Expanding a square

Percentage Accurate: 53.5% → 100.0%

Time: 3.8s

Alternatives: 3

Speedup: 1.7×

• 24.6% 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: 53.5% 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, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

   \[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]
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. Final simplification 100.0%

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

Alternative 2: 97.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (((1.0 + x) * (1.0 + 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 (((1.0d0 + x) * (1.0d0 + 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 (((1.0 + x) * (1.0 + x)) <= 2.0) {
		tmp = x + x;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 + x) * Float64(1.0 + 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 (((1.0 + x) * (1.0 + x)) <= 2.0)
		tmp = x + x;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 + x), $MachinePrecision] * N[(1.0 + x), $MachinePrecision]), $MachinePrecision], 2.0], N[(x + 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 9.5%

      \[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]
   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 97.0

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

   5. Applied rewrites 97.0%

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

   7. Applied rewrites 97.0%

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

   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 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 97.7%

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

Alternative 3: 51.6% accurate, 3.8× speedup

Math

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

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 55.8%

   \[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]
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 49.1

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

5. Applied rewrites 49.1%

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

7. Applied rewrites 49.1%

   \[\leadsto x + \color{blue}{x} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024249 
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1.0) (+ x 1.0)) 1.0))