Numeric.Log:$clog1p from log-domain-0.10.2.1, A

Percentage Accurate: 100.0% → 100.0%

Time: 6.6s

Alternatives: 7

Speedup: 1.5×

• 43.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot 2 + x \cdot x\right) + y \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (+ (* x 2.0) (* x x)) (* y y)))

C

double code(double x, double y) {
	return ((x * 2.0) + (x * x)) + (y * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * 2.0d0) + (x * x)) + (y * y)
end function

Java

public static double code(double x, double y) {
	return ((x * 2.0) + (x * x)) + (y * y);
}

Python

def code(x, y):
	return ((x * 2.0) + (x * x)) + (y * y)

Julia

function code(x, y)
	return Float64(Float64(Float64(x * 2.0) + Float64(x * x)) + Float64(y * y))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * 2.0) + (x * x)) + (y * y);
end

Wolfram

code[x_, y_] := N[(N[(N[(x * 2.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot 2 + x \cdot x\right) + y \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (+ (* x 2.0) (* x x)) (* y y)))

C

double code(double x, double y) {
	return ((x * 2.0) + (x * x)) + (y * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * 2.0d0) + (x * x)) + (y * y)
end function

Java

public static double code(double x, double y) {
	return ((x * 2.0) + (x * x)) + (y * y);
}

Python

def code(x, y):
	return ((x * 2.0) + (x * x)) + (y * y)

Julia

function code(x, y)
	return Float64(Float64(Float64(x * 2.0) + Float64(x * x)) + Float64(y * y))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * 2.0) + (x * x)) + (y * y);
end

Wolfram

code[x_, y_] := N[(N[(N[(x * 2.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, x, \mathsf{fma}\left(y, y, 2 \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma x x (fma y y (* 2.0 x))))

C

double code(double x, double y) {
	return fma(x, x, fma(y, y, (2.0 * x)));
}

Julia

function code(x, y)
	return fma(x, x, fma(y, y, Float64(2.0 * x)))
end

Wolfram

code[x_, y_] := N[(x * x + N[(y * y + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, x \cdot 2 + y \cdot y\right)} \]

   7. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{y \cdot y + x \cdot 2}\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{y \cdot y} + x \cdot 2\right) \]

   9. lower-fma.f64 100.0

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \mathsf{fma}\left(y, y, 2 \cdot x\right)\right)} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x + 2 \cdot x \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(y, y, 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (* x x) (* 2.0 x)) 2e-7) (fma y y (* 2.0 x)) (fma x x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (((x * x) + (2.0 * x)) <= 2e-7) {
		tmp = fma(y, y, (2.0 * x));
	} else {
		tmp = fma(x, x, (y * y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * x) + Float64(2.0 * x)) <= 2e-7)
		tmp = fma(y, y, Float64(2.0 * x));
	else
		tmp = fma(x, x, Float64(y * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], 2e-7], N[(y * y + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x)) < 1.9999999999999999e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot x + {y}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. metadata-eval N/A

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

      4. lft-mult-inverse N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. lft-mult-inverse N/A

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

      17. *-commutative N/A

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

      18. metadata-eval N/A

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

      19. lower-*.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 1.9999999999999999e-7 < (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x))

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, x \cdot 2 + y \cdot y\right)} \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{y \cdot y + x \cdot 2}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{y \cdot y} + x \cdot 2\right) \]

      9. lower-fma.f64 100.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 98.6

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

   7. Applied rewrites 98.6%

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{y \cdot y}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x + 2 \cdot x \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(y, y, 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ y \cdot y + \left(2 \cdot x + x \cdot x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (* y y) (+ (* 2.0 x) (* x x))))

C

double code(double x, double y) {
	return (y * y) + ((2.0 * x) + (x * x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (y * y) + ((2.0d0 * x) + (x * x))
end function

Java

public static double code(double x, double y) {
	return (y * y) + ((2.0 * x) + (x * x));
}

Python

def code(x, y):
	return (y * y) + ((2.0 * x) + (x * x))

Julia

function code(x, y)
	return Float64(Float64(y * y) + Float64(Float64(2.0 * x) + Float64(x * x)))
end

MATLAB

function tmp = code(x, y)
	tmp = (y * y) + ((2.0 * x) + (x * x));
end

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] + N[(N[(2.0 * x), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024230 
(FPCore (x y)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, A"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* y y) (+ (* 2 x) (* x x))))

  (+ (+ (* x 2.0) (* x x)) (* y y)))