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

Percentage Accurate: 75.1% → 100.0%

Time: 5.7s

Alternatives: 4

Speedup: 17.4×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 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 79.1%

   \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
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. lower-*.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. Applied rewrites 100.0%

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

7. Applied rewrites 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 97.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	double tmp;
	if ((pow((x + eps), 2.0) - pow(x, 2.0)) <= 0.0) {
		tmp = eps * (x + 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 + eps) ** 2.0d0) - (x ** 2.0d0)) <= 0.0d0) then
        tmp = eps * (x + x)
    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 = eps * (x + x);
	} 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 = eps * (x + x)
	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(eps * Float64(x + x));
	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 = eps * (x + x);
	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[(eps * N[(x + x), $MachinePrecision]), $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 67.6%

      \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
   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. lower-*.f64 N/A

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

      4. lower-*.f64 98.7

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

   5. Applied rewrites 98.7%

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

   7. Applied rewrites 98.7%

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

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

   1. Initial program 97.8%

      \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
   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. lower-*.f64 93.3

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

   5. Applied rewrites 93.3%

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

4. Final simplification 96.6%

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

Alternative 3: 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 79.1%

   \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
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. lower-*.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. Applied rewrites 100.0%

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

Alternative 4: 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 79.1%

   \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
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. lower-*.f64 76.5

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

5. Applied rewrites 76.5%

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

Reproduce

herbie shell --seed 2024233 
(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)))