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

Percentage Accurate: 75.1% → 100.0%

Time: 5.8s

Alternatives: 5

Speedup: 29.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.1% 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, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

   1. unpow2 79.7%

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

   2. unpow2 79.7%

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

   3. difference-of-squares 79.7%

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

   4. *-commutative 79.7%

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

   5. +-commutative 79.7%

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

   6. associate--l+ 100.0%

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

   7. +-inverses 100.0%

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

   8. +-rgt-identity 100.0%

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

   9. +-commutative 100.0%

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

   10. associate-+r+ 100.0%

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

   11. count-2 100.0%

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

   12. fma-def 100.0%

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

3. Simplified 100.0%

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

   1. fma-udef 100.0%

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

   2. distribute-rgt-in 100.0%

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

   3. fma-def 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 87.4% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-83} \lor \neg \left(x \leq 2.5 \cdot 10^{-38}\right):\\ \;\;\;\;2 \cdot \left(x \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -2.3e-83) (not (<= x 2.5e-38))) (* 2.0 (* x eps)) (* eps eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -2.3e-83) || !(x <= 2.5e-38)) {
		tmp = 2.0 * (x * eps);
	} else {
		tmp = eps * eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-2.3d-83)) .or. (.not. (x <= 2.5d-38))) then
        tmp = 2.0d0 * (x * eps)
    else
        tmp = eps * eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -2.3e-83) || !(x <= 2.5e-38)) {
		tmp = 2.0 * (x * eps);
	} else {
		tmp = eps * eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -2.3e-83) or not (x <= 2.5e-38):
		tmp = 2.0 * (x * eps)
	else:
		tmp = eps * eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -2.3e-83) || !(x <= 2.5e-38))
		tmp = Float64(2.0 * Float64(x * eps));
	else
		tmp = Float64(eps * eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -2.3e-83) || ~((x <= 2.5e-38)))
		tmp = 2.0 * (x * eps);
	else
		tmp = eps * eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -2.3e-83], N[Not[LessEqual[x, 2.5e-38]], $MachinePrecision]], N[(2.0 * N[(x * eps), $MachinePrecision]), $MachinePrecision], N[(eps * eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2999999999999999e-83 or 2.50000000000000017e-38 < x

   1. Initial program 33.5%

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

      1. unpow2 33.5%

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

      2. unpow2 33.5%

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

      3. difference-of-squares 33.6%

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

      4. *-commutative 33.6%

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

      5. +-commutative 33.6%

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

      6. associate--l+ 99.9%

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

      7. +-inverses 99.9%

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

      8. +-rgt-identity 99.9%

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

      9. +-commutative 99.9%

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

      10. associate-+r+ 99.9%

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

      11. count-2 99.9%

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

      12. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in eps around 0 79.7%

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

   if -2.2999999999999999e-83 < x < 2.50000000000000017e-38

   1. Initial program 95.1%

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

      1. unpow2 95.1%

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

      2. unpow2 95.1%

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

      3. difference-of-squares 95.1%

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

      4. *-commutative 95.1%

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

      5. +-commutative 95.1%

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

      6. associate--l+ 100.0%

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

      7. +-inverses 100.0%

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

      8. +-rgt-identity 100.0%

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

      9. +-commutative 100.0%

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

      10. associate-+r+ 100.0%

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

      11. count-2 100.0%

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

      12. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 94.7%

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

      1. unpow2 94.7%

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

   6. Simplified 94.7%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-83} \lor \neg \left(x \leq 2.5 \cdot 10^{-38}\right):\\ \;\;\;\;2 \cdot \left(x \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \]

Alternative 3: 87.4% accurate, 22.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-83}:\\ \;\;\;\;\varepsilon \cdot \left(x + x\right)\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-38}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(x \cdot \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.2e-83)
   (* eps (+ x x))
   (if (<= x 2.5e-38) (* eps eps) (* 2.0 (* x eps)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.2e-83) {
		tmp = eps * (x + x);
	} else if (x <= 2.5e-38) {
		tmp = eps * eps;
	} else {
		tmp = 2.0 * (x * 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 <= (-2.2d-83)) then
        tmp = eps * (x + x)
    else if (x <= 2.5d-38) then
        tmp = eps * eps
    else
        tmp = 2.0d0 * (x * eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.2e-83) {
		tmp = eps * (x + x);
	} else if (x <= 2.5e-38) {
		tmp = eps * eps;
	} else {
		tmp = 2.0 * (x * eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.2e-83:
		tmp = eps * (x + x)
	elif x <= 2.5e-38:
		tmp = eps * eps
	else:
		tmp = 2.0 * (x * eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.2e-83)
		tmp = Float64(eps * Float64(x + x));
	elseif (x <= 2.5e-38)
		tmp = Float64(eps * eps);
	else
		tmp = Float64(2.0 * Float64(x * eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.2e-83)
		tmp = eps * (x + x);
	elseif (x <= 2.5e-38)
		tmp = eps * eps;
	else
		tmp = 2.0 * (x * eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.2e-83], N[(eps * N[(x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.5e-38], N[(eps * eps), $MachinePrecision], N[(2.0 * N[(x * eps), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.20000000000000008e-83

   1. Initial program 38.6%

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

      1. unpow2 38.6%

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

      2. unpow2 38.6%

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

      3. difference-of-squares 38.7%

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

      4. *-commutative 38.7%

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

      5. +-commutative 38.7%

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

      6. associate--l+ 99.9%

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

      7. +-inverses 99.9%

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

      8. +-rgt-identity 99.9%

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

      9. +-commutative 99.9%

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

      10. associate-+r+ 99.9%

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

      11. count-2 99.9%

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

      12. fma-def 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

      2. distribute-rgt-in 99.9%

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

      3. fma-def 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 74.2%

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

      1. *-commutative 74.2%

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

      2. count-2 74.2%

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

      3. distribute-rgt-out 74.2%

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

   8. Simplified 74.2%

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

   if -2.20000000000000008e-83 < x < 2.50000000000000017e-38

   1. Initial program 95.1%

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

      1. unpow2 95.1%

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

      2. unpow2 95.1%

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

      3. difference-of-squares 95.1%

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

      4. *-commutative 95.1%

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

      5. +-commutative 95.1%

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

      6. associate--l+ 100.0%

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

      7. +-inverses 100.0%

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

      8. +-rgt-identity 100.0%

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

      9. +-commutative 100.0%

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

      10. associate-+r+ 100.0%

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

      11. count-2 100.0%

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

      12. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 94.7%

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

      1. unpow2 94.7%

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

   6. Simplified 94.7%

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

   if 2.50000000000000017e-38 < x

   1. Initial program 16.8%

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

      1. unpow2 16.8%

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

      2. unpow2 16.8%

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

      3. difference-of-squares 16.8%

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

      4. *-commutative 16.8%

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

      5. +-commutative 16.8%

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

      6. associate--l+ 99.9%

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

      7. +-inverses 99.9%

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

      8. +-rgt-identity 99.9%

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

      9. +-commutative 99.9%

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

      10. associate-+r+ 99.9%

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

      11. count-2 99.9%

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

      12. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in eps around 0 97.6%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-83}:\\ \;\;\;\;\varepsilon \cdot \left(x + x\right)\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-38}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(x \cdot \varepsilon\right)\\ \end{array} \]

Alternative 4: 100.0% accurate, 29.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

   1. unpow2 79.7%

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

3. Simplified 79.7%

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

   1. unpow2 79.7%

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

   2. +-commutative 79.7%

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

   3. distribute-lft-in 79.7%

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

   4. +-commutative 79.7%

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

   5. add-sqr-sqrt 38.7%

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

   6. sqrt-prod 72.4%

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

   7. unpow2 72.4%

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

   8. pow1/2 72.4%

      \[\leadsto \left(\left(\varepsilon + x\right) \cdot \varepsilon + \color{blue}{{\left({\left(x + \varepsilon\right)}^{2}\right)}^{0.5}} \cdot x\right) - x \cdot x \]

   9. add-sqr-sqrt 38.7%

      \[\leadsto \left(\left(\varepsilon + x\right) \cdot \varepsilon + {\left({\left(x + \varepsilon\right)}^{2}\right)}^{0.5} \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) - x \cdot x \]

   10. sqrt-prod 79.5%

       \[\leadsto \left(\left(\varepsilon + x\right) \cdot \varepsilon + {\left({\left(x + \varepsilon\right)}^{2}\right)}^{0.5} \cdot \color{blue}{\sqrt{x \cdot x}}\right) - x \cdot x \]

   11. pow1/2 79.5%

       \[\leadsto \left(\left(\varepsilon + x\right) \cdot \varepsilon + {\left({\left(x + \varepsilon\right)}^{2}\right)}^{0.5} \cdot \color{blue}{{\left(x \cdot x\right)}^{0.5}}\right) - x \cdot x \]

   12. pow-prod-down 68.6%

       \[\leadsto \left(\left(\varepsilon + x\right) \cdot \varepsilon + \color{blue}{{\left({\left(x + \varepsilon\right)}^{2} \cdot \left(x \cdot x\right)\right)}^{0.5}}\right) - x \cdot x \]

   13. *-commutative 68.6%

       \[\leadsto \left(\left(\varepsilon + x\right) \cdot \varepsilon + {\color{blue}{\left(\left(x \cdot x\right) \cdot {\left(x + \varepsilon\right)}^{2}\right)}}^{0.5}\right) - x \cdot x \]

   14. pow-prod-down 79.5%

       \[\leadsto \left(\left(\varepsilon + x\right) \cdot \varepsilon + \color{blue}{{\left(x \cdot x\right)}^{0.5} \cdot {\left({\left(x + \varepsilon\right)}^{2}\right)}^{0.5}}\right) - x \cdot x \]

   15. pow1/2 79.5%

       \[\leadsto \left(\left(\varepsilon + x\right) \cdot \varepsilon + \color{blue}{\sqrt{x \cdot x}} \cdot {\left({\left(x + \varepsilon\right)}^{2}\right)}^{0.5}\right) - x \cdot x \]

   16. sqrt-prod 38.7%

       \[\leadsto \left(\left(\varepsilon + x\right) \cdot \varepsilon + \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {\left({\left(x + \varepsilon\right)}^{2}\right)}^{0.5}\right) - x \cdot x \]

   17. add-sqr-sqrt 72.4%

       \[\leadsto \left(\left(\varepsilon + x\right) \cdot \varepsilon + \color{blue}{x} \cdot {\left({\left(x + \varepsilon\right)}^{2}\right)}^{0.5}\right) - x \cdot x \]

   18. pow1/2 72.4%

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

   19. unpow2 72.4%

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

   20. sqrt-prod 38.7%

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

   21. add-sqr-sqrt 79.7%

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

   22. +-commutative 79.7%

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

5. Applied egg-rr 79.7%

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

   1. +-commutative 79.7%

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

   2. associate--l+ 79.7%

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

   3. *-commutative 79.7%

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

7. Applied egg-rr 79.7%

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

8. Taylor expanded in x around 0 99.9%

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

   1. unpow2 99.9%

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

   2. +-commutative 99.9%

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

   3. *-commutative 99.9%

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

   4. associate-*r* 100.0%

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

   5. count-2 100.0%

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

   6. distribute-rgt-in 100.0%

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

   7. +-commutative 100.0%

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

10. Simplified 100.0%

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

11. Final simplification 100.0%

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

Alternative 5: 72.5% 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 79.7%

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

   1. unpow2 79.7%

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

   2. unpow2 79.7%

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

   3. difference-of-squares 79.7%

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

   4. *-commutative 79.7%

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

   5. +-commutative 79.7%

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

   6. associate--l+ 100.0%

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

   7. +-inverses 100.0%

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

   8. +-rgt-identity 100.0%

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

   9. +-commutative 100.0%

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

   10. associate-+r+ 100.0%

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

   11. count-2 100.0%

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

   12. fma-def 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in eps around inf 78.3%

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

   1. unpow2 78.3%

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

6. Simplified 78.3%

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

7. Final simplification 78.3%

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

Reproduce

herbie shell --seed 2023238 
(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)))