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

Percentage Accurate: 99.9% → 99.9%

Time: 7.4s

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

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

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. sub-neg N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-in N/A

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

   6. *-lft-identity N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

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

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

   10. lower-*.f64 N/A

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

   11. lower-neg.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 82.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - x \cdot y\right)\\ t_1 := x \cdot \left(-x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+182}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+110}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 (* x y)))) (t_1 (* x (- (* x y)))))
   (if (<= t_0 -5e+182) t_1 (if (<= t_0 1e+110) (* 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 <= -5e+182) {
		tmp = t_1;
	} else if (t_0 <= 1e+110) {
		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 <= (-5d+182)) then
        tmp = t_1
    else if (t_0 <= 1d+110) 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 <= -5e+182) {
		tmp = t_1;
	} else if (t_0 <= 1e+110) {
		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 <= -5e+182:
		tmp = t_1
	elif t_0 <= 1e+110:
		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(-Float64(x * y)))
	tmp = 0.0
	if (t_0 <= -5e+182)
		tmp = t_1;
	elseif (t_0 <= 1e+110)
		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 <= -5e+182)
		tmp = t_1;
	elseif (t_0 <= 1e+110)
		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, -5e+182], t$95$1, If[LessEqual[t$95$0, 1e+110], 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))) < -4.99999999999999973e182 or 1e110 < (*.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. *-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 92.7

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

   5. Applied rewrites 92.7%

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

   if -4.99999999999999973e182 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < 1e110

   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 80.8%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(1 - x \cdot y\right) \leq -5 \cdot 10^{+182}:\\ \;\;\;\;x \cdot \left(-x \cdot y\right)\\ \mathbf{elif}\;x \cdot \left(1 - x \cdot y\right) \leq 10^{+110}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(x * N[((-y) * x + 1.0), $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. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

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

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

   7. lower-fma.f64 N/A

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

   8. lower-neg.f64 99.9

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

4. Applied rewrites 99.9%

   \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(-y, x, 1\right)} \]
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: 52.1% 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 55.1%

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

Reproduce

herbie shell --seed 2024233 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))