Expanding a square

Percentage Accurate: 53.8% → 100.0%

Time: 5.8s

Alternatives: 7

Speedup: 1.7×

• 24.4% 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: 53.8% 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

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

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 57.9%

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

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

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

   2. *-commutative N/A

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

   3. associate--l+ N/A

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

   4. metadata-eval N/A

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

   5. +-rgt-identity N/A

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

   6. distribute-rgt-in N/A

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

   7. *-lft-identity N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. +-lowering-+.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 97.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Wolfram

code[x_] := If[LessEqual[N[(N[(x + 1.0), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 2.0], N[(x * 2.0), $MachinePrecision], N[(x * 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 7.1%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 98.6

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

   5. Simplified 98.6%

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

   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. Step-by-step derivation

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

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

      2. *-commutative N/A

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

      3. associate--l+ N/A

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

      4. metadata-eval N/A

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

      5. +-rgt-identity N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. +-lowering-+.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{x}, x, x\right) \]
   6. Step-by-step derivation

   7. Simplified 98.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{x}, x, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

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

Alternative 3: 97.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

def code(x):
	tmp = 0
	if ((x + 1.0) * (x + 1.0)) <= 2.0:
		tmp = x * 2.0
	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(x * 2.0);
	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 = x * 2.0;
	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[(x * 2.0), $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 7.1%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 98.6

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

   5. Simplified 98.6%

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

   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. *-lowering-*.f64 98.6

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

   5. Simplified 98.6%

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

4. Final simplification 98.6%

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

Alternative 4: 100.0% accurate, 1.7× 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 57.9%

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

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

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

   2. associate--l+ N/A

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

   3. metadata-eval N/A

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

   4. +-rgt-identity N/A

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

   5. *-lowering-*.f64 N/A

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

   6. associate-+l+ N/A

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

   7. metadata-eval N/A

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

   8. +-lowering-+.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 5: 51.3% accurate, 2.5× 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 57.9%

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

3. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 46.5

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

5. Simplified 46.5%

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

6. Final simplification 46.5%

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

Alternative 6: 11.3% accurate, 15.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 57.9%

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

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

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

   2. *-commutative N/A

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

   3. associate--l+ N/A

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

   4. metadata-eval N/A

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

   5. +-rgt-identity N/A

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

   6. distribute-rgt-in N/A

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

   7. *-lft-identity N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. +-lowering-+.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(\color{blue}{x}, x, x\right) \]
6. Step-by-step derivation

7. Simplified 62.4%

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

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x} \]
9. Step-by-step derivation

10. Simplified 10.3%

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

Alternative 7: 3.7% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in x around 0

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

5. Simplified 3.4%

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

   1. metadata-eval 3.4

      \[\leadsto \color{blue}{0} \]

7. Applied egg-rr 3.4%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1.0) (+ x 1.0)) 1.0))