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

Percentage Accurate: 99.9% → 99.9%

Time: 5.5s

Alternatives: 5

Speedup: 1.0×

• 24.8% 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

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

FPCore

(FPCore (x y) :precision binary64 (fma (* (- x) y) x x))

C

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

Julia

function code(x, y)
	return fma(Float64(Float64(-x) * y), x, x)
end

Wolfram

code[x_, y_] := N[(N[((-x) * y), $MachinePrecision] * x + x), $MachinePrecision]

Derivation

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. +-commutative N/A

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

   6. distribute-rgt-in N/A

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

   7. *-lft-identity N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-neg.f64 99.9

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

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-x\right) \cdot y, x, x\right)} \]
5. Add Preprocessing

Alternative 2: 80.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -0.0005 \lor \neg \left(t\_0 \leq 5 \cdot 10^{+198}\right):\\ \;\;\;\;x \cdot \left(\left(-x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 (* x y)))))
   (if (or (<= t_0 -0.0005) (not (<= t_0 5e+198)))
     (* 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 <= -0.0005) || !(t_0 <= 5e+198)) {
		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 <= (-0.0005d0)) .or. (.not. (t_0 <= 5d+198))) 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 <= -0.0005) || !(t_0 <= 5e+198)) {
		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 <= -0.0005) or not (t_0 <= 5e+198):
		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 <= -0.0005) || !(t_0 <= 5e+198))
		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 <= -0.0005) || ~((t_0 <= 5e+198)))
		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, -0.0005], N[Not[LessEqual[t$95$0, 5e+198]], $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))) < -5.0000000000000001e-4 or 5.00000000000000049e198 < (*.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 90.5

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

   5. Applied rewrites 90.5%

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

   if -5.0000000000000001e-4 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < 5.00000000000000049e198

   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 80.9%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(1 - x \cdot y\right) \leq -0.0005 \lor \neg \left(x \cdot \left(1 - x \cdot y\right) \leq 5 \cdot 10^{+198}\right):\\ \;\;\;\;x \cdot \left(\left(-x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -0.0005 \lor \neg \left(t\_0 \leq 5 \cdot 10^{+198}\right):\\ \;\;\;\;\left(\left(-x\right) \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 (* x y)))))
   (if (or (<= t_0 -0.0005) (not (<= t_0 5e+198)))
     (* (* (- 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 <= -0.0005) || !(t_0 <= 5e+198)) {
		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 <= (-0.0005d0)) .or. (.not. (t_0 <= 5d+198))) 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 <= -0.0005) || !(t_0 <= 5e+198)) {
		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 <= -0.0005) or not (t_0 <= 5e+198):
		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 <= -0.0005) || !(t_0 <= 5e+198))
		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 <= -0.0005) || ~((t_0 <= 5e+198)))
		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, -0.0005], N[Not[LessEqual[t$95$0, 5e+198]], $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))) < -5.0000000000000001e-4 or 5.00000000000000049e198 < (*.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. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 58.4

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

   4. Applied rewrites 58.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 58.4

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

   6. Applied rewrites 58.4%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \left({x}^{2} \cdot y\right)} \]
   8. 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 84.3

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

   9. Applied rewrites 84.3%

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

   if -5.0000000000000001e-4 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < 5.00000000000000049e198

   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 80.9%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(1 - x \cdot y\right) \leq -0.0005 \lor \neg \left(x \cdot \left(1 - x \cdot y\right) \leq 5 \cdot 10^{+198}\right):\\ \;\;\;\;\left(\left(-x\right) \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]
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: 51.2% 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 47.6%

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

Reproduce

herbie shell --seed 2024320 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))