Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 94.0% → 99.9%

Time: 3.6s

Alternatives: 4

Speedup: 1.5×

• 43.1% 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: 94.0% 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: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.3%

   \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
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: 67.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-110}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.5e-110) (* x x) (* (fma 2.0 x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.5e-110) {
		tmp = x * x;
	} else {
		tmp = fma(2.0, x, y) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.5e-110)
		tmp = Float64(x * x);
	else
		tmp = Float64(fma(2.0, x, y) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.5e-110], N[(x * x), $MachinePrecision], N[(N[(2.0 * x + y), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.5000000000000001e-110

   1. Initial program 95.2%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   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 42.2

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

   5. Applied rewrites 42.2%

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

   6. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 67.0

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

   8. Applied rewrites 67.0%

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

   if -4.5000000000000001e-110 < x

   1. Initial program 95.3%

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

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

      2. *-inverses N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 69.3%

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

Alternative 3: 67.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-124}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 6.8e-124) (* x x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.8e-124) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 6.8d-124) then
        tmp = x * x
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6.8e-124) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.8e-124:
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.8e-124)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.8e-124)
		tmp = x * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.8e-124], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.8000000000000001e-124

   1. Initial program 94.8%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   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 54.9

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

   5. Applied rewrites 54.9%

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

   6. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.1

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

   8. Applied rewrites 61.1%

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

   if 6.8000000000000001e-124 < y

   1. Initial program 96.0%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 67.4

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

   5. Applied rewrites 67.4%

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

Alternative 4: 57.7% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ x \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.3%

   \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
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 59.9

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

5. Applied rewrites 59.9%

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

6. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 54.1

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

8. Applied rewrites 54.1%

   \[\leadsto \color{blue}{x \cdot x} \]
9. 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 2024318 
(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)))