Expanding a square

Percentage Accurate: 55.0% → 100.0%

Time: 2.7s

Alternatives: 3

Speedup: 1.8×

• 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: 55.0% 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 53.6%

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

   1. difference-of-sqr-1 53.6%

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

   2. *-commutative 53.6%

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

   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: 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 53.6%

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

   1. difference-of-sqr-1 53.6%

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

   2. *-commutative 53.6%

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

   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 3: 50.0% 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 53.6%

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

   1. difference-of-sqr-1 53.6%

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

   2. *-commutative 53.6%

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

   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 51.6%

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

6. Final simplification 51.6%

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

Reproduce

herbie shell --seed 2024106 
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1.0) (+ x 1.0)) 1.0))