Examples.Basics.BasicTests:f3 from sbv-4.4

Percentage Accurate: 100.0% → 100.0%

Time: 2.9s

Alternatives: 4

Speedup: 1.0×

• 44.0% 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: 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]

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 68.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.7 \cdot 10^{-92}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.7e-92) (* x (+ x (* y 2.0))) (* y (+ y (* x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.7e-92) {
		tmp = x * (x + (y * 2.0));
	} else {
		tmp = y * (y + (x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.7d-92) then
        tmp = x * (x + (y * 2.0d0))
    else
        tmp = y * (y + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.7e-92) {
		tmp = x * (x + (y * 2.0));
	} else {
		tmp = y * (y + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.7e-92:
		tmp = x * (x + (y * 2.0))
	else:
		tmp = y * (y + (x * 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.7e-92)
		tmp = Float64(x * Float64(x + Float64(y * 2.0)));
	else
		tmp = Float64(y * Float64(y + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.7e-92)
		tmp = x * (x + (y * 2.0));
	else
		tmp = y * (y + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.7e-92], N[(x * N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.70000000000000009e-92

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 67.4%

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

      1. *-commutative 67.4%

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

      2. associate-*r* 67.4%

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

      3. *-commutative 67.4%

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

      4. unpow2 67.4%

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

      5. distribute-lft-in 69.0%

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

      6. *-commutative 69.0%

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

      7. fma-define 69.0%

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

   5. Simplified 69.0%

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

   6. Taylor expanded in x around 0 69.0%

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

   if 5.70000000000000009e-92 < 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 74.9%

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

      1. +-commutative 74.9%

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

      2. unpow2 74.9%

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

      3. associate-*r* 74.9%

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

      4. distribute-rgt-in 77.8%

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

   5. Simplified 77.8%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.7 \cdot 10^{-92}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return x * (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 * 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0 56.8%

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

   1. *-commutative 56.8%

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

   2. associate-*r* 56.8%

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

   3. *-commutative 56.8%

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

   4. unpow2 56.8%

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

   5. distribute-lft-in 59.5%

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

   6. *-commutative 59.5%

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

   7. fma-define 59.5%

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

5. Simplified 59.5%

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

6. Taylor expanded in x around 0 59.5%

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

7. Final simplification 59.5%

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

Alternative 4: 15.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0 56.8%

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

   1. *-commutative 56.8%

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

   2. associate-*r* 56.8%

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

   3. *-commutative 56.8%

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

   4. unpow2 56.8%

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

   5. distribute-lft-in 59.5%

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

   6. *-commutative 59.5%

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

   7. fma-define 59.5%

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

5. Simplified 59.5%

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

6. Taylor expanded in x around 0 14.3%

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

   1. associate-*r* 14.3%

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

8. Simplified 14.3%

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

9. Final simplification 14.3%

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

Developer target: 93.7% accurate, 0.5× 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 2024066 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f3 from sbv-4.4"
  :precision binary64

  :alt
  (+ (* x x) (+ (* y y) (* 2.0 (* y x))))

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