Expanding a square

Percentage Accurate: 54.4% → 100.0%

Time: 3.5s

Alternatives: 4

Speedup: 1.8×

• 25.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x + 1\right) \cdot \left(x + 1\right) - 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (* (+ x 1.0) (+ x 1.0)) 1.0))

C

double code(double x) {
	return ((x + 1.0) * (x + 1.0)) - 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x + 1.0d0) * (x + 1.0d0)) - 1.0d0
end function

Java

public static double code(double x) {
	return ((x + 1.0) * (x + 1.0)) - 1.0;
}

Python

def code(x):
	return ((x + 1.0) * (x + 1.0)) - 1.0

Julia

function code(x)
	return Float64(Float64(Float64(x + 1.0) * Float64(x + 1.0)) - 1.0)
end

MATLAB

function tmp = code(x)
	tmp = ((x + 1.0) * (x + 1.0)) - 1.0;
end

Wolfram

code[x_] := N[(N[(N[(x + 1.0), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 54.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x + 1\right) \cdot \left(x + 1\right) - 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (* (+ x 1.0) (+ x 1.0)) 1.0))

C

double code(double x) {
	return ((x + 1.0) * (x + 1.0)) - 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x + 1.0d0) * (x + 1.0d0)) - 1.0d0
end function

Java

public static double code(double x) {
	return ((x + 1.0) * (x + 1.0)) - 1.0;
}

Python

def code(x):
	return ((x + 1.0) * (x + 1.0)) - 1.0

Julia

function code(x)
	return Float64(Float64(Float64(x + 1.0) * Float64(x + 1.0)) - 1.0)
end

MATLAB

function tmp = code(x)
	tmp = ((x + 1.0) * (x + 1.0)) - 1.0;
end

Wolfram

code[x_] := N[(N[(N[(x + 1.0), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x + x \cdot \left(x + 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ x (* x (+ x 1.0))))

C

double code(double x) {
	return x + (x * (x + 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x + (x * (x + 1.0d0))
end function

Java

public static double code(double x) {
	return x + (x * (x + 1.0));
}

Python

def code(x):
	return x + (x * (x + 1.0))

Julia

function code(x)
	return Float64(x + Float64(x * Float64(x + 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = x + (x * (x + 1.0));
end

Wolfram

code[x_] := N[(x + N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.5%

   \[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]
2. Step-by-step derivation

   1. difference-of-sqr-1 N/A

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

   2. associate--l+ N/A

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

   3. metadata-eval N/A

      \[\leadsto \left(\left(x + 1\right) + 1\right) \cdot \left(x + 0\right) \]

   4. +-rgt-identity N/A

      \[\leadsto \left(\left(x + 1\right) + 1\right) \cdot x \]

   5. *-lft-identity N/A

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

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(x + 1\right) + 1\right), \color{blue}{\left(1 \cdot x\right)}\right) \]

   7. associate-+l+ N/A

      \[\leadsto \mathsf{*.f64}\left(\left(x + \left(1 + 1\right)\right), \left(\color{blue}{1} \cdot x\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot x + \left(1 + 1\right)\right), \left(1 \cdot x\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(1 \cdot x\right), \left(1 + 1\right)\right), \left(\color{blue}{1} \cdot x\right)\right) \]

   10. *-lft-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(1 + 1\right)\right), \left(1 \cdot x\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), \left(1 \cdot x\right)\right) \]

   12. *-lft-identity 100.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right) \]

3. Simplified 100.0%

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

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

   3. associate-+l+ N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

      \[\leadsto x \cdot \left(x + 1\right) + x \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(x + 1\right)\right), \color{blue}{x}\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x + 1\right)\right), x\right) \]

   8. +-lowering-+.f64 100.0%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), x\right) \]

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

   \[\leadsto x + x \cdot \left(x + 1\right) \]
8. Add Preprocessing

Alternative 2: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.0) (* x x) (if (<= x 2.0) (* x 2.0) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= -2.0) {
		tmp = x * x;
	} else if (x <= 2.0) {
		tmp = x * 2.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

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

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.0) {
		tmp = x * x;
	} else if (x <= 2.0) {
		tmp = x * 2.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.0:
		tmp = x * x
	elif x <= 2.0:
		tmp = x * 2.0
	else:
		tmp = x * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.0)
		tmp = Float64(x * x);
	elseif (x <= 2.0)
		tmp = Float64(x * 2.0);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.0)
		tmp = x * x;
	elseif (x <= 2.0)
		tmp = x * 2.0;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.0], N[(x * x), $MachinePrecision], If[LessEqual[x, 2.0], N[(x * 2.0), $MachinePrecision], N[(x * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2 or 2 < x

   1. Initial program 100.0%

      \[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]
   2. Step-by-step derivation

      1. difference-of-sqr-1 N/A

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

      2. associate--l+ N/A

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

      3. metadata-eval N/A

         \[\leadsto \left(\left(x + 1\right) + 1\right) \cdot \left(x + 0\right) \]

      4. +-rgt-identity N/A

         \[\leadsto \left(\left(x + 1\right) + 1\right) \cdot x \]

      5. *-lft-identity N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x + 1\right) + 1\right), \color{blue}{\left(1 \cdot x\right)}\right) \]

      7. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x + \left(1 + 1\right)\right), \left(\color{blue}{1} \cdot x\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot x + \left(1 + 1\right)\right), \left(1 \cdot x\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(1 \cdot x\right), \left(1 + 1\right)\right), \left(\color{blue}{1} \cdot x\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(1 + 1\right)\right), \left(1 \cdot x\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), \left(1 \cdot x\right)\right) \]

      12. *-lft-identity 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(x + 2\right) \cdot x} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 98.0%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   7. Simplified 98.0%

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

   if -2 < x < 2

   1. Initial program 6.3%

      \[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]
   2. Step-by-step derivation

      1. difference-of-sqr-1 N/A

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

      2. associate--l+ N/A

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

      3. metadata-eval N/A

         \[\leadsto \left(\left(x + 1\right) + 1\right) \cdot \left(x + 0\right) \]

      4. +-rgt-identity N/A

         \[\leadsto \left(\left(x + 1\right) + 1\right) \cdot x \]

      5. *-lft-identity N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x + 1\right) + 1\right), \color{blue}{\left(1 \cdot x\right)}\right) \]

      7. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x + \left(1 + 1\right)\right), \left(\color{blue}{1} \cdot x\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot x + \left(1 + 1\right)\right), \left(1 \cdot x\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(1 \cdot x\right), \left(1 + 1\right)\right), \left(\color{blue}{1} \cdot x\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(1 + 1\right)\right), \left(1 \cdot x\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), \left(1 \cdot x\right)\right) \]

      12. *-lft-identity 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(x + 2\right) \cdot x} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 99.2%

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{x}\right) \]

   7. Simplified 99.2%

      \[\leadsto \color{blue}{2 \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.5%

   \[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]
2. Step-by-step derivation

   1. difference-of-sqr-1 N/A

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

   2. associate--l+ N/A

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

   3. metadata-eval N/A

      \[\leadsto \left(\left(x + 1\right) + 1\right) \cdot \left(x + 0\right) \]

   4. +-rgt-identity N/A

      \[\leadsto \left(\left(x + 1\right) + 1\right) \cdot x \]

   5. *-lft-identity N/A

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

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(x + 1\right) + 1\right), \color{blue}{\left(1 \cdot x\right)}\right) \]

   7. associate-+l+ N/A

      \[\leadsto \mathsf{*.f64}\left(\left(x + \left(1 + 1\right)\right), \left(\color{blue}{1} \cdot x\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot x + \left(1 + 1\right)\right), \left(1 \cdot x\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(1 \cdot x\right), \left(1 + 1\right)\right), \left(\color{blue}{1} \cdot x\right)\right) \]

   10. *-lft-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(1 + 1\right)\right), \left(1 \cdot x\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), \left(1 \cdot x\right)\right) \]

   12. *-lft-identity 100.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\left(x + 2\right) \cdot x} \]
4. Add Preprocessing

5. Final simplification 100.0%

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

Alternative 4: 50.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 2 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * 2.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.5%

   \[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]
2. Step-by-step derivation

   1. difference-of-sqr-1 N/A

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

   2. associate--l+ N/A

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

   3. metadata-eval N/A

      \[\leadsto \left(\left(x + 1\right) + 1\right) \cdot \left(x + 0\right) \]

   4. +-rgt-identity N/A

      \[\leadsto \left(\left(x + 1\right) + 1\right) \cdot x \]

   5. *-lft-identity N/A

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

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(x + 1\right) + 1\right), \color{blue}{\left(1 \cdot x\right)}\right) \]

   7. associate-+l+ N/A

      \[\leadsto \mathsf{*.f64}\left(\left(x + \left(1 + 1\right)\right), \left(\color{blue}{1} \cdot x\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot x + \left(1 + 1\right)\right), \left(1 \cdot x\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(1 \cdot x\right), \left(1 + 1\right)\right), \left(\color{blue}{1} \cdot x\right)\right) \]

   10. *-lft-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(1 + 1\right)\right), \left(1 \cdot x\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), \left(1 \cdot x\right)\right) \]

   12. *-lft-identity 100.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\left(x + 2\right) \cdot x} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 51.0%

      \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{x}\right) \]

7. Simplified 51.0%

   \[\leadsto \color{blue}{2 \cdot x} \]

8. Final simplification 51.0%

   \[\leadsto x \cdot 2 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024161 
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1.0) (+ x 1.0)) 1.0))