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

Percentage Accurate: 83.4% → 97.0%

Time: 5.9s

Alternatives: 5

Speedup: 29.6×

• could not determine a ground truth

  1. x = 2.1825576252958148e-287
  2. eps = 1.1078634623551513e-91

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: 83.4% 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: 97.0% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-108}:\\ \;\;\;\;\left(x + \left(x + \varepsilon\right)\right) \cdot \left(\left(x + \varepsilon\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon + x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -6.2e-108)
   (* (+ x (+ x eps)) (- (+ x eps) x))
   (* eps (+ eps (* x 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -6.2e-108) {
		tmp = (x + (x + eps)) * ((x + eps) - x);
	} else {
		tmp = eps * (eps + (x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-6.2d-108)) then
        tmp = (x + (x + eps)) * ((x + eps) - x)
    else
        tmp = eps * (eps + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -6.2e-108) {
		tmp = (x + (x + eps)) * ((x + eps) - x);
	} else {
		tmp = eps * (eps + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -6.2e-108:
		tmp = (x + (x + eps)) * ((x + eps) - x)
	else:
		tmp = eps * (eps + (x * 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -6.2e-108)
		tmp = Float64(Float64(x + Float64(x + eps)) * Float64(Float64(x + eps) - x));
	else
		tmp = Float64(eps * Float64(eps + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -6.2e-108)
		tmp = (x + (x + eps)) * ((x + eps) - x);
	else
		tmp = eps * (eps + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6.2e-108], N[(N[(x + N[(x + eps), $MachinePrecision]), $MachinePrecision] * N[(N[(x + eps), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(eps * N[(eps + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.20000000000000028e-108

   1. Initial program 86.2%

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

      1. unpow2 86.2%

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

      2. unpow2 86.2%

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

      3. difference-of-squares 86.2%

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

      4. +-commutative 86.2%

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

      5. +-commutative 86.2%

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

   4. Applied egg-rr 86.2%

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

   if -6.20000000000000028e-108 < x

   1. Initial program 83.0%

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

      1. +-commutative 83.0%

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

      2. unpow2 83.0%

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

      3. unpow2 83.0%

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

      4. difference-of-squares 83.1%

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

      5. *-commutative 83.1%

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

      6. associate--l+ 99.9%

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

      7. +-inverses 99.9%

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

      8. +-rgt-identity 99.9%

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

      9. associate-+l+ 99.9%

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

      10. *-lft-identity 99.9%

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

      11. metadata-eval 99.9%

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

      12. distribute-rgt-out 99.9%

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

      13. distribute-lft-out 99.9%

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

      14. metadata-eval 99.9%

          \[\leadsto \varepsilon \cdot \left(\varepsilon + x \cdot \left(\frac{2}{2} + \color{blue}{1}\right)\right) \]

      15. metadata-eval 99.9%

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-108}:\\ \;\;\;\;\left(x + \left(x + \varepsilon\right)\right) \cdot \left(\left(x + \varepsilon\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon + x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.1% accurate, 14.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-54}:\\ \;\;\;\;\left(\left(x + \varepsilon\right) - x\right) \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon + x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.2e-54) (* (- (+ x eps) x) (* x 2.0)) (* eps (+ eps (* x 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.2e-54) {
		tmp = ((x + eps) - x) * (x * 2.0);
	} else {
		tmp = eps * (eps + (x * 2.0));
	}
	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-54)) then
        tmp = ((x + eps) - x) * (x * 2.0d0)
    else
        tmp = eps * (eps + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -3.2e-54) {
		tmp = ((x + eps) - x) * (x * 2.0);
	} else {
		tmp = eps * (eps + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -3.2e-54:
		tmp = ((x + eps) - x) * (x * 2.0)
	else:
		tmp = eps * (eps + (x * 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.2e-54)
		tmp = Float64(Float64(Float64(x + eps) - x) * Float64(x * 2.0));
	else
		tmp = Float64(eps * Float64(eps + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -3.2e-54)
		tmp = ((x + eps) - x) * (x * 2.0);
	else
		tmp = eps * (eps + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.2e-54], N[(N[(N[(x + eps), $MachinePrecision] - x), $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[(eps * N[(eps + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.19999999999999998e-54

   1. Initial program 84.2%

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

      1. unpow2 84.2%

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

      2. unpow2 84.2%

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

      3. difference-of-squares 84.2%

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

      4. +-commutative 84.2%

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

      5. +-commutative 84.2%

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

   4. Applied egg-rr 84.2%

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

   5. Taylor expanded in eps around 0 81.2%

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

   if -3.19999999999999998e-54 < x

   1. Initial program 83.5%

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

      1. +-commutative 83.5%

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

      2. unpow2 83.5%

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

      3. unpow2 83.5%

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

      4. difference-of-squares 83.5%

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

      5. *-commutative 83.5%

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

      6. associate--l+ 99.1%

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

      7. +-inverses 99.1%

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

      8. +-rgt-identity 99.1%

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

      9. associate-+l+ 99.1%

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

      10. *-lft-identity 99.1%

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

      11. metadata-eval 99.1%

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

      12. distribute-rgt-out 99.1%

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

      13. distribute-lft-out 99.1%

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

      14. metadata-eval 99.1%

          \[\leadsto \varepsilon \cdot \left(\varepsilon + x \cdot \left(\frac{2}{2} + \color{blue}{1}\right)\right) \]

      15. metadata-eval 99.1%

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

      16. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-54}:\\ \;\;\;\;\left(\left(x + \varepsilon\right) - x\right) \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon + x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.0% 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 83.5%

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

   1. +-commutative 83.5%

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

   2. unpow2 83.5%

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

   3. unpow2 83.5%

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

   4. difference-of-squares 83.6%

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

   5. *-commutative 83.6%

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

   6. associate--l+ 92.0%

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

   7. +-inverses 92.0%

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

   8. +-rgt-identity 92.0%

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

   9. associate-+l+ 92.0%

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

   10. *-lft-identity 92.0%

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

   11. metadata-eval 92.0%

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

   12. distribute-rgt-out 92.0%

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

   13. distribute-lft-out 92.0%

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

   14. metadata-eval 92.0%

       \[\leadsto \varepsilon \cdot \left(\varepsilon + x \cdot \left(\frac{2}{2} + \color{blue}{1}\right)\right) \]

   15. metadata-eval 92.0%

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

   16. metadata-eval 92.0%

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

3. Simplified 92.0%

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

5. Final simplification 92.0%

   \[\leadsto \varepsilon \cdot \left(\varepsilon + x \cdot 2\right) \]
6. Add Preprocessing

Alternative 4: 55.4% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.5%

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

   1. +-commutative 83.5%

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

   2. unpow2 83.5%

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

   3. unpow2 83.5%

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

   4. difference-of-squares 83.6%

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

   5. *-commutative 83.6%

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

   6. associate--l+ 92.0%

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

   7. +-inverses 92.0%

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

   8. +-rgt-identity 92.0%

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

   9. associate-+l+ 92.0%

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

   10. *-lft-identity 92.0%

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

   11. metadata-eval 92.0%

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

   12. distribute-rgt-out 92.0%

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

   13. distribute-lft-out 92.0%

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

   14. metadata-eval 92.0%

       \[\leadsto \varepsilon \cdot \left(\varepsilon + x \cdot \left(\frac{2}{2} + \color{blue}{1}\right)\right) \]

   15. metadata-eval 92.0%

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

   16. metadata-eval 92.0%

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

3. Simplified 92.0%

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

5. Taylor expanded in eps around 0 55.0%

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

6. Final simplification 55.0%

   \[\leadsto 2 \cdot \left(x \cdot \varepsilon\right) \]
7. Add Preprocessing

Alternative 5: 55.4% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.5%

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

   1. +-commutative 83.5%

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

   2. unpow2 83.5%

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

   3. unpow2 83.5%

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

   4. difference-of-squares 83.6%

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

   5. *-commutative 83.6%

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

   6. associate--l+ 92.0%

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

   7. +-inverses 92.0%

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

   8. +-rgt-identity 92.0%

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

   9. associate-+l+ 92.0%

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

   10. *-lft-identity 92.0%

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

   11. metadata-eval 92.0%

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

   12. distribute-rgt-out 92.0%

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

   13. distribute-lft-out 92.0%

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

   14. metadata-eval 92.0%

       \[\leadsto \varepsilon \cdot \left(\varepsilon + x \cdot \left(\frac{2}{2} + \color{blue}{1}\right)\right) \]

   15. metadata-eval 92.0%

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

   16. metadata-eval 92.0%

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

3. Simplified 92.0%

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

5. Taylor expanded in eps around 0 55.0%

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

   1. associate-*r* 55.0%

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

7. Simplified 55.0%

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

8. Final simplification 55.0%

   \[\leadsto x \cdot \left(\varepsilon \cdot 2\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(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)))