Examples.Basics.BasicTests:f3 from sbv-4.4

Percentage Accurate: 100.0% → 100.0%

Time: 4.8s

Alternatives: 5

Speedup: 1.0×

• 42.9% 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: 67.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-102} \lor \neg \left(y \leq 6.8 \cdot 10^{-18}\right) \land y \leq 1.35 \cdot 10^{+24}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 1.15e-102) (and (not (<= y 6.8e-18)) (<= y 1.35e+24)))
   (* x (+ x (* y 2.0)))
   (* y (+ y (* x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 1.15e-102) || (!(y <= 6.8e-18) && (y <= 1.35e+24))) {
		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 <= 1.15d-102) .or. (.not. (y <= 6.8d-18)) .and. (y <= 1.35d+24)) 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 <= 1.15e-102) || (!(y <= 6.8e-18) && (y <= 1.35e+24))) {
		tmp = x * (x + (y * 2.0));
	} else {
		tmp = y * (y + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 1.15e-102) or (not (y <= 6.8e-18) and (y <= 1.35e+24)):
		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 <= 1.15e-102) || (!(y <= 6.8e-18) && (y <= 1.35e+24)))
		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 <= 1.15e-102) || (~((y <= 6.8e-18)) && (y <= 1.35e+24)))
		tmp = x * (x + (y * 2.0));
	else
		tmp = y * (y + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 1.15e-102], And[N[Not[LessEqual[y, 6.8e-18]], $MachinePrecision], LessEqual[y, 1.35e+24]]], 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 < 1.14999999999999993e-102 or 6.80000000000000002e-18 < y < 1.35e24

   1. Initial program 100.0%

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

   3. Taylor expanded in y around -inf 65.7%

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

      1. mul-1-neg 65.7%

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

      2. unsub-neg 65.7%

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

      3. mul-1-neg 65.7%

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

      4. unsub-neg 65.7%

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

      5. *-commutative 65.7%

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

      6. unpow2 65.7%

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

      7. associate-/l* 65.7%

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

      8. distribute-lft-out-- 65.7%

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

   5. Simplified 65.7%

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

   6. Taylor expanded in y around 0 67.9%

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

      1. associate-*r* 67.9%

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

      2. *-commutative 67.9%

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

      3. associate-*r* 67.9%

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

      4. unpow2 67.9%

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

      5. distribute-lft-in 69.4%

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

      6. +-commutative 69.4%

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

   8. Simplified 69.4%

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

   if 1.14999999999999993e-102 < y < 6.80000000000000002e-18 or 1.35e24 < 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 82.6%

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

      1. +-commutative 82.6%

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

      2. unpow2 82.6%

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

      3. associate-*r* 82.6%

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

      4. distribute-rgt-in 87.0%

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

   5. Simplified 87.0%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-102} \lor \neg \left(y \leq 6.8 \cdot 10^{-18}\right) \land y \leq 1.35 \cdot 10^{+24}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.2% 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 -inf 74.0%

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

   1. mul-1-neg 74.0%

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

   2. unsub-neg 74.0%

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

   3. mul-1-neg 74.0%

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

   4. unsub-neg 74.0%

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

   5. *-commutative 74.0%

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

   6. unpow2 74.0%

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

   7. associate-/l* 74.0%

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

   8. distribute-lft-out-- 74.0%

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

5. Simplified 74.0%

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

6. Taylor expanded in y around 0 54.6%

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

   1. associate-*r* 54.6%

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

   2. *-commutative 54.6%

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

   3. associate-*r* 54.6%

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

   4. unpow2 54.6%

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

   5. distribute-lft-in 56.5%

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

   6. +-commutative 56.5%

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

8. Simplified 56.5%

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

9. Final simplification 56.5%

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

Alternative 4: 14.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0 50.4%

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

   1. +-commutative 50.4%

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

   2. unpow2 50.4%

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

   3. associate-*r* 50.4%

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

   4. distribute-rgt-in 53.6%

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

5. Simplified 53.6%

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

6. Taylor expanded in y around 0 10.5%

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

7. Final simplification 10.5%

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

Alternative 5: 14.8% 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 x around 0 50.4%

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

   1. +-commutative 50.4%

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

   2. unpow2 50.4%

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

   3. associate-*r* 50.4%

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

   4. distribute-rgt-in 53.6%

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

5. Simplified 53.6%

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

6. Taylor expanded in y around 0 10.5%

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

   1. associate-*r* 10.5%

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

   2. *-commutative 10.5%

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

8. Simplified 10.5%

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

9. Final simplification 10.5%

   \[\leadsto y \cdot \left(x \cdot 2\right) \]
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 2024082 
(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)))