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

Percentage Accurate: 75.3% → 100.0%

Time: 7.8s

Alternatives: 8

Speedup: 23.2×

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: 75.3% 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, 13.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.5%

   \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-rgt-identity N/A

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

   4. *-inverses N/A

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

   5. associate-/l* N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. unpow2 N/A

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

   10. associate-*l/ N/A

       \[\leadsto \color{blue}{\frac{2 \cdot x}{\varepsilon} \cdot {\varepsilon}^{2}} + {\varepsilon}^{2} \]

   11. associate-*r/ N/A

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

   12. *-commutative N/A

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

   13. *-rgt-identity N/A

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

   14. distribute-lft-in N/A

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

   15. +-commutative N/A

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

   16. unpow2 N/A

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

   17. associate-*l* N/A

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

   18. *-lowering-*.f64 N/A

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

   19. +-commutative N/A

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

   20. distribute-rgt-in N/A

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

5. Simplified 100.0%

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

   1. distribute-rgt-in N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. *-commutative N/A

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

   4. count-2 N/A

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

   5. +-lowering-+.f64 N/A

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

   6. *-lowering-*.f64 100.0

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

7. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x + x, \varepsilon, \varepsilon \cdot \varepsilon\right)} \]
8. Add Preprocessing

Alternative 2: 97.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{2} - {x}^{2} \leq 2 \cdot 10^{-306}:\\ \;\;\;\;\left(x + x\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x + \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- (pow (+ x eps) 2.0) (pow x 2.0)) 2e-306)
   (* (+ x x) eps)
   (* eps (+ x eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((pow((x + eps), 2.0) - pow(x, 2.0)) <= 2e-306) {
		tmp = (x + x) * eps;
	} else {
		tmp = eps * (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 + eps) ** 2.0d0) - (x ** 2.0d0)) <= 2d-306) then
        tmp = (x + x) * eps
    else
        tmp = eps * (x + eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((Math.pow((x + eps), 2.0) - Math.pow(x, 2.0)) <= 2e-306) {
		tmp = (x + x) * eps;
	} else {
		tmp = eps * (x + eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (math.pow((x + eps), 2.0) - math.pow(x, 2.0)) <= 2e-306:
		tmp = (x + x) * eps
	else:
		tmp = eps * (x + eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64((Float64(x + eps) ^ 2.0) - (x ^ 2.0)) <= 2e-306)
		tmp = Float64(Float64(x + x) * eps);
	else
		tmp = Float64(eps * Float64(x + eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((((x + eps) ^ 2.0) - (x ^ 2.0)) <= 2e-306)
		tmp = (x + x) * eps;
	else
		tmp = eps * (x + eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[Power[N[(x + eps), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], 2e-306], N[(N[(x + x), $MachinePrecision] * eps), $MachinePrecision], N[(eps * N[(x + eps), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 2 binary64)) (pow.f64 x #s(literal 2 binary64))) < 2.00000000000000006e-306

   1. Initial program 58.0%

      \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 98.0

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

   5. Simplified 98.0%

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-commutative N/A

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

      4. count-2 N/A

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

      5. +-lowering-+.f64 98.0

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

   7. Applied egg-rr 98.0%

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

   if 2.00000000000000006e-306 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 2 binary64)) (pow.f64 x #s(literal 2 binary64)))

   1. Initial program 98.3%

      \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. *-inverses N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{2 \cdot x}{\varepsilon} \cdot {\varepsilon}^{2}} + {\varepsilon}^{2} \]

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. *-lowering-*.f64 N/A

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

      19. +-commutative N/A

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

      20. distribute-rgt-in N/A

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

   5. Simplified 100.0%

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

      1. *-commutative N/A

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

      2. count-2 N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in eps around inf

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

   10. Simplified 94.7%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{2} - {x}^{2} \leq 2 \cdot 10^{-306}:\\ \;\;\;\;\left(x + x\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x + \varepsilon\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{2} - {x}^{2} \leq 0:\\ \;\;\;\;x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- (pow (+ x eps) 2.0) (pow x 2.0)) 0.0) (* x eps) (* eps eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((pow((x + eps), 2.0) - pow(x, 2.0)) <= 0.0) {
		tmp = 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 + eps) ** 2.0d0) - (x ** 2.0d0)) <= 0.0d0) then
        tmp = x * eps
    else
        tmp = eps * eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((Math.pow((x + eps), 2.0) - Math.pow(x, 2.0)) <= 0.0) {
		tmp = x * eps;
	} else {
		tmp = eps * eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (math.pow((x + eps), 2.0) - math.pow(x, 2.0)) <= 0.0:
		tmp = x * eps
	else:
		tmp = eps * eps
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[Power[N[(x + eps), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], 0.0], N[(x * eps), $MachinePrecision], N[(eps * eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 2 binary64)) (pow.f64 x #s(literal 2 binary64))) < 0.0

   1. Initial program 57.8%

      \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. *-inverses N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{2 \cdot x}{\varepsilon} \cdot {\varepsilon}^{2}} + {\varepsilon}^{2} \]

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. *-lowering-*.f64 N/A

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

      19. +-commutative N/A

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

      20. distribute-rgt-in N/A

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

   5. Simplified 100.0%

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

      1. *-commutative N/A

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

      2. count-2 N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in eps around inf

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

   10. Simplified 61.8%

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

   11. Taylor expanded in eps around 0

       \[\leadsto \color{blue}{\varepsilon \cdot x} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 61.8

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

   13. Simplified 61.8%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 2 binary64)) (pow.f64 x #s(literal 2 binary64)))

   1. Initial program 97.9%

      \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 93.5

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

   5. Simplified 93.5%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{2} - {x}^{2} \leq 0:\\ \;\;\;\;x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 100.0% accurate, 17.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.5%

   \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-rgt-identity N/A

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

   4. *-inverses N/A

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

   5. associate-/l* N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. unpow2 N/A

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

   10. associate-*l/ N/A

       \[\leadsto \color{blue}{\frac{2 \cdot x}{\varepsilon} \cdot {\varepsilon}^{2}} + {\varepsilon}^{2} \]

   11. associate-*r/ N/A

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

   12. *-commutative N/A

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

   13. *-rgt-identity N/A

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

   14. distribute-lft-in N/A

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

   15. +-commutative N/A

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

   16. unpow2 N/A

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

   17. associate-*l* N/A

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

   18. *-lowering-*.f64 N/A

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

   19. +-commutative N/A

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

   20. distribute-rgt-in N/A

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

5. Simplified 100.0%

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

Alternative 5: 100.0% accurate, 17.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

   \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-rgt-identity N/A

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

   4. *-inverses N/A

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

   5. associate-/l* N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. unpow2 N/A

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

   10. associate-*l/ N/A

       \[\leadsto \color{blue}{\frac{2 \cdot x}{\varepsilon} \cdot {\varepsilon}^{2}} + {\varepsilon}^{2} \]

   11. associate-*r/ N/A

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

   12. *-commutative N/A

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

   13. *-rgt-identity N/A

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

   14. distribute-lft-in N/A

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

   15. +-commutative N/A

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

   16. unpow2 N/A

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

   17. associate-*l* N/A

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

   18. *-lowering-*.f64 N/A

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

   19. +-commutative N/A

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

   20. distribute-rgt-in N/A

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

5. Simplified 100.0%

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

   1. *-commutative N/A

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

   2. count-2 N/A

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

   3. associate-+r+ N/A

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

   4. +-commutative N/A

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

   5. +-lowering-+.f64 N/A

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

   6. +-commutative N/A

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

   7. +-lowering-+.f64 100.0

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

7. Applied egg-rr 100.0%

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

8. Final simplification 100.0%

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

Alternative 6: 76.2% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

   \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-rgt-identity N/A

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

   4. *-inverses N/A

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

   5. associate-/l* N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. unpow2 N/A

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

   10. associate-*l/ N/A

       \[\leadsto \color{blue}{\frac{2 \cdot x}{\varepsilon} \cdot {\varepsilon}^{2}} + {\varepsilon}^{2} \]

   11. associate-*r/ N/A

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

   12. *-commutative N/A

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

   13. *-rgt-identity N/A

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

   14. distribute-lft-in N/A

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

   15. +-commutative N/A

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

   16. unpow2 N/A

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

   17. associate-*l* N/A

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

   18. *-lowering-*.f64 N/A

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

   19. +-commutative N/A

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

   20. distribute-rgt-in N/A

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

5. Simplified 100.0%

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

   1. *-commutative N/A

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

   2. count-2 N/A

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

   3. associate-+r+ N/A

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

   4. +-commutative N/A

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

   5. +-lowering-+.f64 N/A

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

   6. +-commutative N/A

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

   7. +-lowering-+.f64 100.0

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

7. Applied egg-rr 100.0%

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

8. Taylor expanded in eps around inf

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

10. Simplified 75.2%

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

11. Final simplification 75.2%

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

Alternative 7: 72.9% accurate, 34.8× 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 74.5%

   \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 71.7

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

5. Simplified 71.7%

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

Alternative 8: 38.5% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 74.5%

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

   1. sub-neg N/A

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

   2. unpow2 N/A

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

   3. distribute-lft-in N/A

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

   4. associate-+l+ N/A

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

   5. *-commutative N/A

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

   6. distribute-rgt-in N/A

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

   7. +-commutative N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. +-lowering-+.f64 N/A

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

   10. distribute-rgt-in N/A

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

   11. +-commutative N/A

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

   12. accelerator-lowering-fma.f64 N/A

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

   13. +-lowering-+.f64 N/A

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

   14. unpow2 N/A

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

   15. distribute-rgt-neg-in N/A

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

   16. *-lowering-*.f64 N/A

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

   17. neg-lowering-neg.f64 66.7

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

4. Applied egg-rr 66.7%

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

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{-1 \cdot {x}^{2} + {x}^{2}} \]
6. Step-by-step derivation

   1. distribute-lft1-in N/A

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

   2. metadata-eval N/A

      \[\leadsto \color{blue}{0} \cdot {x}^{2} \]

   3. mul0-lft 34.8

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

7. Simplified 34.8%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(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)))