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

Percentage Accurate: 75.2% → 100.0%

Time: 5.9s

Alternatives: 7

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: 75.2% 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, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot 2, \varepsilon, \varepsilon \cdot \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (fma (* x 2.0) eps (* eps eps)))

C

double code(double x, double eps) {
	return fma((x * 2.0), eps, (eps * eps));
}

Julia

function code(x, eps)
	return fma(Float64(x * 2.0), eps, Float64(eps * eps))
end

Wolfram

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

Derivation

1. Initial program 72.3%

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

   1. +-commutative 72.3%

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

   2. unpow2 72.3%

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

   3. unpow2 72.3%

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

   4. difference-of-squares 72.3%

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

   5. sub-neg 72.3%

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

   6. distribute-lft-in 72.3%

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

   7. +-commutative 72.3%

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

   8. distribute-lft-in 72.3%

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

   9. associate-+l+ 72.3%

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

   10. remove-double-neg 72.3%

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

   11. sub-neg 72.3%

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

   12. +-commutative 72.3%

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

   13. associate--l+ 100.0%

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

   14. +-inverses 100.0%

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

   15. +-rgt-identity 100.0%

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

   16. *-commutative 100.0%

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

   17. associate-+l+ 100.0%

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

   18. count-2 100.0%

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

   19. *-commutative 100.0%

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

3. Simplified 100.0%

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

   1. +-commutative 100.0%

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

   2. distribute-rgt-in 100.0%

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

   3. fma-define 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot 2, \varepsilon, \varepsilon \cdot \varepsilon\right)} \]

   4. pow2 100.0%

      \[\leadsto \mathsf{fma}\left(x \cdot 2, \varepsilon, \color{blue}{{\varepsilon}^{2}}\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot 2, \varepsilon, {\varepsilon}^{2}\right)} \]
7. Step-by-step derivation

   1. pow2 100.0%

      \[\leadsto \mathsf{fma}\left(x \cdot 2, \varepsilon, \color{blue}{\varepsilon \cdot \varepsilon}\right) \]

8. Applied egg-rr 100.0%

   \[\leadsto \mathsf{fma}\left(x \cdot 2, \varepsilon, \color{blue}{\varepsilon \cdot \varepsilon}\right) \]
9. Add Preprocessing

Alternative 2: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon, \varepsilon, x \cdot \left(2 \cdot \varepsilon\right)\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (fma eps eps (* x (* 2.0 eps))))

C

double code(double x, double eps) {
	return fma(eps, eps, (x * (2.0 * eps)));
}

Julia

function code(x, eps)
	return fma(eps, eps, Float64(x * Float64(2.0 * eps)))
end

Wolfram

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

Derivation

1. Initial program 72.3%

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

   1. +-commutative 72.3%

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

   2. unpow2 72.3%

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

   3. unpow2 72.3%

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

   4. difference-of-squares 72.3%

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

   5. sub-neg 72.3%

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

   6. distribute-lft-in 72.3%

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

   7. +-commutative 72.3%

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

   8. distribute-lft-in 72.3%

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

   9. associate-+l+ 72.3%

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

   10. remove-double-neg 72.3%

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

   11. sub-neg 72.3%

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

   12. +-commutative 72.3%

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

   13. associate--l+ 100.0%

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

   14. +-inverses 100.0%

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

   15. +-rgt-identity 100.0%

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

   16. *-commutative 100.0%

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

   17. associate-+l+ 100.0%

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

   18. count-2 100.0%

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

   19. *-commutative 100.0%

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

3. Simplified 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. fma-define 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon, \left(x \cdot 2\right) \cdot \varepsilon\right)} \]

   3. associate-*l* 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, \varepsilon, \color{blue}{x \cdot \left(2 \cdot \varepsilon\right)}\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon, x \cdot \left(2 \cdot \varepsilon\right)\right)} \]
7. Add Preprocessing

Alternative 3: 89.9% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-130} \lor \neg \left(x \leq 7.5 \cdot 10^{-120}\right):\\ \;\;\;\;\varepsilon \cdot \left(x + x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -3.4e-130) (not (<= x 7.5e-120))) (* eps (+ x x)) (* eps eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -3.4e-130) || !(x <= 7.5e-120)) {
		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 <= (-3.4d-130)) .or. (.not. (x <= 7.5d-120))) 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 <= -3.4e-130) || !(x <= 7.5e-120)) {
		tmp = eps * (x + x);
	} else {
		tmp = eps * eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -3.4e-130) or not (x <= 7.5e-120):
		tmp = eps * (x + x)
	else:
		tmp = eps * eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -3.4e-130) || !(x <= 7.5e-120))
		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 <= -3.4e-130) || ~((x <= 7.5e-120)))
		tmp = eps * (x + x);
	else
		tmp = eps * eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -3.4e-130], N[Not[LessEqual[x, 7.5e-120]], $MachinePrecision]], N[(eps * N[(x + x), $MachinePrecision]), $MachinePrecision], N[(eps * eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.40000000000000005e-130 or 7.5000000000000004e-120 < x

   1. Initial program 39.9%

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

      1. sub-neg 39.9%

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

   4. Applied egg-rr 39.9%

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

      1. sub-neg 39.9%

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

      2. unpow2 39.8%

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

      3. unpow2 39.9%

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

      4. difference-of-squares 39.9%

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

      5. *-commutative 39.9%

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

      6. +-commutative 39.9%

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

      7. +-commutative 39.9%

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

      8. associate--l+ 99.9%

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

      9. +-inverses 99.9%

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

      10. +-rgt-identity 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around inf 83.9%

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

   if -3.40000000000000005e-130 < x < 7.5000000000000004e-120

   1. Initial program 98.8%

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

      1. +-commutative 98.8%

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

      2. unpow2 98.8%

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

      3. unpow2 98.8%

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

      4. difference-of-squares 98.8%

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

      5. sub-neg 98.8%

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

      6. distribute-lft-in 98.8%

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

      7. +-commutative 98.8%

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

      8. distribute-lft-in 98.8%

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

      9. associate-+l+ 98.8%

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

      10. remove-double-neg 98.8%

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

      11. sub-neg 98.8%

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

      12. +-commutative 98.8%

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

      13. associate--l+ 100.0%

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

      14. +-inverses 100.0%

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

      15. +-rgt-identity 100.0%

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

      16. *-commutative 100.0%

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

      17. associate-+l+ 100.0%

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

      18. count-2 100.0%

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

      19. *-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon + x \cdot 2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 98.0%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-130} \lor \neg \left(x \leq 7.5 \cdot 10^{-120}\right):\\ \;\;\;\;\varepsilon \cdot \left(x + x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.9% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-130} \lor \neg \left(x \leq 8 \cdot 10^{-120}\right):\\ \;\;\;\;2 \cdot \left(x \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -3.2e-130) (not (<= x 8e-120))) (* 2.0 (* x eps)) (* eps eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -3.2e-130) || !(x <= 8e-120)) {
		tmp = 2.0 * (x * eps);
	} 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 <= (-3.2d-130)) .or. (.not. (x <= 8d-120))) then
        tmp = 2.0d0 * (x * eps)
    else
        tmp = eps * eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -3.2e-130) || !(x <= 8e-120)) {
		tmp = 2.0 * (x * eps);
	} else {
		tmp = eps * eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -3.2e-130) or not (x <= 8e-120):
		tmp = 2.0 * (x * eps)
	else:
		tmp = eps * eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -3.2e-130) || !(x <= 8e-120))
		tmp = Float64(2.0 * Float64(x * eps));
	else
		tmp = Float64(eps * eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -3.2e-130) || ~((x <= 8e-120)))
		tmp = 2.0 * (x * eps);
	else
		tmp = eps * eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -3.2e-130], N[Not[LessEqual[x, 8e-120]], $MachinePrecision]], N[(2.0 * N[(x * eps), $MachinePrecision]), $MachinePrecision], N[(eps * eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.2e-130 or 7.99999999999999983e-120 < x

   1. Initial program 39.9%

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

      1. +-commutative 39.9%

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

      2. unpow2 39.9%

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

      3. unpow2 39.9%

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

      4. difference-of-squares 39.9%

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

      5. sub-neg 39.9%

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

      6. distribute-lft-in 39.8%

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

      7. +-commutative 39.8%

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

      8. distribute-lft-in 39.9%

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

      9. associate-+l+ 39.9%

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

      10. remove-double-neg 39.9%

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

      11. sub-neg 39.9%

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

      12. +-commutative 39.9%

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

      13. associate--l+ 99.9%

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

      14. +-inverses 99.9%

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

      15. +-rgt-identity 99.9%

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

      16. *-commutative 99.9%

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

      17. associate-+l+ 100.0%

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

      18. count-2 100.0%

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

      19. *-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon + x \cdot 2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 83.9%

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

      1. *-commutative 83.9%

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

   7. Simplified 83.9%

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

   if -3.2e-130 < x < 7.99999999999999983e-120

   1. Initial program 98.8%

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

      1. +-commutative 98.8%

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

      2. unpow2 98.8%

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

      3. unpow2 98.8%

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

      4. difference-of-squares 98.8%

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

      5. sub-neg 98.8%

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

      6. distribute-lft-in 98.8%

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

      7. +-commutative 98.8%

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

      8. distribute-lft-in 98.8%

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

      9. associate-+l+ 98.8%

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

      10. remove-double-neg 98.8%

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

      11. sub-neg 98.8%

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

      12. +-commutative 98.8%

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

      13. associate--l+ 100.0%

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

      14. +-inverses 100.0%

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

      15. +-rgt-identity 100.0%

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

      16. *-commutative 100.0%

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

      17. associate-+l+ 100.0%

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

      18. count-2 100.0%

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

      19. *-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon + x \cdot 2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 98.0%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-130} \lor \neg \left(x \leq 8 \cdot 10^{-120}\right):\\ \;\;\;\;2 \cdot \left(x \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 100.0% accurate, 29.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.3%

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

   1. +-commutative 72.3%

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

   2. unpow2 72.3%

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

   3. unpow2 72.3%

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

   4. difference-of-squares 72.3%

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

   5. sub-neg 72.3%

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

   6. distribute-lft-in 72.3%

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

   7. +-commutative 72.3%

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

   8. distribute-lft-in 72.3%

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

   9. associate-+l+ 72.3%

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

   10. remove-double-neg 72.3%

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

   11. sub-neg 72.3%

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

   12. +-commutative 72.3%

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

   13. associate--l+ 100.0%

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

   14. +-inverses 100.0%

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

   15. +-rgt-identity 100.0%

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

   16. *-commutative 100.0%

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

   17. associate-+l+ 100.0%

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

   18. count-2 100.0%

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

   19. *-commutative 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon + x \cdot 2\right)} \]
4. Add Preprocessing

5. Final simplification 100.0%

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

Alternative 6: 72.9% 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 72.3%

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

   1. +-commutative 72.3%

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

   2. unpow2 72.3%

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

   3. unpow2 72.3%

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

   4. difference-of-squares 72.3%

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

   5. sub-neg 72.3%

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

   6. distribute-lft-in 72.3%

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

   7. +-commutative 72.3%

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

   8. distribute-lft-in 72.3%

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

   9. associate-+l+ 72.3%

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

   10. remove-double-neg 72.3%

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

   11. sub-neg 72.3%

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

   12. +-commutative 72.3%

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

   13. associate--l+ 100.0%

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

   14. +-inverses 100.0%

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

   15. +-rgt-identity 100.0%

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

   16. *-commutative 100.0%

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

   17. associate-+l+ 100.0%

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

   18. count-2 100.0%

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

   19. *-commutative 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon + x \cdot 2\right)} \]
4. Add Preprocessing

5. Taylor expanded in eps around inf 69.8%

   \[\leadsto \varepsilon \cdot \color{blue}{\varepsilon} \]
6. Add Preprocessing

Alternative 7: 2.9% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ -8 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 -8.0)

C

double code(double x, double eps) {
	return -8.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = -8.0d0
end function

Java

public static double code(double x, double eps) {
	return -8.0;
}

Python

def code(x, eps):
	return -8.0

Julia

function code(x, eps)
	return -8.0
end

MATLAB

function tmp = code(x, eps)
	tmp = -8.0;
end

Wolfram

code[x_, eps_] := -8.0

Derivation

1. Initial program 72.3%

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

   1. sub-neg 72.3%

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

4. Applied egg-rr 72.3%

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

   1. sub-neg 72.3%

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

   2. unpow2 72.3%

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

   3. unpow2 72.3%

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

   4. difference-of-squares 72.3%

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

   5. *-commutative 72.3%

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

   6. +-commutative 72.3%

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

   7. +-commutative 72.3%

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

   8. associate--l+ 100.0%

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

   9. +-inverses 100.0%

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

   10. +-rgt-identity 100.0%

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

6. Simplified 100.0%

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

7. Taylor expanded in x around inf 62.8%

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

8. Taylor expanded in eps around 0 62.8%

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

10. Simplified 2.9%

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

Reproduce

herbie shell --seed 2024152 
(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)))