Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 94.1% → 100.0%

Time: 3.3s

Alternatives: 4

Speedup: 1.5×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (fma (fma y 2.0 x) x (* y y)))

C

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

Julia

x, y = sort([x, y])
function code(x, y)
	return fma(fma(y, 2.0, x), x, Float64(y * y))
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(y * 2.0 + x), $MachinePrecision] * x + N[(y * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.0%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. distribute-lft-out N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 87.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(y, 2, x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.5e-33) (* (fma y 2.0 x) x) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.5e-33) {
		tmp = fma(y, 2.0, x) * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.5e-33)
		tmp = Float64(fma(y, 2.0, x) * x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.5e-33], N[(N[(y * 2.0 + x), $MachinePrecision] * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.4999999999999999e-33

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. *-inverses N/A

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

      13. *-rgt-identity N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. metadata-eval N/A

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

      18. lower-fma.f64 87.1

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

   5. Applied rewrites 87.1%

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

   if 3.4999999999999999e-33 < y

   1. Initial program 87.6%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

         \[\leadsto \color{blue}{y \cdot y} \]

      2. lower-*.f64 87.9

         \[\leadsto \color{blue}{y \cdot y} \]

   5. Applied rewrites 87.9%

      \[\leadsto \color{blue}{y \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Developer Target 1: 94.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024230 
(FPCore (x y)
  :name "Examples.Basics.ProofTests:f4 from sbv-4.4"
  :precision binary64

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

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