Expanding a square

Percentage Accurate: 54.8% → 100.0%

Time: 4.2s

Alternatives: 4

Speedup: 1.7×

• 25.6% 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: 54.8% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.0%

   \[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]
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. lift-+.f64 N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. associate--l+ N/A

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

   7. lift-+.f64 N/A

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

   8. associate--l+ N/A

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

   9. metadata-eval N/A

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

   10. +-rgt-identity N/A

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

   11. lower-fma.f64 100.0

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. 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. Add Preprocessing

Alternative 2: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x + 1\right) \cdot \left(x + 1\right) - 1 \leq 0.005:\\ \;\;\;\;x + x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (* (+ x 1.0) (+ x 1.0)) 1.0) 0.005) (+ x x) (* x x)))

C

double code(double x) {
	double tmp;
	if ((((x + 1.0) * (x + 1.0)) - 1.0) <= 0.005) {
		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 + 1.0d0) * (x + 1.0d0)) - 1.0d0) <= 0.005d0) 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 + 1.0) * (x + 1.0)) - 1.0) <= 0.005) {
		tmp = x + x;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (((x + 1.0) * (x + 1.0)) - 1.0) <= 0.005:
		tmp = x + x
	else:
		tmp = x * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(Float64(x + 1.0) * Float64(x + 1.0)) - 1.0) <= 0.005)
		tmp = Float64(x + 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)) - 1.0) <= 0.005)
		tmp = x + x;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[(x + 1.0), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], 0.005], N[(x + x), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 x #s(literal 1 binary64)) (+.f64 x #s(literal 1 binary64))) #s(literal 1 binary64)) < 0.0050000000000000001

   1. Initial program 7.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 98.9

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

   5. Applied rewrites 98.9%

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

   7. Applied rewrites 98.9%

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

   if 0.0050000000000000001 < (-.f64 (*.f64 (+.f64 x #s(literal 1 binary64)) (+.f64 x #s(literal 1 binary64))) #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 98.3

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

   5. Applied rewrites 98.3%

      \[\leadsto \color{blue}{x \cdot x} \]
3. Recombined 2 regimes into one program.
4. 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 51.0%

   \[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]
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: 50.2% 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 51.0%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 54.0

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

5. Applied rewrites 54.0%

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

7. Applied rewrites 54.0%

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

Reproduce

herbie shell --seed 2024337 
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1.0) (+ x 1.0)) 1.0))