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

Percentage Accurate: 75.0% → 100.0%

Time: 4.5s

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

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

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 74.7%

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

   1. unpow2 74.7%

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

   2. unpow2 74.7%

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

   3. difference-of-squares 74.7%

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

   4. *-commutative 74.7%

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

   5. +-commutative 74.7%

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

   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 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. Final simplification 100.0%

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

Alternative 2: 90.6% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-81} \lor \neg \left(x \leq 4.1 \cdot 10^{-98}\right):\\ \;\;\;\;2 \cdot \left(x \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -6e-81) (not (<= x 4.1e-98))) (* 2.0 (* x eps)) (* eps eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -6e-81) || !(x <= 4.1e-98)) {
		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 <= (-6d-81)) .or. (.not. (x <= 4.1d-98))) 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 <= -6e-81) || !(x <= 4.1e-98)) {
		tmp = 2.0 * (x * eps);
	} else {
		tmp = eps * eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -6e-81) or not (x <= 4.1e-98):
		tmp = 2.0 * (x * eps)
	else:
		tmp = eps * eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -6e-81) || !(x <= 4.1e-98))
		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 <= -6e-81) || ~((x <= 4.1e-98)))
		tmp = 2.0 * (x * eps);
	else
		tmp = eps * eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -6e-81], N[Not[LessEqual[x, 4.1e-98]], $MachinePrecision]], N[(2.0 * N[(x * eps), $MachinePrecision]), $MachinePrecision], N[(eps * eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.9999999999999998e-81 or 4.0999999999999998e-98 < x

   1. Initial program 28.4%

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

      1. unpow2 28.4%

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

      2. unpow2 28.4%

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

      3. difference-of-squares 28.4%

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

      4. *-commutative 28.4%

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

      5. +-commutative 28.4%

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

      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 88.8%

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

   if -5.9999999999999998e-81 < x < 4.0999999999999998e-98

   1. Initial program 94.9%

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

      1. unpow2 94.9%

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

      2. unpow2 94.9%

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

      3. difference-of-squares 95.0%

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

      4. *-commutative 95.0%

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

      5. +-commutative 95.0%

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

      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 93.7%

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

      1. unpow2 93.7%

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

   6. Simplified 93.7%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-81} \lor \neg \left(x \leq 4.1 \cdot 10^{-98}\right):\\ \;\;\;\;2 \cdot \left(x \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \]

Alternative 3: 90.5% accurate, 22.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{-79}:\\ \;\;\;\;x \cdot \left(\varepsilon + \varepsilon\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-97}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(x \cdot \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.16e-79)
   (* x (+ eps eps))
   (if (<= x 1.4e-97) (* eps eps) (* 2.0 (* x eps)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.16e-79) {
		tmp = x * (eps + eps);
	} else if (x <= 1.4e-97) {
		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 <= (-1.16d-79)) then
        tmp = x * (eps + eps)
    else if (x <= 1.4d-97) 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 <= -1.16e-79) {
		tmp = x * (eps + eps);
	} else if (x <= 1.4e-97) {
		tmp = eps * eps;
	} else {
		tmp = 2.0 * (x * eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.16e-79:
		tmp = x * (eps + eps)
	elif x <= 1.4e-97:
		tmp = eps * eps
	else:
		tmp = 2.0 * (x * eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.16e-79)
		tmp = Float64(x * Float64(eps + eps));
	elseif (x <= 1.4e-97)
		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 <= -1.16e-79)
		tmp = x * (eps + eps);
	elseif (x <= 1.4e-97)
		tmp = eps * eps;
	else
		tmp = 2.0 * (x * eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.16e-79], N[(x * N[(eps + eps), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.4e-97], N[(eps * eps), $MachinePrecision], N[(2.0 * N[(x * eps), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.15999999999999999e-79

   1. Initial program 32.8%

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

      1. unpow2 32.8%

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

      2. unpow2 32.8%

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

      3. difference-of-squares 32.6%

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

      4. *-commutative 32.6%

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

      5. +-commutative 32.6%

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

      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. Taylor expanded in x around inf 91.1%

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

      1. *-commutative 91.1%

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

      2. count-2 91.1%

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

      3. distribute-lft-out 91.1%

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

   8. Simplified 91.1%

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

   if -1.15999999999999999e-79 < x < 1.4000000000000001e-97

   1. Initial program 94.9%

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

      1. unpow2 94.9%

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

      2. unpow2 94.9%

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

      3. difference-of-squares 95.0%

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

      4. *-commutative 95.0%

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

      5. +-commutative 95.0%

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

      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 93.7%

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

      1. unpow2 93.7%

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

   6. Simplified 93.7%

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

   if 1.4000000000000001e-97 < x

   1. Initial program 25.3%

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

      1. unpow2 25.3%

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

      2. unpow2 25.3%

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

      3. difference-of-squares 25.5%

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

      4. *-commutative 25.5%

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

      5. +-commutative 25.5%

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

      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 87.2%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{-79}:\\ \;\;\;\;x \cdot \left(\varepsilon + \varepsilon\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-97}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(x \cdot \varepsilon\right)\\ \end{array} \]

Alternative 4: 100.0% accurate, 29.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. unpow2 74.7%

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

3. Simplified 74.7%

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

   1. unpow2 74.7%

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

   2. difference-of-squares 74.7%

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

   3. +-commutative 74.7%

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

   4. +-commutative 74.7%

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

5. Applied egg-rr 74.7%

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

6. Taylor expanded in eps around 0 100.0%

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

7. Taylor expanded in eps around inf 100.0%

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

8. Final simplification 100.0%

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

Alternative 5: 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 74.7%

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

   1. unpow2 74.7%

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

   2. unpow2 74.7%

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

   3. difference-of-squares 74.7%

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

   4. *-commutative 74.7%

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

   5. +-commutative 74.7%

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

   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 71.8%

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

   1. unpow2 71.8%

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

6. Simplified 71.8%

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

7. Final simplification 71.8%

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

Reproduce

herbie shell --seed 2023178 
(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)))