Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 93.9% → 100.0%

Time: 3.5s

Alternatives: 5

Speedup: 1.1×

• 44.0% 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: 93.9% 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, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(y, y, x \cdot \mathsf{fma}\left(2, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2e+154) (fma y y (* x (fma 2.0 y x))) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2e+154) {
		tmp = fma(y, y, (x * fma(2.0, y, x)));
	} else {
		tmp = y * y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2e+154)
		tmp = fma(y, y, Float64(x * fma(2.0, y, 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, 2e+154], N[(y * y + N[(x * N[(2.0 * y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.00000000000000007e154

   1. Initial program 97.4%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 97.4

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. distribute-lft-out N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if 2.00000000000000007e154 < y

   1. Initial program 87.9%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 2: 87.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-95}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(2, x, y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.6e-95) (* x x) (* y (fma 2.0 x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.6e-95) {
		tmp = x * x;
	} else {
		tmp = y * fma(2.0, x, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.6e-95)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * fma(2.0, x, 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[x, -3.6e-95], N[(x * x), $MachinePrecision], N[(y * N[(2.0 * x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.6e-95

   1. Initial program 88.9%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 83.1

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

   5. Applied rewrites 83.1%

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

   if -3.6e-95 < x

   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 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-rgt-identity N/A

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

      5. *-inverses N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

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

      9. *-lft-identity N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-lft1-in N/A

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

      16. associate-*r/ N/A

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

      17. associate-*l/ N/A

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

      18. associate-/l* N/A

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

      19. *-inverses N/A

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

      20. *-rgt-identity N/A

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

      21. lower-fma.f64 93.4

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

   5. Applied rewrites 93.4%

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

Developer Target 1: 94.0% 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 2024228 
(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)))