Expanding a square

Percentage Accurate: 54.4% → 100.0%

Time: 2.1s

Alternatives: 4

Speedup: 1.8×

• 24.3% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.4%

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

   1. difference-of-sqr-1 59.4%

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

   2. *-commutative 59.4%

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

   3. associate--l+ 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \left(x + \color{blue}{0}\right) \cdot \left(\left(x + 1\right) + 1\right) \]

   5. +-rgt-identity 100.0%

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

   6. associate-+l+ 100.0%

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

   7. metadata-eval 100.0%

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

   8. metadata-eval 100.0%

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

   9. metadata-eval 100.0%

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

   10. metadata-eval 100.0%

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

   11. metadata-eval 100.0%

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

   12. sub-neg 100.0%

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

   13. metadata-eval 100.0%

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

   14. metadata-eval 100.0%

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

   15. metadata-eval 100.0%

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

3. Simplified 100.0%

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

   1. sub-neg 100.0%

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

   2. distribute-lft-in 100.0%

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

   3. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

   \[\leadsto x \cdot x + x \cdot 2 \]

Alternative 2: 97.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -2.0) {
		tmp = x * x;
	} else if (x <= 2.0) {
		tmp = x + x;
	} 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 + x
    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 + x;
	} 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 + x
	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 + x);
	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 + x;
	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 + x), $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-1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate--l+ 100.0%

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

      4. metadata-eval 100.0%

         \[\leadsto \left(x + \color{blue}{0}\right) \cdot \left(\left(x + 1\right) + 1\right) \]

      5. +-rgt-identity 100.0%

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

      6. associate-+l+ 100.0%

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

      7. metadata-eval 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

      15. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 96.4%

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

      1. unpow2 96.4%

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

   6. Simplified 96.4%

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

   if -2 < x < 2

   1. Initial program 8.9%

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

      1. difference-of-sqr-1 8.9%

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

      2. *-commutative 8.9%

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

      3. associate--l+ 100.0%

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

      4. metadata-eval 100.0%

         \[\leadsto \left(x + \color{blue}{0}\right) \cdot \left(\left(x + 1\right) + 1\right) \]

      5. +-rgt-identity 100.0%

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

      6. associate-+l+ 100.0%

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

      7. metadata-eval 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

      15. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 97.5%

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

      1. count-2 97.5%

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

   6. Simplified 97.5%

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

4. Final simplification 96.9%

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

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 59.4%

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

   1. difference-of-sqr-1 59.4%

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

   2. *-commutative 59.4%

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

   3. associate--l+ 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \left(x + \color{blue}{0}\right) \cdot \left(\left(x + 1\right) + 1\right) \]

   5. +-rgt-identity 100.0%

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

   6. associate-+l+ 100.0%

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

   7. metadata-eval 100.0%

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

   8. metadata-eval 100.0%

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

   9. metadata-eval 100.0%

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

   10. metadata-eval 100.0%

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

   11. metadata-eval 100.0%

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

   12. sub-neg 100.0%

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

   13. metadata-eval 100.0%

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

   14. metadata-eval 100.0%

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

   15. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto x \cdot \left(x - -2\right) \]

Alternative 4: 51.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.4%

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

   1. difference-of-sqr-1 59.4%

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

   2. *-commutative 59.4%

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

   3. associate--l+ 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \left(x + \color{blue}{0}\right) \cdot \left(\left(x + 1\right) + 1\right) \]

   5. +-rgt-identity 100.0%

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

   6. associate-+l+ 100.0%

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

   7. metadata-eval 100.0%

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

   8. metadata-eval 100.0%

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

   9. metadata-eval 100.0%

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

   10. metadata-eval 100.0%

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

   11. metadata-eval 100.0%

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

   12. sub-neg 100.0%

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

   13. metadata-eval 100.0%

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

   14. metadata-eval 100.0%

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

   15. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around inf 56.0%

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

   1. unpow2 56.0%

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

6. Simplified 56.0%

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

7. Final simplification 56.0%

   \[\leadsto x \cdot x \]

Reproduce

herbie shell --seed 2023181 
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1.0) (+ x 1.0)) 1.0))