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

Percentage Accurate: 99.9% → 99.9%

Time: 8.6s

Alternatives: 3

Speedup: 1.0×

• 24.6% 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: 82.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - x \cdot y\right)\\ t_1 := x \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+69}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+84}:\\ \;\;\;\;-\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 (* x y)))) (t_1 (* x (* y (- x)))))
   (if (<= t_0 -5e+69) t_1 (if (<= t_0 1e+84) (- (- x)) t_1))))

C

double code(double x, double y) {
	double t_0 = x * (1.0 - (x * y));
	double t_1 = x * (y * -x);
	double tmp;
	if (t_0 <= -5e+69) {
		tmp = t_1;
	} else if (t_0 <= 1e+84) {
		tmp = -(-x);
	} 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 * (y * -x)
    if (t_0 <= (-5d+69)) then
        tmp = t_1
    else if (t_0 <= 1d+84) then
        tmp = -(-x)
    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 * (y * -x);
	double tmp;
	if (t_0 <= -5e+69) {
		tmp = t_1;
	} else if (t_0 <= 1e+84) {
		tmp = -(-x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (1.0 - (x * y))
	t_1 = x * (y * -x)
	tmp = 0
	if t_0 <= -5e+69:
		tmp = t_1
	elif t_0 <= 1e+84:
		tmp = -(-x)
	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(y * Float64(-x)))
	tmp = 0.0
	if (t_0 <= -5e+69)
		tmp = t_1;
	elseif (t_0 <= 1e+84)
		tmp = Float64(-Float64(-x));
	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 * (y * -x);
	tmp = 0.0;
	if (t_0 <= -5e+69)
		tmp = t_1;
	elseif (t_0 <= 1e+84)
		tmp = -(-x);
	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[(y * (-x)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+69], t$95$1, If[LessEqual[t$95$0, 1e+84], (-(-x)), t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < -5.00000000000000036e69 or 1.00000000000000006e84 < (*.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 90.2

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

   5. Applied rewrites 90.2%

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

   if -5.00000000000000036e69 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < 1.00000000000000006e84

   1. Initial program 99.9%

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

   4. Applied rewrites 83.4%

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

   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-*r* 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 7.3

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

   7. Applied rewrites 7.3%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

   9. Applied rewrites 17.7%

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

   10. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 83.8

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

   12. Applied rewrites 83.8%

       \[\leadsto -\color{blue}{\left(-x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 51.2% accurate, 2.8× speedup

Math

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

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 Float64(-Float64(-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

4. Applied rewrites 67.3%

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

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-*r* 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 22.6

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

7. Applied rewrites 22.6%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. distribute-frac-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

9. Applied rewrites 33.8%

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

10. Taylor expanded in x around 0

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 50.2

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

12. Applied rewrites 50.2%

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

Reproduce

herbie shell --seed 2024234 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))