Examples.Basics.BasicTests:f3 from sbv-4.4

Percentage Accurate: 100.0% → 99.1%

Time: 4.4s

Alternatives: 4

Speedup: 1.0×

• 44.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-158}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(2, x, y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= (+ x y) -1e-158) (* x x) (* y (fma 2.0 x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if ((x + y) <= -1e-158) {
		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 (Float64(x + y) <= -1e-158)
		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[N[(x + y), $MachinePrecision], -1e-158], N[(x * x), $MachinePrecision], N[(y * N[(2.0 * x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -1.00000000000000006e-158

   1. Initial program 100.0%

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

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

   5. Applied rewrites 100.0%

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

   if -1.00000000000000006e-158 < (+.f64 x y)

   1. Initial program 100.0%

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

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

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

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

      9. distribute-lft1-in N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. *-commutative N/A

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

      18. associate-*r/ N/A

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

      19. associate-*l/ N/A

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

      20. associate-/l* N/A

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

      21. *-inverses N/A

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

      22. *-rgt-identity N/A

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

      23. *-rgt-identity N/A

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

      24. +-commutative N/A

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

      25. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 2: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-158}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= (+ x y) -1e-158) (* x x) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if ((x + y) <= -1e-158) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x + y) <= (-1d-158)) then
        tmp = x * x
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if ((x + y) <= -1e-158) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if (x + y) <= -1e-158:
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (Float64(x + y) <= -1e-158)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x + y) <= -1e-158)
		tmp = x * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[N[(x + y), $MachinePrecision], -1e-158], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -1.00000000000000006e-158

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(x + y\right) \]
   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 98.8

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

   5. Applied rewrites 98.8%

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

   if -1.00000000000000006e-158 < (+.f64 x y)

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(x + y\right) \]
   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 98.9

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

   5. Applied rewrites 98.9%

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

Developer Target 1: 93.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024229 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f3 from sbv-4.4"
  :precision binary64

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

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