cdot 10^{-53}\right)\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.6e+115)
         (and (not (<= x -5.6e+60)) (or (<= x -1.3e-50) (not (<= x 9e-53)))))
   (* x (* x (- y)))
   x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.6e+115) || (!(x <= -5.6e+60) && ((x <= -1.3e-50) || !(x <= 9e-53)))) {
		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 <= (-2.6d+115)) .or. (.not. (x <= (-5.6d+60))) .and. (x <= (-1.3d-50)) .or. (.not. (x <= 9d-53))) 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 <= -2.6e+115) || (!(x <= -5.6e+60) && ((x <= -1.3e-50) || !(x <= 9e-53)))) {
		tmp = x * (x * -y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.6e+115) or (not (x <= -5.6e+60) and ((x <= -1.3e-50) or not (x <= 9e-53))):
		tmp = x * (x * -y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.6e+115) || (!(x <= -5.6e+60) && ((x <= -1.3e-50) || !(x <= 9e-53))))
		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 <= -2.6e+115) || (~((x <= -5.6e+60)) && ((x <= -1.3e-50) || ~((x <= 9e-53)))))
		tmp = x * (x * -y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.6e+115], And[N[Not[LessEqual[x, -5.6e+60]], $MachinePrecision], Or[LessEqual[x, -1.3e-50], N[Not[LessEqual[x, 9e-53]], $MachinePrecision]]]], N[(x * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -2.6e115 or -5.6e60 < x < -1.3000000000000001e-50 or 8.9999999999999997e-53 < x

   1. Initial program 99.8%

      \[x \cdot \left(1 - x \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 78.9%

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

      1. mul-1-neg 78.9%

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

      2. *-commutative 78.9%

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

      3. distribute-rgt-neg-in 78.9%

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

   5. Simplified 78.9%

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

   if -2.6e115 < x < -5.6e60 or -1.3000000000000001e-50 < x < 8.9999999999999997e-53

   1. Initial program 100.0%

      \[x \cdot \left(1 - x \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 87.4%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+115} \lor \neg \left(x \leq -5.6 \cdot 10^{+60}\right) \land \left(x \leq -1.3 \cdot 10^{-50} \lor \neg \left(x \leq 9 \cdot 10^{-53}\right)\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{+115}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{+60}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-54}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-x\right)\right)\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-52}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* x (- y)))))
   (if (<= x -2.6e+115)
     t_0
     (if (<= x -7.5e+60)
       x
       (if (<= x -2e-54) (* y (* x (- x))) (if (<= x 1.25e-52) x t_0))))))

C

double code(double x, double y) {
	double t_0 = x * (x * -y);
	double tmp;
	if (x <= -2.6e+115) {
		tmp = t_0;
	} else if (x <= -7.5e+60) {
		tmp = x;
	} else if (x <= -2e-54) {
		tmp = y * (x * -x);
	} else if (x <= 1.25e-52) {
		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 (x <= (-2.6d+115)) then
        tmp = t_0
    else if (x <= (-7.5d+60)) then
        tmp = x
    else if (x <= (-2d-54)) then
        tmp = y * (x * -x)
    else if (x <= 1.25d-52) 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 (x <= -2.6e+115) {
		tmp = t_0;
	} else if (x <= -7.5e+60) {
		tmp = x;
	} else if (x <= -2e-54) {
		tmp = y * (x * -x);
	} else if (x <= 1.25e-52) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (x * -y)
	tmp = 0
	if x <= -2.6e+115:
		tmp = t_0
	elif x <= -7.5e+60:
		tmp = x
	elif x <= -2e-54:
		tmp = y * (x * -x)
	elif x <= 1.25e-52:
		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 (x <= -2.6e+115)
		tmp = t_0;
	elseif (x <= -7.5e+60)
		tmp = x;
	elseif (x <= -2e-54)
		tmp = Float64(y * Float64(x * Float64(-x)));
	elseif (x <= 1.25e-52)
		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 (x <= -2.6e+115)
		tmp = t_0;
	elseif (x <= -7.5e+60)
		tmp = x;
	elseif (x <= -2e-54)
		tmp = y * (x * -x);
	elseif (x <= 1.25e-52)
		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[x, -2.6e+115], t$95$0, If[LessEqual[x, -7.5e+60], x, If[LessEqual[x, -2e-54], N[(y * N[(x * (-x)), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.25e-52], x, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6e115 or 1.25e-52 < x

   1. Initial program 99.9%

      \[x \cdot \left(1 - x \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 79.4%

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

      1. mul-1-neg 79.4%

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

      2. *-commutative 79.4%

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

      3. distribute-rgt-neg-in 79.4%

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

   5. Simplified 79.4%

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

   if -2.6e115 < x < -7.5e60 or -2.0000000000000001e-54 < x < 1.25e-52

   1. Initial program 100.0%

      \[x \cdot \left(1 - x \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 87.4%

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

   if -7.5e60 < x < -2.0000000000000001e-54

   1. Initial program 99.7%

      \[x \cdot \left(1 - x \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 75.9%

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

      1. associate-*r* 75.9%

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

      2. mul-1-neg 75.9%

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

   5. Simplified 75.9%

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

      1. unpow2 75.9%

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

      2. distribute-lft-neg-in 75.9%

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

   7. Applied egg-rr 75.9%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+115}:\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{+60}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-54}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-x\right)\right)\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-52}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \end{array} \]
5. Add Preprocessing

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. Add Preprocessing
3. Add Preprocessing

Alternative 5: 50.3% 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. Add Preprocessing

3. Taylor expanded in x around 0 51.5%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))