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

Percentage Accurate: 75.6% → 100.0%

Time: 5.6s

Alternatives: 4

Speedup: 29.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. +-commutative 75.5%

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

   2. sqr-pow 75.5%

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

   3. sqr-pow 75.5%

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

   4. unpow2 75.5%

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

   5. unpow2 75.5%

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

   6. difference-of-squares 75.5%

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

   7. associate--l+ 100.0%

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

   8. +-inverses 100.0%

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

   9. +-rgt-identity 100.0%

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

   10. *-commutative 100.0%

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

   11. associate-+l+ 100.0%

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

   12. *-lft-identity 100.0%

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

   13. metadata-eval 100.0%

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

   14. distribute-rgt-out 100.0%

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

   15. distribute-lft-out 100.0%

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

   16. metadata-eval 100.0%

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

   17. metadata-eval 100.0%

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

   18. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in eps around 0 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-*r* 100.0%

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

   3. fma-def 100.0%

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

   4. *-commutative 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. +-commutative 75.5%

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

   2. sqr-pow 75.5%

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

   3. sqr-pow 75.5%

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

   4. unpow2 75.5%

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

   5. unpow2 75.5%

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

   6. difference-of-squares 75.5%

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

   7. associate--l+ 100.0%

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

   8. +-inverses 100.0%

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

   9. +-rgt-identity 100.0%

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

   10. *-commutative 100.0%

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

   11. associate-+l+ 100.0%

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

   12. *-lft-identity 100.0%

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

   13. metadata-eval 100.0%

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

   14. distribute-rgt-out 100.0%

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

   15. distribute-lft-out 100.0%

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

   16. metadata-eval 100.0%

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

   17. metadata-eval 100.0%

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

   18. metadata-eval 100.0%

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

3. Simplified 100.0%

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

   1. distribute-lft-in 100.0%

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

   2. fma-def 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 29.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. +-commutative 75.5%

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

   2. sqr-pow 75.5%

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

   3. sqr-pow 75.5%

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

   4. unpow2 75.5%

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

   5. unpow2 75.5%

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

   6. difference-of-squares 75.5%

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

   7. associate--l+ 100.0%

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

   8. +-inverses 100.0%

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

   9. +-rgt-identity 100.0%

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

   10. *-commutative 100.0%

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

   11. associate-+l+ 100.0%

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

   12. *-lft-identity 100.0%

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

   13. metadata-eval 100.0%

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

   14. distribute-rgt-out 100.0%

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

   15. distribute-lft-out 100.0%

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

   16. metadata-eval 100.0%

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

   17. metadata-eval 100.0%

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

   18. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 4: 63.7% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. +-commutative 75.5%

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

   2. sqr-pow 75.5%

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

   3. sqr-pow 75.5%

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

   4. unpow2 75.5%

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

   5. unpow2 75.5%

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

   6. difference-of-squares 75.5%

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

   7. associate--l+ 100.0%

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

   8. +-inverses 100.0%

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

   9. +-rgt-identity 100.0%

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

   10. *-commutative 100.0%

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

   11. associate-+l+ 100.0%

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

   12. *-lft-identity 100.0%

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

   13. metadata-eval 100.0%

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

   14. distribute-rgt-out 100.0%

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

   15. distribute-lft-out 100.0%

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

   16. metadata-eval 100.0%

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

   17. metadata-eval 100.0%

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

   18. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in eps around 0 64.5%

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

6. Final simplification 64.5%

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

Reproduce

herbie shell --seed 2024027 
(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)))