ENA, Section 1.4, Exercise 4b, n=2

Percentage Accurate: 74.9% → 100.0%

Time: 6.8s

Alternatives: 5

Speedup: 29.6×

Specification

\[\left(-1000000000 \leq x \land x \leq 1000000000\right) \land \left(-1 \leq \varepsilon \land \varepsilon \leq 1\right)\]

Math

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

FPCore

(FPCore (x eps) :precision binary64 (- (pow (+ x eps) 2.0) (pow x 2.0)))

C

double code(double x, double eps) {
	return pow((x + eps), 2.0) - pow(x, 2.0);
}

Fortran

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

Java

public static double code(double x, double eps) {
	return Math.pow((x + eps), 2.0) - Math.pow(x, 2.0);
}

Python

def code(x, eps):
	return math.pow((x + eps), 2.0) - math.pow(x, 2.0)

Julia

function code(x, eps)
	return Float64((Float64(x + eps) ^ 2.0) - (x ^ 2.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = ((x + eps) ^ 2.0) - (x ^ 2.0);
end

Wolfram

code[x_, eps_] := N[(N[Power[N[(x + eps), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]

Initial Program: 74.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (- (pow (+ x eps) 2.0) (pow x 2.0)))

C

double code(double x, double eps) {
	return pow((x + eps), 2.0) - pow(x, 2.0);
}

Fortran

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

Java

public static double code(double x, double eps) {
	return Math.pow((x + eps), 2.0) - Math.pow(x, 2.0);
}

Python

def code(x, eps):
	return math.pow((x + eps), 2.0) - math.pow(x, 2.0)

Julia

function code(x, eps)
	return Float64((Float64(x + eps) ^ 2.0) - (x ^ 2.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = ((x + eps) ^ 2.0) - (x ^ 2.0);
end

Wolfram

code[x_, eps_] := N[(N[Power[N[(x + eps), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 29.6× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps * (eps + (x + x));
}

Fortran

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

Java

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

Python

def code(x, eps):
	return eps * (eps + (x + x))

Julia

function code(x, eps)
	return Float64(eps * Float64(eps + Float64(x + x)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (eps + (x + x));
end

Wolfram

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

Derivation

1. Initial program 76.9%

   \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
2. Step-by-step derivation

   1. unpow2 76.9%

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

   2. unpow2 76.9%

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

   3. difference-of-squares 76.9%

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

   4. *-commutative 76.9%

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

   5. associate--l+ 76.9%

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

   6. unsub-neg 76.9%

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

   7. +-commutative 76.9%

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

   8. +-commutative 76.9%

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

   9. remove-double-neg 76.9%

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

   10. sub-neg 76.9%

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

   11. +-commutative 76.9%

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

   12. associate--l+ 100.0%

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

   13. +-inverses 100.0%

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

   14. +-rgt-identity 100.0%

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

   15. +-commutative 100.0%

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

   16. associate-+l+ 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \varepsilon \cdot \left(\varepsilon + \left(x + x\right)\right) \]

Alternative 2: 90.0% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{-127} \lor \neg \left(x \leq 4.3 \cdot 10^{-126}\right):\\ \;\;\;\;\varepsilon \cdot \left(x + x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -5.6e-127) (not (<= x 4.3e-126))) (* eps (+ x x)) (* eps eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -5.6e-127) || !(x <= 4.3e-126)) {
		tmp = eps * (x + x);
	} else {
		tmp = eps * eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-5.6d-127)) .or. (.not. (x <= 4.3d-126))) then
        tmp = eps * (x + x)
    else
        tmp = eps * eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -5.6e-127) || !(x <= 4.3e-126)) {
		tmp = eps * (x + x);
	} else {
		tmp = eps * eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -5.6e-127) or not (x <= 4.3e-126):
		tmp = eps * (x + x)
	else:
		tmp = eps * eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -5.6e-127) || !(x <= 4.3e-126))
		tmp = Float64(eps * Float64(x + x));
	else
		tmp = Float64(eps * eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -5.6e-127) || ~((x <= 4.3e-126)))
		tmp = eps * (x + x);
	else
		tmp = eps * eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -5.6e-127], N[Not[LessEqual[x, 4.3e-126]], $MachinePrecision]], N[(eps * N[(x + x), $MachinePrecision]), $MachinePrecision], N[(eps * eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.59999999999999999e-127 or 4.30000000000000033e-126 < x

   1. Initial program 42.9%

      \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
   2. Step-by-step derivation

      1. unpow2 42.9%

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

      2. unpow2 42.9%

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

      3. difference-of-squares 42.9%

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

      4. *-commutative 42.9%

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

      5. associate--l+ 42.9%

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

      6. unsub-neg 42.9%

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

      7. +-commutative 42.9%

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

      8. +-commutative 42.9%

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

      9. remove-double-neg 42.9%

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

      10. sub-neg 42.9%

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

      11. +-commutative 42.9%

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

      12. associate--l+ 100.0%

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

      13. +-inverses 100.0%

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

      14. +-rgt-identity 100.0%

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

      15. +-commutative 100.0%

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

      16. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in eps around 0 85.3%

      \[\leadsto \color{blue}{2 \cdot \left(\varepsilon \cdot x\right)} \]
   5. Step-by-step derivation

      1. count-2 85.3%

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

      2. distribute-lft-in 85.3%

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

   6. Simplified 85.3%

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

   if -5.59999999999999999e-127 < x < 4.30000000000000033e-126

   1. Initial program 98.7%

      \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
   2. Step-by-step derivation

      1. unpow2 98.7%

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

      2. unpow2 98.7%

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

      3. difference-of-squares 98.7%

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

      4. *-commutative 98.7%

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

      5. associate--l+ 98.7%

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

      6. unsub-neg 98.7%

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

      7. +-commutative 98.7%

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

      8. +-commutative 98.7%

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

      9. remove-double-neg 98.7%

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

      10. sub-neg 98.7%

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

      11. +-commutative 98.7%

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

      12. associate--l+ 100.0%

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

      13. +-inverses 100.0%

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

      14. +-rgt-identity 100.0%

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

      15. +-commutative 100.0%

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

      16. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 98.3%

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

      1. unpow2 98.3%

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

   6. Simplified 98.3%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{-127} \lor \neg \left(x \leq 4.3 \cdot 10^{-126}\right):\\ \;\;\;\;\varepsilon \cdot \left(x + x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \]

Alternative 3: 72.0% accurate, 69.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \varepsilon \end{array} \]

FPCore

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

C

double code(double x, double eps) {
	return eps * eps;
}

Fortran

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

Java

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

Python

def code(x, eps):
	return eps * eps

Julia

function code(x, eps)
	return Float64(eps * eps)
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * eps;
end

Wolfram

code[x_, eps_] := N[(eps * eps), $MachinePrecision]

Derivation

1. Initial program 76.9%

   \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
2. Step-by-step derivation

   1. unpow2 76.9%

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

   2. unpow2 76.9%

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

   3. difference-of-squares 76.9%

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

   4. *-commutative 76.9%

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

   5. associate--l+ 76.9%

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

   6. unsub-neg 76.9%

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

   7. +-commutative 76.9%

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

   8. +-commutative 76.9%

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

   9. remove-double-neg 76.9%

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

   10. sub-neg 76.9%

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

   11. +-commutative 76.9%

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

   12. associate--l+ 100.0%

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

   13. +-inverses 100.0%

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

   14. +-rgt-identity 100.0%

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

   15. +-commutative 100.0%

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

   16. associate-+l+ 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in eps around inf 74.8%

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

   1. unpow2 74.8%

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

6. Simplified 74.8%

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

7. Final simplification 74.8%

   \[\leadsto \varepsilon \cdot \varepsilon \]

Alternative 4: 2.9% accurate, 207.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return -1.0

Julia

function code(x, eps)
	return -1.0
end

MATLAB

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

Wolfram

code[x_, eps_] := -1.0

Derivation

1. Initial program 76.9%

   \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
2. Step-by-step derivation

   1. unpow2 76.9%

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

   2. unpow2 76.9%

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

   3. difference-of-squares 76.9%

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

   4. *-commutative 76.9%

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

   5. associate--l+ 76.9%

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

   6. unsub-neg 76.9%

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

   7. +-commutative 76.9%

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

   8. +-commutative 76.9%

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

   9. remove-double-neg 76.9%

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

   10. sub-neg 76.9%

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

   11. +-commutative 76.9%

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

   12. associate--l+ 100.0%

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

   13. +-inverses 100.0%

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

   14. +-rgt-identity 100.0%

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

   15. +-commutative 100.0%

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

   16. associate-+l+ 100.0%

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

3. Simplified 100.0%

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

5. Applied egg-rr 2.9%

   \[\leadsto \color{blue}{-1} \]

6. Final simplification 2.9%

   \[\leadsto -1 \]

Alternative 5: 37.8% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 76.9%

   \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
2. Step-by-step derivation

   1. unpow2 76.9%

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

   2. unpow2 76.9%

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

   3. difference-of-squares 76.9%

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

   4. *-commutative 76.9%

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

   5. associate--l+ 76.9%

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

   6. unsub-neg 76.9%

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

   7. +-commutative 76.9%

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

   8. +-commutative 76.9%

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

   9. remove-double-neg 76.9%

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

   10. sub-neg 76.9%

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

   11. +-commutative 76.9%

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

   12. associate--l+ 100.0%

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

   13. +-inverses 100.0%

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

   14. +-rgt-identity 100.0%

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

   15. +-commutative 100.0%

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

   16. associate-+l+ 100.0%

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

3. Simplified 100.0%

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

5. Applied egg-rr 39.9%

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

6. Final simplification 39.9%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023297 
(FPCore (x eps)
  :name "ENA, Section 1.4, Exercise 4b, n=2"
  :precision binary64
  :pre (and (and (<= -1000000000.0 x) (<= x 1000000000.0)) (and (<= -1.0 eps) (<= eps 1.0)))
  (- (pow (+ x eps) 2.0) (pow x 2.0)))