Expanding a square

Percentage Accurate: 54.5% → 100.0%

Time: 3.4s

Alternatives: 4

Speedup: 1.8×

• 25.0% 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: 54.5% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.4%

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

   1. difference-of-sqr-1 55.4%

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

   2. *-commutative 55.4%

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

   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%

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

3. Simplified 100.0%

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

   1. distribute-lft-in 100.0%

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

   2. fma-define 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, x \cdot 2\right)} \]
7. Add Preprocessing

Alternative 2: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -2.0) (not (<= x 2.0))) (* x x) (* x 2.0)))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-2.0d0)) .or. (.not. (x <= 2.0d0))) then
        tmp = x * x
    else
        tmp = x * 2.0d0
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if (x <= -2.0) or not (x <= 2.0):
		tmp = x * x
	else:
		tmp = x * 2.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[Or[LessEqual[x, -2.0], N[Not[LessEqual[x, 2.0]], $MachinePrecision]], N[(x * x), $MachinePrecision], N[(x * 2.0), $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%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.0%

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

   if -2 < x < 2

   1. Initial program 12.8%

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

      1. difference-of-sqr-1 12.8%

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

      2. *-commutative 12.8%

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

      3. associate--l+ 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-rgt-identity 99.9%

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

      6. associate-+l+ 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 94.9%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2\\ \end{array} \]
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 55.4%

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

   1. difference-of-sqr-1 55.4%

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

   2. *-commutative 55.4%

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

   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%

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

3. Simplified 100.0%

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

Alternative 4: 50.6% 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 55.4%

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

   1. difference-of-sqr-1 55.4%

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

   2. *-commutative 55.4%

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

   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%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 50.7%

   \[\leadsto x \cdot \color{blue}{2} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024137 
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1.0) (+ x 1.0)) 1.0))