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

Percentage Accurate: 75.4% → 100.0%

Time: 3.4s

Alternatives: 5

Speedup: 23.0×

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: 75.4% 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, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

   1. unpow2 75.0%

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

   2. unpow2 75.0%

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

   3. difference-of-squares 75.1%

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

   4. *-commutative 75.1%

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

   5. +-commutative 75.1%

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

   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. Final simplification 100.0%

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

Alternative 2: 90.8% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-96} \lor \neg \left(x \leq 4.2 \cdot 10^{-110}\right):\\ \;\;\;\;2 \cdot \left(\varepsilon \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -3.2e-96) (not (<= x 4.2e-110))) (* 2.0 (* eps x)) (* eps eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -3.2e-96) || !(x <= 4.2e-110)) {
		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 <= (-3.2d-96)) .or. (.not. (x <= 4.2d-110))) 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 <= -3.2e-96) || !(x <= 4.2e-110)) {
		tmp = 2.0 * (eps * x);
	} else {
		tmp = eps * eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -3.2e-96) or not (x <= 4.2e-110):
		tmp = 2.0 * (eps * x)
	else:
		tmp = eps * eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -3.2e-96) || !(x <= 4.2e-110))
		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 <= -3.2e-96) || ~((x <= 4.2e-110)))
		tmp = 2.0 * (eps * x);
	else
		tmp = eps * eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -3.2e-96], N[Not[LessEqual[x, 4.2e-110]], $MachinePrecision]], N[(2.0 * N[(eps * x), $MachinePrecision]), $MachinePrecision], N[(eps * eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.20000000000000012e-96 or 4.20000000000000004e-110 < x

   1. Initial program 33.6%

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

      1. unpow2 33.6%

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

      2. unpow2 33.6%

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

      3. difference-of-squares 33.7%

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

      4. *-commutative 33.7%

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

      5. +-commutative 33.7%

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

      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+ 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 0 83.0%

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

   if -3.20000000000000012e-96 < x < 4.20000000000000004e-110

   1. Initial program 97.9%

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

      1. unpow2 97.9%

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

      2. unpow2 97.9%

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

      3. difference-of-squares 97.9%

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

      4. *-commutative 97.9%

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

      5. +-commutative 97.9%

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

      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 97.1%

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

      1. unpow2 97.1%

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

   6. Simplified 97.1%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-96} \lor \neg \left(x \leq 4.2 \cdot 10^{-110}\right):\\ \;\;\;\;2 \cdot \left(\varepsilon \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \]

Alternative 3: 90.8% accurate, 22.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-96}:\\ \;\;\;\;\varepsilon \cdot \left(x + x\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-112}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\varepsilon \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.8e-96)
   (* eps (+ x x))
   (if (<= x 3e-112) (* eps eps) (* 2.0 (* eps x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.8e-96) {
		tmp = eps * (x + x);
	} else if (x <= 3e-112) {
		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 <= (-3.8d-96)) then
        tmp = eps * (x + x)
    else if (x <= 3d-112) 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 <= -3.8e-96) {
		tmp = eps * (x + x);
	} else if (x <= 3e-112) {
		tmp = eps * eps;
	} else {
		tmp = 2.0 * (eps * x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -3.8e-96:
		tmp = eps * (x + x)
	elif x <= 3e-112:
		tmp = eps * eps
	else:
		tmp = 2.0 * (eps * x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.8e-96)
		tmp = Float64(eps * Float64(x + x));
	elseif (x <= 3e-112)
		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 <= -3.8e-96)
		tmp = eps * (x + x);
	elseif (x <= 3e-112)
		tmp = eps * eps;
	else
		tmp = 2.0 * (eps * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.8e-96], N[(eps * N[(x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3e-112], N[(eps * eps), $MachinePrecision], N[(2.0 * N[(eps * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.8000000000000001e-96

   1. Initial program 31.0%

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

      1. unpow2 31.0%

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

      2. unpow2 31.0%

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

      3. difference-of-squares 30.9%

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

      4. *-commutative 30.9%

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

      5. +-commutative 30.9%

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

      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+ 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%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around inf 80.1%

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

      1. count-2 80.1%

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

      2. distribute-lft-out 80.1%

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

   8. Simplified 80.1%

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

   if -3.8000000000000001e-96 < x < 3.0000000000000001e-112

   1. Initial program 97.9%

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

      1. unpow2 97.9%

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

      2. unpow2 97.9%

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

      3. difference-of-squares 97.9%

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

      4. *-commutative 97.9%

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

      5. +-commutative 97.9%

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

      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 97.1%

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

      1. unpow2 97.1%

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

   6. Simplified 97.1%

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

   if 3.0000000000000001e-112 < x

   1. Initial program 35.7%

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

      1. unpow2 35.7%

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

      2. unpow2 35.7%

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

      3. difference-of-squares 35.9%

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

      4. *-commutative 35.9%

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

      5. +-commutative 35.9%

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

      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 0 85.3%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-96}:\\ \;\;\;\;\varepsilon \cdot \left(x + x\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-112}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\varepsilon \cdot x\right)\\ \end{array} \]

Alternative 4: 100.0% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

   1. unpow2 75.0%

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

   2. unpow2 75.0%

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

   3. difference-of-squares 75.1%

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

   4. *-commutative 75.1%

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

   5. +-commutative 75.1%

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

   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. 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%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 5: 72.8% accurate, 69.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \varepsilon \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

   1. unpow2 75.0%

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

   2. unpow2 75.0%

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

   3. difference-of-squares 75.1%

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

   4. *-commutative 75.1%

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

   5. +-commutative 75.1%

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

   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 72.3%

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

   1. unpow2 72.3%

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

6. Simplified 72.3%

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

7. Final simplification 72.3%

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

Reproduce

herbie shell --seed 2023192 
(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)))