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

Percentage Accurate: 75.0% → 100.0%

Time: 5.9s

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: 75.0% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.6%

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

   1. +-commutative 78.6%

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

   2. unpow2 78.6%

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

   3. unpow2 78.6%

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

   4. difference-of-squares 78.6%

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

   5. sub-neg 78.6%

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

   6. distribute-lft-in 78.6%

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

   7. +-commutative 78.6%

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

   8. distribute-lft-in 78.6%

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

   9. +-commutative 78.6%

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

   10. sub-neg 78.6%

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

   11. associate--l+ 99.9%

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

   12. +-inverses 99.9%

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

   13. +-rgt-identity 99.9%

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

   14. *-commutative 99.9%

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

   15. associate-+l+ 100.0%

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

   16. count-2 100.0%

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

   17. *-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 99.9%

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

   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. Final simplification 100.0%

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

Alternative 2: 90.9% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-101} \lor \neg \left(x \leq 4.2 \cdot 10^{-85}\right):\\ \;\;\;\;2 \cdot \left(\varepsilon \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -6.2e-101) (not (<= x 4.2e-85))) (* 2.0 (* eps x)) (* eps eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -6.2e-101) || !(x <= 4.2e-85)) {
		tmp = 2.0 * (eps * 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-101)) .or. (.not. (x <= 4.2d-85))) then
        tmp = 2.0d0 * (eps * 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-101) || !(x <= 4.2e-85)) {
		tmp = 2.0 * (eps * x);
	} else {
		tmp = eps * eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -6.2e-101) or not (x <= 4.2e-85):
		tmp = 2.0 * (eps * x)
	else:
		tmp = eps * eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -6.2e-101) || !(x <= 4.2e-85))
		tmp = Float64(2.0 * Float64(eps * x));
	else
		tmp = Float64(eps * eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -6.2e-101) || ~((x <= 4.2e-85)))
		tmp = 2.0 * (eps * x);
	else
		tmp = eps * eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -6.2e-101], N[Not[LessEqual[x, 4.2e-85]], $MachinePrecision]], N[(2.0 * N[(eps * x), $MachinePrecision]), $MachinePrecision], N[(eps * eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.19999999999999946e-101 or 4.2e-85 < x

   1. Initial program 37.5%

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

      1. +-commutative 37.5%

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

      2. unpow2 37.5%

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

      3. unpow2 37.5%

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

      4. difference-of-squares 37.6%

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

      5. sub-neg 37.6%

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

      6. distribute-lft-in 37.5%

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

      7. +-commutative 37.5%

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

      8. distribute-lft-in 37.6%

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

      9. +-commutative 37.6%

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

      10. sub-neg 37.6%

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

      11. associate--l+ 99.8%

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

      12. +-inverses 99.8%

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

      13. +-rgt-identity 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-+l+ 99.9%

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

      16. count-2 99.9%

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

      17. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in eps around 0 82.5%

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

      1. *-commutative 82.5%

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

   7. Simplified 82.5%

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

   if -6.19999999999999946e-101 < x < 4.2e-85

   1. Initial program 98.0%

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

      1. +-commutative 98.0%

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

      2. unpow2 98.0%

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

      3. unpow2 98.0%

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

      4. difference-of-squares 98.0%

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

      5. sub-neg 98.0%

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

      6. distribute-lft-in 97.9%

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

      7. +-commutative 97.9%

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

      8. distribute-lft-in 98.0%

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

      9. +-commutative 98.0%

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

      10. sub-neg 98.0%

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

      11. associate--l+ 100.0%

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

      12. +-inverses 100.0%

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

      13. +-rgt-identity 100.0%

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

      14. *-commutative 100.0%

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

      15. associate-+l+ 100.0%

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

      16. count-2 100.0%

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

      17. *-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 95.8%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-101} \lor \neg \left(x \leq 4.2 \cdot 10^{-85}\right):\\ \;\;\;\;2 \cdot \left(\varepsilon \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.8% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-98}:\\ \;\;\;\;x \cdot \left(\varepsilon \cdot 2\right)\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-84}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\varepsilon \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.1e-98)
   (* x (* eps 2.0))
   (if (<= x 2.3e-84) (* eps eps) (* 2.0 (* eps x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.1e-98) {
		tmp = x * (eps * 2.0);
	} else if (x <= 2.3e-84) {
		tmp = eps * eps;
	} else {
		tmp = 2.0 * (eps * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.1d-98)) then
        tmp = x * (eps * 2.0d0)
    else if (x <= 2.3d-84) then
        tmp = eps * eps
    else
        tmp = 2.0d0 * (eps * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.1e-98) {
		tmp = x * (eps * 2.0);
	} else if (x <= 2.3e-84) {
		tmp = eps * eps;
	} else {
		tmp = 2.0 * (eps * x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.1e-98:
		tmp = x * (eps * 2.0)
	elif x <= 2.3e-84:
		tmp = eps * eps
	else:
		tmp = 2.0 * (eps * x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.1e-98)
		tmp = Float64(x * Float64(eps * 2.0));
	elseif (x <= 2.3e-84)
		tmp = Float64(eps * eps);
	else
		tmp = Float64(2.0 * Float64(eps * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.1e-98)
		tmp = x * (eps * 2.0);
	elseif (x <= 2.3e-84)
		tmp = eps * eps;
	else
		tmp = 2.0 * (eps * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.1e-98], N[(x * N[(eps * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.3e-84], N[(eps * eps), $MachinePrecision], N[(2.0 * N[(eps * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.09999999999999992e-98

   1. Initial program 42.6%

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

      1. +-commutative 42.6%

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

      2. unpow2 42.6%

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

      3. unpow2 42.6%

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

      4. difference-of-squares 42.6%

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

      5. sub-neg 42.6%

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

      6. distribute-lft-in 42.5%

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

      7. +-commutative 42.5%

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

      8. distribute-lft-in 42.6%

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

      9. +-commutative 42.6%

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

      10. sub-neg 42.6%

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

      11. associate--l+ 99.8%

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

      12. +-inverses 99.8%

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

      13. +-rgt-identity 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-+l+ 99.9%

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

      16. count-2 99.9%

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

      17. *-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. fma-define 99.9%

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

      3. associate-*l* 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in eps around 0 80.0%

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

      1. *-commutative 80.0%

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

      2. associate-*r* 80.0%

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

      3. *-commutative 80.0%

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

      4. associate-*r* 80.0%

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

   9. Simplified 80.0%

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

   if -2.09999999999999992e-98 < x < 2.29999999999999981e-84

   1. Initial program 98.0%

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

      1. +-commutative 98.0%

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

      2. unpow2 98.0%

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

      3. unpow2 98.0%

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

      4. difference-of-squares 98.0%

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

      5. sub-neg 98.0%

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

      6. distribute-lft-in 97.9%

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

      7. +-commutative 97.9%

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

      8. distribute-lft-in 98.0%

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

      9. +-commutative 98.0%

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

      10. sub-neg 98.0%

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

      11. associate--l+ 100.0%

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

      12. +-inverses 100.0%

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

      13. +-rgt-identity 100.0%

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

      14. *-commutative 100.0%

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

      15. associate-+l+ 100.0%

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

      16. count-2 100.0%

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

      17. *-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 95.8%

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

   if 2.29999999999999981e-84 < x

   1. Initial program 30.4%

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

      1. +-commutative 30.4%

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

      2. unpow2 30.4%

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

      3. unpow2 30.4%

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

      4. difference-of-squares 30.4%

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

      5. sub-neg 30.4%

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

      6. distribute-lft-in 30.3%

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

      7. +-commutative 30.3%

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

      8. distribute-lft-in 30.4%

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

      9. +-commutative 30.4%

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

      10. sub-neg 30.4%

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

      11. associate--l+ 99.9%

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

      12. +-inverses 99.9%

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

      13. +-rgt-identity 99.9%

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

      14. *-commutative 99.9%

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

      15. associate-+l+ 99.9%

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

      16. count-2 99.9%

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

      17. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in eps around 0 86.1%

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

      1. *-commutative 86.1%

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

   7. Simplified 86.1%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-98}:\\ \;\;\;\;x \cdot \left(\varepsilon \cdot 2\right)\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-84}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\varepsilon \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 78.6%

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

   1. +-commutative 78.6%

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

   2. unpow2 78.6%

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

   3. unpow2 78.6%

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

   4. difference-of-squares 78.6%

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

   5. sub-neg 78.6%

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

   6. distribute-lft-in 78.6%

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

   7. +-commutative 78.6%

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

   8. distribute-lft-in 78.6%

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

   9. +-commutative 78.6%

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

   10. sub-neg 78.6%

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

   11. associate--l+ 99.9%

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

   12. +-inverses 99.9%

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

   13. +-rgt-identity 99.9%

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

   14. *-commutative 99.9%

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

   15. associate-+l+ 100.0%

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

   16. count-2 100.0%

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

   17. *-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. Add Preprocessing

Alternative 5: 72.7% 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 78.6%

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

   1. +-commutative 78.6%

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

   2. unpow2 78.6%

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

   3. unpow2 78.6%

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

   4. difference-of-squares 78.6%

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

   5. sub-neg 78.6%

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

   6. distribute-lft-in 78.6%

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

   7. +-commutative 78.6%

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

   8. distribute-lft-in 78.6%

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

   9. +-commutative 78.6%

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

   10. sub-neg 78.6%

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

   11. associate--l+ 99.9%

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

   12. +-inverses 99.9%

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

   13. +-rgt-identity 99.9%

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

   14. *-commutative 99.9%

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

   15. associate-+l+ 100.0%

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

   16. count-2 100.0%

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

   17. *-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 73.9%

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

Reproduce

herbie shell --seed 2024132 
(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)))