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

Percentage Accurate: 99.9% → 99.9%

Time: 7.8s

Alternatives: 4

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. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. distribute-lft-out-- N/A

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

   4. *-rgt-identity N/A

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

   5. associate-*l* N/A

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

   6. unpow2 N/A

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

   7. lower--.f64 N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 2: 82.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - x \cdot y\right)\\ t_1 := x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+78}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 (* x y)))) (t_1 (* x (* x (- y)))))
   (if (<= t_0 -2e+50) t_1 (if (<= t_0 1e+78) (* x 1.0) t_1))))

C

double code(double x, double y) {
	double t_0 = x * (1.0 - (x * y));
	double t_1 = x * (x * -y);
	double tmp;
	if (t_0 <= -2e+50) {
		tmp = t_1;
	} else if (t_0 <= 1e+78) {
		tmp = x * 1.0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = x * (1.0d0 - (x * y))
    t_1 = x * (x * -y)
    if (t_0 <= (-2d+50)) then
        tmp = t_1
    else if (t_0 <= 1d+78) then
        tmp = x * 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (1.0 - (x * y));
	double t_1 = x * (x * -y);
	double tmp;
	if (t_0 <= -2e+50) {
		tmp = t_1;
	} else if (t_0 <= 1e+78) {
		tmp = x * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (1.0 - (x * y))
	t_1 = x * (x * -y)
	tmp = 0
	if t_0 <= -2e+50:
		tmp = t_1
	elif t_0 <= 1e+78:
		tmp = x * 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(1.0 - Float64(x * y)))
	t_1 = Float64(x * Float64(x * Float64(-y)))
	tmp = 0.0
	if (t_0 <= -2e+50)
		tmp = t_1;
	elseif (t_0 <= 1e+78)
		tmp = Float64(x * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (1.0 - (x * y));
	t_1 = x * (x * -y);
	tmp = 0.0;
	if (t_0 <= -2e+50)
		tmp = t_1;
	elseif (t_0 <= 1e+78)
		tmp = x * 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(1.0 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * (-y)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+50], t$95$1, If[LessEqual[t$95$0, 1e+78], N[(x * 1.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < -2.0000000000000002e50 or 1.00000000000000001e78 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y)))

   1. Initial program 99.8%

      \[x \cdot \left(1 - x \cdot y\right) \]
   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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 88.4

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

   5. Applied rewrites 88.4%

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

   if -2.0000000000000002e50 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < 1.00000000000000001e78

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 88.0%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(1 - x \cdot y\right) \leq -2 \cdot 10^{+50}:\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;x \cdot \left(1 - x \cdot y\right) \leq 10^{+78}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - x \cdot y\right)\\ t_1 := \left(x \cdot x\right) \cdot \left(-y\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+78}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 (* x y)))) (t_1 (* (* x x) (- y))))
   (if (<= t_0 -2e+50) t_1 (if (<= t_0 1e+78) (* x 1.0) t_1))))

C

double code(double x, double y) {
	double t_0 = x * (1.0 - (x * y));
	double t_1 = (x * x) * -y;
	double tmp;
	if (t_0 <= -2e+50) {
		tmp = t_1;
	} else if (t_0 <= 1e+78) {
		tmp = x * 1.0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = x * (1.0d0 - (x * y))
    t_1 = (x * x) * -y
    if (t_0 <= (-2d+50)) then
        tmp = t_1
    else if (t_0 <= 1d+78) then
        tmp = x * 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (1.0 - (x * y));
	double t_1 = (x * x) * -y;
	double tmp;
	if (t_0 <= -2e+50) {
		tmp = t_1;
	} else if (t_0 <= 1e+78) {
		tmp = x * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (1.0 - (x * y))
	t_1 = (x * x) * -y
	tmp = 0
	if t_0 <= -2e+50:
		tmp = t_1
	elif t_0 <= 1e+78:
		tmp = x * 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(1.0 - Float64(x * y)))
	t_1 = Float64(Float64(x * x) * Float64(-y))
	tmp = 0.0
	if (t_0 <= -2e+50)
		tmp = t_1;
	elseif (t_0 <= 1e+78)
		tmp = Float64(x * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (1.0 - (x * y));
	t_1 = (x * x) * -y;
	tmp = 0.0;
	if (t_0 <= -2e+50)
		tmp = t_1;
	elseif (t_0 <= 1e+78)
		tmp = x * 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(1.0 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * (-y)), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+50], t$95$1, If[LessEqual[t$95$0, 1e+78], N[(x * 1.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < -2.0000000000000002e50 or 1.00000000000000001e78 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y)))

   1. Initial program 99.8%

      \[x \cdot \left(1 - x \cdot y\right) \]
   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. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 40.9

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

   7. Applied rewrites 40.9%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{neg}\left(x \cdot \left(x \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\right)}{\color{blue}{x \cdot y}} \]
   9. Step-by-step derivation

      1. lower-*.f64 40.9

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

   10. Applied rewrites 40.9%

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

   11. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. mul-1-neg N/A

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

       5. unpow2 N/A

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

       6. distribute-rgt-neg-in N/A

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

       7. mul-1-neg N/A

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

       8. lower-*.f64 N/A

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

       9. mul-1-neg N/A

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

       10. lower-neg.f64 82.4

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

   13. Applied rewrites 82.4%

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

   if -2.0000000000000002e50 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < 1.00000000000000001e78

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 88.0%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(1 - x \cdot y\right) \leq -2 \cdot 10^{+50}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(-y\right)\\ \mathbf{elif}\;x \cdot \left(1 - x \cdot y\right) \leq 10^{+78}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.6% 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 48.2%

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

Reproduce

herbie shell --seed 2024231 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))