Expanding a square

Percentage Accurate: 54.5% → 100.0%

Time: 4.3s

Alternatives: 5

Speedup: 1.7×

• 25.1% 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, 1.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	return fma((x + 1.0), x, x);
}

Julia

function code(x)
	return fma(Float64(x + 1.0), x, x)
end

Wolfram

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

Derivation

1. Initial program 51.1%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. lift-+.f64 N/A

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

   5. associate--l+ N/A

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

   6. metadata-eval N/A

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

   7. +-rgt-identity N/A

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

   8. *-commutative N/A

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

   9. distribute-lft-in N/A

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

   10. *-rgt-identity N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 100.0

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(x + 1, x, x\right) \]
6. Add Preprocessing

Alternative 2: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x + 1\right) \cdot \left(x + 1\right) \leq 2:\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (+ x 1.0) (+ x 1.0)) 2.0) (* 2.0 x) (* x x)))

C

double code(double x) {
	double tmp;
	if (((x + 1.0) * (x + 1.0)) <= 2.0) {
		tmp = 2.0 * x;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((x + 1.0d0) * (x + 1.0d0)) <= 2.0d0) then
        tmp = 2.0d0 * x
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((x + 1.0) * (x + 1.0)) <= 2.0) {
		tmp = 2.0 * x;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((x + 1.0) * (x + 1.0)) <= 2.0:
		tmp = 2.0 * x
	else:
		tmp = x * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x + 1.0) * Float64(x + 1.0)) <= 2.0)
		tmp = Float64(2.0 * x);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((x + 1.0) * (x + 1.0)) <= 2.0)
		tmp = 2.0 * x;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x + 1.0), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 2.0], N[(2.0 * x), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (+.f64 x #s(literal 1 binary64)) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 11.2%

      \[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 95.8

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

   5. Applied rewrites 95.8%

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

   if 2 < (*.f64 (+.f64 x #s(literal 1 binary64)) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 100.0%

      \[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + 1\right) \cdot \left(x + 1\right) \leq 2:\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.1%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. lift-+.f64 N/A

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

   5. associate--l+ N/A

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

   6. metadata-eval N/A

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

   7. +-rgt-identity N/A

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

   8. lower-*.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. associate-+l+ N/A

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

   11. metadata-eval N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 4: 50.6% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 2.0 * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.1%

   \[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 54.4

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

5. Applied rewrites 54.4%

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

6. Final simplification 54.4%

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

Alternative 5: 3.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 1 - 1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 - 1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.1%

   \[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} - 1 \]
4. Step-by-step derivation

5. Applied rewrites 4.0%

   \[\leadsto \color{blue}{1} - 1 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024255 
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1.0) (+ x 1.0)) 1.0))