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

Percentage Accurate: 74.1% → 100.0%

Time: 5.5s

Alternatives: 4

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: 74.1% 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, 14.8× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{-1} \cdot \left(2 \cdot \left(-x\right)\right) - \varepsilon \cdot \frac{\varepsilon}{-1} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (- (* (/ eps -1.0) (* 2.0 (- x))) (* eps (/ eps -1.0))))

C

double code(double x, double eps) {
	return ((eps / -1.0) * (2.0 * -x)) - (eps * (eps / -1.0));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((eps / (-1.0d0)) * (2.0d0 * -x)) - (eps * (eps / (-1.0d0)))
end function

Java

public static double code(double x, double eps) {
	return ((eps / -1.0) * (2.0 * -x)) - (eps * (eps / -1.0));
}

Python

def code(x, eps):
	return ((eps / -1.0) * (2.0 * -x)) - (eps * (eps / -1.0))

Julia

function code(x, eps)
	return Float64(Float64(Float64(eps / -1.0) * Float64(2.0 * Float64(-x))) - Float64(eps * Float64(eps / -1.0)))
end

MATLAB

function tmp = code(x, eps)
	tmp = ((eps / -1.0) * (2.0 * -x)) - (eps * (eps / -1.0));
end

Wolfram

code[x_, eps_] := N[(N[(N[(eps / -1.0), $MachinePrecision] * N[(2.0 * (-x)), $MachinePrecision]), $MachinePrecision] - N[(eps * N[(eps / -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.2%

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

   1. +-commutative 75.2%

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

   2. unpow2 75.2%

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

   3. unpow2 75.2%

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

   4. difference-of-squares 75.2%

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

   5. *-commutative 75.2%

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

   6. sub-neg 75.2%

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

   7. +-commutative 75.2%

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

   8. associate-+l+ 75.2%

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

   9. remove-double-neg 75.2%

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

   10. sub-neg 75.2%

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

   11. +-commutative 75.2%

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

   12. associate--l+ 100.0%

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

   13. +-inverses 100.0%

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

   14. +-rgt-identity 100.0%

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

   15. remove-double-neg 100.0%

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

   16. sub-neg 100.0%

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

   17. remove-double-neg 100.0%

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

   18. sub-neg 100.0%

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

   19. associate--l- 100.0%

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

   20. neg-mul-1 100.0%

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

   21. neg-mul-1 100.0%

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

   22. distribute-rgt-out 100.0%

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

   23. metadata-eval 100.0%

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

3. Simplified 100.0%

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

   1. flip3-- 67.5%

      \[\leadsto \varepsilon \cdot \color{blue}{\frac{{\varepsilon}^{3} - {\left(x \cdot -2\right)}^{3}}{\varepsilon \cdot \varepsilon + \left(\left(x \cdot -2\right) \cdot \left(x \cdot -2\right) + \varepsilon \cdot \left(x \cdot -2\right)\right)}} \]

   2. associate-*r/ 54.4%

      \[\leadsto \color{blue}{\frac{\varepsilon \cdot \left({\varepsilon}^{3} - {\left(x \cdot -2\right)}^{3}\right)}{\varepsilon \cdot \varepsilon + \left(\left(x \cdot -2\right) \cdot \left(x \cdot -2\right) + \varepsilon \cdot \left(x \cdot -2\right)\right)}} \]

   3. associate-/l* 67.4%

      \[\leadsto \color{blue}{\frac{\varepsilon}{\frac{\varepsilon \cdot \varepsilon + \left(\left(x \cdot -2\right) \cdot \left(x \cdot -2\right) + \varepsilon \cdot \left(x \cdot -2\right)\right)}{{\varepsilon}^{3} - {\left(x \cdot -2\right)}^{3}}}} \]

   4. clear-num 67.4%

      \[\leadsto \frac{\varepsilon}{\color{blue}{\frac{1}{\frac{{\varepsilon}^{3} - {\left(x \cdot -2\right)}^{3}}{\varepsilon \cdot \varepsilon + \left(\left(x \cdot -2\right) \cdot \left(x \cdot -2\right) + \varepsilon \cdot \left(x \cdot -2\right)\right)}}}} \]

   5. flip3-- 99.6%

      \[\leadsto \frac{\varepsilon}{\frac{1}{\color{blue}{\varepsilon - x \cdot -2}}} \]

   6. sub-neg 99.6%

      \[\leadsto \frac{\varepsilon}{\frac{1}{\color{blue}{\varepsilon + \left(-x \cdot -2\right)}}} \]

   7. distribute-rgt-neg-in 99.6%

      \[\leadsto \frac{\varepsilon}{\frac{1}{\varepsilon + \color{blue}{x \cdot \left(--2\right)}}} \]

   8. metadata-eval 99.6%

      \[\leadsto \frac{\varepsilon}{\frac{1}{\varepsilon + x \cdot \color{blue}{2}}} \]

5. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{\varepsilon}{\frac{1}{\varepsilon + x \cdot 2}}} \]
6. Step-by-step derivation

   1. frac-2neg 99.6%

      \[\leadsto \frac{\varepsilon}{\color{blue}{\frac{-1}{-\left(\varepsilon + x \cdot 2\right)}}} \]

   2. associate-/r/ 100.0%

      \[\leadsto \color{blue}{\frac{\varepsilon}{-1} \cdot \left(-\left(\varepsilon + x \cdot 2\right)\right)} \]

   3. +-commutative 100.0%

      \[\leadsto \frac{\varepsilon}{-1} \cdot \left(-\color{blue}{\left(x \cdot 2 + \varepsilon\right)}\right) \]

   4. distribute-neg-in 100.0%

      \[\leadsto \frac{\varepsilon}{-1} \cdot \color{blue}{\left(\left(-x \cdot 2\right) + \left(-\varepsilon\right)\right)} \]

   5. distribute-lft-in 100.0%

      \[\leadsto \color{blue}{\frac{\varepsilon}{-1} \cdot \left(-x \cdot 2\right) + \frac{\varepsilon}{-1} \cdot \left(-\varepsilon\right)} \]

   6. metadata-eval 100.0%

      \[\leadsto \frac{\varepsilon}{\color{blue}{-1}} \cdot \left(-x \cdot 2\right) + \frac{\varepsilon}{-1} \cdot \left(-\varepsilon\right) \]

   7. *-commutative 100.0%

      \[\leadsto \frac{\varepsilon}{-1} \cdot \left(-\color{blue}{2 \cdot x}\right) + \frac{\varepsilon}{-1} \cdot \left(-\varepsilon\right) \]

   8. metadata-eval 100.0%

      \[\leadsto \frac{\varepsilon}{-1} \cdot \left(-2 \cdot x\right) + \frac{\varepsilon}{\color{blue}{-1}} \cdot \left(-\varepsilon\right) \]

7. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{\varepsilon}{-1} \cdot \left(-2 \cdot x\right) + \frac{\varepsilon}{-1} \cdot \left(-\varepsilon\right)} \]

8. Final simplification 100.0%

   \[\leadsto \frac{\varepsilon}{-1} \cdot \left(2 \cdot \left(-x\right)\right) - \varepsilon \cdot \frac{\varepsilon}{-1} \]

Alternative 2: 89.7% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-83} \lor \neg \left(x \leq 1.55 \cdot 10^{-77}\right):\\ \;\;\;\;\varepsilon \cdot \left(2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -6.2e-83) (not (<= x 1.55e-77))) (* eps (* 2.0 x)) (* eps eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -6.2e-83) || !(x <= 1.55e-77)) {
		tmp = eps * (2.0 * 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 <= (-6.2d-83)) .or. (.not. (x <= 1.55d-77))) then
        tmp = eps * (2.0d0 * x)
    else
        tmp = eps * eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -6.2e-83) || !(x <= 1.55e-77)) {
		tmp = eps * (2.0 * x);
	} else {
		tmp = eps * eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -6.2e-83) or not (x <= 1.55e-77):
		tmp = eps * (2.0 * x)
	else:
		tmp = eps * eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -6.2e-83) || !(x <= 1.55e-77))
		tmp = Float64(eps * Float64(2.0 * x));
	else
		tmp = Float64(eps * eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -6.2e-83) || ~((x <= 1.55e-77)))
		tmp = eps * (2.0 * x);
	else
		tmp = eps * eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -6.2e-83], N[Not[LessEqual[x, 1.55e-77]], $MachinePrecision]], N[(eps * N[(2.0 * x), $MachinePrecision]), $MachinePrecision], N[(eps * eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.19999999999999985e-83 or 1.55000000000000004e-77 < x

   1. Initial program 26.9%

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

      1. +-commutative 26.9%

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

      2. unpow2 26.9%

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

      3. unpow2 26.9%

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

      4. difference-of-squares 26.9%

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

      5. *-commutative 26.9%

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

      6. sub-neg 26.9%

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

      7. +-commutative 26.9%

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

      8. associate-+l+ 26.9%

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

      9. remove-double-neg 26.9%

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

      10. sub-neg 26.9%

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

      11. +-commutative 26.9%

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

      12. associate--l+ 99.9%

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

      13. +-inverses 99.9%

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

      14. +-rgt-identity 99.9%

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

      15. remove-double-neg 99.9%

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

      16. sub-neg 99.9%

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

      17. remove-double-neg 99.9%

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

      18. sub-neg 99.9%

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

      19. associate--l- 99.9%

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

      20. neg-mul-1 99.9%

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

      21. neg-mul-1 99.9%

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

      22. distribute-rgt-out 99.9%

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

      23. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in eps around 0 88.6%

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

      1. *-commutative 88.6%

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

      2. associate-*r* 88.6%

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

      3. *-commutative 88.6%

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

   6. Simplified 88.6%

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

   if -6.19999999999999985e-83 < x < 1.55000000000000004e-77

   1. Initial program 97.1%

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

      1. +-commutative 97.1%

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

      2. unpow2 97.1%

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

      3. unpow2 97.1%

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

      4. difference-of-squares 97.1%

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

      5. *-commutative 97.1%

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

      6. sub-neg 97.1%

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

      7. +-commutative 97.1%

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

      8. associate-+l+ 97.1%

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

      9. remove-double-neg 97.1%

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

      10. sub-neg 97.1%

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

      11. +-commutative 97.1%

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

      12. associate--l+ 100.0%

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

      13. +-inverses 100.0%

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

      14. +-rgt-identity 100.0%

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

      15. remove-double-neg 100.0%

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

      16. sub-neg 100.0%

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

      17. remove-double-neg 100.0%

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

      18. sub-neg 100.0%

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

      19. associate--l- 100.0%

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

      20. neg-mul-1 100.0%

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

      21. neg-mul-1 100.0%

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

      22. distribute-rgt-out 100.0%

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

      23. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. flip3-- 53.0%

         \[\leadsto \varepsilon \cdot \color{blue}{\frac{{\varepsilon}^{3} - {\left(x \cdot -2\right)}^{3}}{\varepsilon \cdot \varepsilon + \left(\left(x \cdot -2\right) \cdot \left(x \cdot -2\right) + \varepsilon \cdot \left(x \cdot -2\right)\right)}} \]

      2. associate-*r/ 44.1%

         \[\leadsto \color{blue}{\frac{\varepsilon \cdot \left({\varepsilon}^{3} - {\left(x \cdot -2\right)}^{3}\right)}{\varepsilon \cdot \varepsilon + \left(\left(x \cdot -2\right) \cdot \left(x \cdot -2\right) + \varepsilon \cdot \left(x \cdot -2\right)\right)}} \]

      3. associate-/l* 52.8%

         \[\leadsto \color{blue}{\frac{\varepsilon}{\frac{\varepsilon \cdot \varepsilon + \left(\left(x \cdot -2\right) \cdot \left(x \cdot -2\right) + \varepsilon \cdot \left(x \cdot -2\right)\right)}{{\varepsilon}^{3} - {\left(x \cdot -2\right)}^{3}}}} \]

      4. clear-num 52.9%

         \[\leadsto \frac{\varepsilon}{\color{blue}{\frac{1}{\frac{{\varepsilon}^{3} - {\left(x \cdot -2\right)}^{3}}{\varepsilon \cdot \varepsilon + \left(\left(x \cdot -2\right) \cdot \left(x \cdot -2\right) + \varepsilon \cdot \left(x \cdot -2\right)\right)}}}} \]

      5. flip3-- 99.7%

         \[\leadsto \frac{\varepsilon}{\frac{1}{\color{blue}{\varepsilon - x \cdot -2}}} \]

      6. sub-neg 99.7%

         \[\leadsto \frac{\varepsilon}{\frac{1}{\color{blue}{\varepsilon + \left(-x \cdot -2\right)}}} \]

      7. distribute-rgt-neg-in 99.7%

         \[\leadsto \frac{\varepsilon}{\frac{1}{\varepsilon + \color{blue}{x \cdot \left(--2\right)}}} \]

      8. metadata-eval 99.7%

         \[\leadsto \frac{\varepsilon}{\frac{1}{\varepsilon + x \cdot \color{blue}{2}}} \]

   5. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\varepsilon}{\frac{1}{\varepsilon + x \cdot 2}}} \]

   6. Taylor expanded in eps around inf 94.5%

      \[\leadsto \frac{\varepsilon}{\color{blue}{\frac{1}{\varepsilon}}} \]
   7. Step-by-step derivation

      1. div-inv 94.7%

         \[\leadsto \color{blue}{\varepsilon \cdot \frac{1}{\frac{1}{\varepsilon}}} \]

      2. associate-/r/ 94.8%

         \[\leadsto \varepsilon \cdot \color{blue}{\left(\frac{1}{1} \cdot \varepsilon\right)} \]

      3. metadata-eval 94.8%

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

      4. *-un-lft-identity 94.8%

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

   8. Applied egg-rr 94.8%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-83} \lor \neg \left(x \leq 1.55 \cdot 10^{-77}\right):\\ \;\;\;\;\varepsilon \cdot \left(2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \]

Alternative 3: 100.0% accurate, 29.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.2%

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

   1. +-commutative 75.2%

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

   2. unpow2 75.2%

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

   3. unpow2 75.2%

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

   4. difference-of-squares 75.2%

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

   5. *-commutative 75.2%

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

   6. sub-neg 75.2%

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

   7. +-commutative 75.2%

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

   8. associate-+l+ 75.2%

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

   9. remove-double-neg 75.2%

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

   10. sub-neg 75.2%

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

   11. +-commutative 75.2%

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

   12. associate--l+ 100.0%

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

   13. +-inverses 100.0%

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

   14. +-rgt-identity 100.0%

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

   15. remove-double-neg 100.0%

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

   16. sub-neg 100.0%

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

   17. remove-double-neg 100.0%

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

   18. sub-neg 100.0%

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

   19. associate--l- 100.0%

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

   20. neg-mul-1 100.0%

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

   21. neg-mul-1 100.0%

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

   22. distribute-rgt-out 100.0%

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

   23. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \varepsilon \cdot \left(\varepsilon - x \cdot -2\right) \]

Alternative 4: 71.5% 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 75.2%

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

   1. +-commutative 75.2%

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

   2. unpow2 75.2%

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

   3. unpow2 75.2%

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

   4. difference-of-squares 75.2%

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

   5. *-commutative 75.2%

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

   6. sub-neg 75.2%

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

   7. +-commutative 75.2%

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

   8. associate-+l+ 75.2%

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

   9. remove-double-neg 75.2%

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

   10. sub-neg 75.2%

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

   11. +-commutative 75.2%

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

   12. associate--l+ 100.0%

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

   13. +-inverses 100.0%

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

   14. +-rgt-identity 100.0%

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

   15. remove-double-neg 100.0%

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

   16. sub-neg 100.0%

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

   17. remove-double-neg 100.0%

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

   18. sub-neg 100.0%

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

   19. associate--l- 100.0%

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

   20. neg-mul-1 100.0%

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

   21. neg-mul-1 100.0%

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

   22. distribute-rgt-out 100.0%

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

   23. metadata-eval 100.0%

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

3. Simplified 100.0%

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

   1. flip3-- 67.5%

      \[\leadsto \varepsilon \cdot \color{blue}{\frac{{\varepsilon}^{3} - {\left(x \cdot -2\right)}^{3}}{\varepsilon \cdot \varepsilon + \left(\left(x \cdot -2\right) \cdot \left(x \cdot -2\right) + \varepsilon \cdot \left(x \cdot -2\right)\right)}} \]

   2. associate-*r/ 54.4%

      \[\leadsto \color{blue}{\frac{\varepsilon \cdot \left({\varepsilon}^{3} - {\left(x \cdot -2\right)}^{3}\right)}{\varepsilon \cdot \varepsilon + \left(\left(x \cdot -2\right) \cdot \left(x \cdot -2\right) + \varepsilon \cdot \left(x \cdot -2\right)\right)}} \]

   3. associate-/l* 67.4%

      \[\leadsto \color{blue}{\frac{\varepsilon}{\frac{\varepsilon \cdot \varepsilon + \left(\left(x \cdot -2\right) \cdot \left(x \cdot -2\right) + \varepsilon \cdot \left(x \cdot -2\right)\right)}{{\varepsilon}^{3} - {\left(x \cdot -2\right)}^{3}}}} \]

   4. clear-num 67.4%

      \[\leadsto \frac{\varepsilon}{\color{blue}{\frac{1}{\frac{{\varepsilon}^{3} - {\left(x \cdot -2\right)}^{3}}{\varepsilon \cdot \varepsilon + \left(\left(x \cdot -2\right) \cdot \left(x \cdot -2\right) + \varepsilon \cdot \left(x \cdot -2\right)\right)}}}} \]

   5. flip3-- 99.6%

      \[\leadsto \frac{\varepsilon}{\frac{1}{\color{blue}{\varepsilon - x \cdot -2}}} \]

   6. sub-neg 99.6%

      \[\leadsto \frac{\varepsilon}{\frac{1}{\color{blue}{\varepsilon + \left(-x \cdot -2\right)}}} \]

   7. distribute-rgt-neg-in 99.6%

      \[\leadsto \frac{\varepsilon}{\frac{1}{\varepsilon + \color{blue}{x \cdot \left(--2\right)}}} \]

   8. metadata-eval 99.6%

      \[\leadsto \frac{\varepsilon}{\frac{1}{\varepsilon + x \cdot \color{blue}{2}}} \]

5. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{\varepsilon}{\frac{1}{\varepsilon + x \cdot 2}}} \]

6. Taylor expanded in eps around inf 72.2%

   \[\leadsto \frac{\varepsilon}{\color{blue}{\frac{1}{\varepsilon}}} \]
7. Step-by-step derivation

   1. div-inv 72.3%

      \[\leadsto \color{blue}{\varepsilon \cdot \frac{1}{\frac{1}{\varepsilon}}} \]

   2. associate-/r/ 72.4%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\frac{1}{1} \cdot \varepsilon\right)} \]

   3. metadata-eval 72.4%

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

   4. *-un-lft-identity 72.4%

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

8. Applied egg-rr 72.4%

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

9. Final simplification 72.4%

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

Reproduce

herbie shell --seed 2023301 
(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)))