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

Percentage Accurate: 99.9% → 99.9%

Time: 5.9s

Alternatives: 4

Speedup: 1.0×

• 25.5% 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

\[\begin{array}{l} \\ x \cdot \left(1 - x \cdot y\right) \end{array} \]

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 2: 81.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+140} \lor \neg \left(t\_0 \leq 0.004\right):\\ \;\;\;\;x \cdot \left(\left(-x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 (* x y)))))
   (if (or (<= t_0 -5e+140) (not (<= t_0 0.004)))
     (* x (* (- x) y))
     (* x 1.0))))

C

double code(double x, double y) {
	double t_0 = x * (1.0 - (x * y));
	double tmp;
	if ((t_0 <= -5e+140) || !(t_0 <= 0.004)) {
		tmp = x * (-x * y);
	} else {
		tmp = x * 1.0;
	}
	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 * (1.0d0 - (x * y))
    if ((t_0 <= (-5d+140)) .or. (.not. (t_0 <= 0.004d0))) then
        tmp = x * (-x * y)
    else
        tmp = x * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (1.0 - (x * y));
	double tmp;
	if ((t_0 <= -5e+140) || !(t_0 <= 0.004)) {
		tmp = x * (-x * y);
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (1.0 - (x * y))
	tmp = 0
	if (t_0 <= -5e+140) or not (t_0 <= 0.004):
		tmp = x * (-x * y)
	else:
		tmp = x * 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(1.0 - Float64(x * y)))
	tmp = 0.0
	if ((t_0 <= -5e+140) || !(t_0 <= 0.004))
		tmp = Float64(x * Float64(Float64(-x) * y));
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (1.0 - (x * y));
	tmp = 0.0;
	if ((t_0 <= -5e+140) || ~((t_0 <= 0.004)))
		tmp = x * (-x * y);
	else
		tmp = x * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(1.0 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e+140], N[Not[LessEqual[t$95$0, 0.004]], $MachinePrecision]], N[(x * N[((-x) * y), $MachinePrecision]), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < -5.00000000000000008e140 or 0.0040000000000000001 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y)))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 88.6

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

   5. Applied rewrites 88.6%

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

   if -5.00000000000000008e140 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < 0.0040000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 82.8%

      \[\leadsto x \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(1 - x \cdot y\right) \leq -5 \cdot 10^{+140} \lor \neg \left(x \cdot \left(1 - x \cdot y\right) \leq 0.004\right):\\ \;\;\;\;x \cdot \left(\left(-x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+140} \lor \neg \left(t\_0 \leq 0.004\right):\\ \;\;\;\;\left(\left(-x\right) \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 (* x y)))))
   (if (or (<= t_0 -5e+140) (not (<= t_0 0.004)))
     (* (* (- x) x) y)
     (* x 1.0))))

C

double code(double x, double y) {
	double t_0 = x * (1.0 - (x * y));
	double tmp;
	if ((t_0 <= -5e+140) || !(t_0 <= 0.004)) {
		tmp = (-x * x) * y;
	} else {
		tmp = x * 1.0;
	}
	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 * (1.0d0 - (x * y))
    if ((t_0 <= (-5d+140)) .or. (.not. (t_0 <= 0.004d0))) then
        tmp = (-x * x) * y
    else
        tmp = x * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (1.0 - (x * y));
	double tmp;
	if ((t_0 <= -5e+140) || !(t_0 <= 0.004)) {
		tmp = (-x * x) * y;
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (1.0 - (x * y))
	tmp = 0
	if (t_0 <= -5e+140) or not (t_0 <= 0.004):
		tmp = (-x * x) * y
	else:
		tmp = x * 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(1.0 - Float64(x * y)))
	tmp = 0.0
	if ((t_0 <= -5e+140) || !(t_0 <= 0.004))
		tmp = Float64(Float64(Float64(-x) * x) * y);
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (1.0 - (x * y));
	tmp = 0.0;
	if ((t_0 <= -5e+140) || ~((t_0 <= 0.004)))
		tmp = (-x * x) * y;
	else
		tmp = x * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(1.0 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e+140], N[Not[LessEqual[t$95$0, 0.004]], $MachinePrecision]], N[(N[((-x) * x), $MachinePrecision] * y), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < -5.00000000000000008e140 or 0.0040000000000000001 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y)))

   1. Initial program 99.9%

      \[x \cdot \left(1 - x \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. distribute-rgt-in N/A

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

      6. distribute-lft-neg-out N/A

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

      7. lift-*.f64 N/A

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

      8. fp-cancel-sub-sign N/A

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

      9. *-lft-identity N/A

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

      10. lower--.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 89.7

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

   4. Applied rewrites 89.7%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 84.8

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

   7. Applied rewrites 84.8%

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

   if -5.00000000000000008e140 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < 0.0040000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 82.8%

      \[\leadsto x \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(1 - x \cdot y\right) \leq -5 \cdot 10^{+140} \lor \neg \left(x \cdot \left(1 - x \cdot y\right) \leq 0.004\right):\\ \;\;\;\;\left(\left(-x\right) \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ x \cdot 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x 1.0))

C

double code(double x, double y) {
	return x * 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * 1.0d0
end function

Java

public static double code(double x, double y) {
	return x * 1.0;
}

Python

def code(x, y):
	return x * 1.0

Julia

function code(x, y)
	return Float64(x * 1.0)
end

MATLAB

function tmp = code(x, y)
	tmp = x * 1.0;
end

Wolfram

code[x_, y_] := N[(x * 1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

   \[\leadsto x \cdot \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 45.1%

   \[\leadsto x \cdot \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024338 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))