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

Percentage Accurate: 75.2% → 100.0%

Time: 6.0s

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: 75.2% 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, 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 73.4%

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

   1. +-commutative 73.4%

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

   2. unpow2 73.4%

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

   3. unpow2 73.4%

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

   4. difference-of-squares 73.4%

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

   5. sub-neg 73.4%

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

   6. distribute-lft-in 73.3%

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

   7. +-commutative 73.3%

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

   8. distribute-lft-in 73.4%

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

   9. associate-+l+ 73.4%

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

   10. remove-double-neg 73.4%

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

   11. sub-neg 73.4%

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

   12. +-commutative 73.4%

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

   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. Add Preprocessing

Alternative 2: 90.2% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{-89} \lor \neg \left(x \leq 8.5 \cdot 10^{-91}\right):\\ \;\;\;\;2 \cdot \left(\varepsilon \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.16e-89) (not (<= x 8.5e-91))) (* 2.0 (* eps x)) (* eps eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.16e-89) || !(x <= 8.5e-91)) {
		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 <= (-1.16d-89)) .or. (.not. (x <= 8.5d-91))) 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 <= -1.16e-89) || !(x <= 8.5e-91)) {
		tmp = 2.0 * (eps * x);
	} else {
		tmp = eps * eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -1.16e-89) or not (x <= 8.5e-91):
		tmp = 2.0 * (eps * x)
	else:
		tmp = eps * eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -1.16e-89) || !(x <= 8.5e-91))
		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 <= -1.16e-89) || ~((x <= 8.5e-91)))
		tmp = 2.0 * (eps * x);
	else
		tmp = eps * eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -1.16e-89], N[Not[LessEqual[x, 8.5e-91]], $MachinePrecision]], N[(2.0 * N[(eps * x), $MachinePrecision]), $MachinePrecision], N[(eps * eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.15999999999999993e-89 or 8.49999999999999985e-91 < x

   1. Initial program 19.3%

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

      1. +-commutative 19.3%

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

      2. unpow2 19.3%

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

      3. unpow2 19.3%

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

      4. difference-of-squares 19.3%

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

      5. sub-neg 19.3%

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

      6. distribute-lft-in 19.3%

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

      7. +-commutative 19.3%

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

      8. distribute-lft-in 19.3%

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

      9. associate-+l+ 19.3%

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

      10. remove-double-neg 19.3%

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

      11. sub-neg 19.3%

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

      12. +-commutative 19.3%

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

      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 0 96.7%

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

      1. *-commutative 96.7%

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

   7. Simplified 96.7%

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

   if -1.15999999999999993e-89 < x < 8.49999999999999985e-91

   1. Initial program 95.3%

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

      1. +-commutative 95.3%

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

      2. unpow2 95.3%

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

      3. unpow2 95.3%

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

      4. difference-of-squares 95.3%

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

      5. sub-neg 95.3%

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

      6. distribute-lft-in 95.3%

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

      7. +-commutative 95.3%

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

      8. distribute-lft-in 95.3%

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

      9. associate-+l+ 95.3%

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

      10. remove-double-neg 95.3%

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

      11. sub-neg 95.3%

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

      12. +-commutative 95.3%

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

      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 93.0%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{-89} \lor \neg \left(x \leq 8.5 \cdot 10^{-91}\right):\\ \;\;\;\;2 \cdot \left(\varepsilon \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.2% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-89}:\\ \;\;\;\;\varepsilon \cdot \left(x + x\right)\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-91}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\varepsilon \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.35e-89)
   (* eps (+ x x))
   (if (<= x 8.5e-91) (* eps eps) (* 2.0 (* eps x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.35e-89) {
		tmp = eps * (x + x);
	} else if (x <= 8.5e-91) {
		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 <= (-1.35d-89)) then
        tmp = eps * (x + x)
    else if (x <= 8.5d-91) 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 <= -1.35e-89) {
		tmp = eps * (x + x);
	} else if (x <= 8.5e-91) {
		tmp = eps * eps;
	} else {
		tmp = 2.0 * (eps * x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.35e-89:
		tmp = eps * (x + x)
	elif x <= 8.5e-91:
		tmp = eps * eps
	else:
		tmp = 2.0 * (eps * x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.35e-89)
		tmp = Float64(eps * Float64(x + x));
	elseif (x <= 8.5e-91)
		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 <= -1.35e-89)
		tmp = eps * (x + x);
	elseif (x <= 8.5e-91)
		tmp = eps * eps;
	else
		tmp = 2.0 * (eps * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.35e-89], N[(eps * N[(x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.5e-91], N[(eps * eps), $MachinePrecision], N[(2.0 * N[(eps * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.34999999999999994e-89

   1. Initial program 19.8%

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

      1. sub-neg 19.8%

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

   4. Applied egg-rr 19.8%

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

      1. sub-neg 19.8%

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

      2. unpow2 19.8%

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

      3. unpow2 19.8%

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

      4. difference-of-squares 19.9%

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

      5. *-commutative 19.9%

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

      6. +-commutative 19.9%

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

      7. +-commutative 19.9%

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

      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 97.0%

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

   if -1.34999999999999994e-89 < x < 8.49999999999999985e-91

   1. Initial program 95.3%

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

      1. +-commutative 95.3%

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

      2. unpow2 95.3%

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

      3. unpow2 95.3%

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

      4. difference-of-squares 95.3%

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

      5. sub-neg 95.3%

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

      6. distribute-lft-in 95.3%

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

      7. +-commutative 95.3%

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

      8. distribute-lft-in 95.3%

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

      9. associate-+l+ 95.3%

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

      10. remove-double-neg 95.3%

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

      11. sub-neg 95.3%

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

      12. +-commutative 95.3%

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

      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 93.0%

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

   if 8.49999999999999985e-91 < x

   1. Initial program 18.8%

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

      1. +-commutative 18.8%

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

      2. unpow2 18.8%

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

      3. unpow2 18.8%

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

      4. difference-of-squares 18.7%

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

      5. sub-neg 18.7%

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

      6. distribute-lft-in 18.8%

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

      7. +-commutative 18.8%

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

      8. distribute-lft-in 18.7%

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

      9. associate-+l+ 18.7%

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

      10. remove-double-neg 18.7%

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

      11. sub-neg 18.7%

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

      12. +-commutative 18.7%

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

      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 0 96.3%

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

      1. *-commutative 96.3%

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

   7. Simplified 96.3%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-89}:\\ \;\;\;\;\varepsilon \cdot \left(x + x\right)\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-91}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\varepsilon \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.6% 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 73.4%

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

   1. +-commutative 73.4%

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

   2. unpow2 73.4%

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

   3. unpow2 73.4%

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

   4. difference-of-squares 73.4%

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

   5. sub-neg 73.4%

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

   6. distribute-lft-in 73.3%

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

   7. +-commutative 73.3%

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

   8. distribute-lft-in 73.4%

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

   9. associate-+l+ 73.4%

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

   10. remove-double-neg 73.4%

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

   11. sub-neg 73.4%

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

   12. +-commutative 73.4%

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

   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 71.0%

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

Alternative 5: 2.9% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return -2.0

Julia

function code(x, eps)
	return -2.0
end

MATLAB

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

Wolfram

code[x_, eps_] := -2.0

Derivation

1. Initial program 73.4%

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

   1. sub-neg 73.4%

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

4. Applied egg-rr 73.4%

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

   1. sub-neg 73.4%

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

   2. unpow2 73.4%

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

   3. unpow2 73.4%

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

   4. difference-of-squares 73.4%

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

   5. *-commutative 73.4%

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

   6. +-commutative 73.4%

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

   7. +-commutative 73.4%

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

   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 63.3%

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

   1. add-log-exp 36.1%

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

   2. exp-prod 36.1%

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

   3. flip-+ 35.9%

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

   4. div-inv 35.9%

      \[\leadsto \log \left({\left(e^{\varepsilon}\right)}^{\color{blue}{\left(\left(x \cdot x - x \cdot x\right) \cdot \frac{1}{x - x}\right)}}\right) \]

   5. +-inverses 35.9%

      \[\leadsto \log \left({\left(e^{\varepsilon}\right)}^{\left(\color{blue}{0} \cdot \frac{1}{x - x}\right)}\right) \]

   6. +-inverses 35.9%

      \[\leadsto \log \left({\left(e^{\varepsilon}\right)}^{\left(\color{blue}{\left(x - x\right)} \cdot \frac{1}{x - x}\right)}\right) \]

   7. pow-unpow 36.1%

      \[\leadsto \log \color{blue}{\left({\left({\left(e^{\varepsilon}\right)}^{\left(x - x\right)}\right)}^{\left(\frac{1}{x - x}\right)}\right)} \]

   8. +-inverses 36.1%

      \[\leadsto \log \left({\left({\left(e^{\varepsilon}\right)}^{\color{blue}{0}}\right)}^{\left(\frac{1}{x - x}\right)}\right) \]

   9. metadata-eval 36.1%

      \[\leadsto \log \left({\color{blue}{1}}^{\left(\frac{1}{x - x}\right)}\right) \]

   10. metadata-eval 36.1%

       \[\leadsto \log \left({\color{blue}{\left({\left(e^{1}\right)}^{0}\right)}}^{\left(\frac{1}{x - x}\right)}\right) \]

   11. +-inverses 36.1%

       \[\leadsto \log \left({\left({\left(e^{1}\right)}^{\color{blue}{\left(x - x\right)}}\right)}^{\left(\frac{1}{x - x}\right)}\right) \]

   12. pow-unpow 0.0%

       \[\leadsto \log \color{blue}{\left({\left(e^{1}\right)}^{\left(\left(x - x\right) \cdot \frac{1}{x - x}\right)}\right)} \]

   13. +-inverses 0.0%

       \[\leadsto \log \left({\left(e^{1}\right)}^{\left(\color{blue}{0} \cdot \frac{1}{x - x}\right)}\right) \]

   14. +-inverses 0.0%

       \[\leadsto \log \left({\left(e^{1}\right)}^{\left(\color{blue}{\left(x \cdot x - x \cdot x\right)} \cdot \frac{1}{x - x}\right)}\right) \]

   15. div-inv 0.0%

       \[\leadsto \log \left({\left(e^{1}\right)}^{\color{blue}{\left(\frac{x \cdot x - x \cdot x}{x - x}\right)}}\right) \]

   16. flip-+ 36.0%

       \[\leadsto \log \left({\left(e^{1}\right)}^{\color{blue}{\left(x + x\right)}}\right) \]

   17. exp-prod 36.0%

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

   18. *-un-lft-identity 36.0%

       \[\leadsto \log \left(e^{\color{blue}{x + x}}\right) \]

   19. *-un-lft-identity 36.0%

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

   20. log-prod 36.0%

       \[\leadsto \color{blue}{\log 1 + \log \left(e^{x + x}\right)} \]

   21. metadata-eval 36.0%

       \[\leadsto \color{blue}{0} + \log \left(e^{x + x}\right) \]

9. Applied egg-rr 0.0%

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

11. Simplified 2.9%

    \[\leadsto \color{blue}{-2} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024188 
(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)))