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

Percentage Accurate: 75.2% → 100.0%

Time: 5.1s

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: 75.2% 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, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.3%

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

   1. unpow2 N/A

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

   2. unpow2 N/A

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

   3. difference-of-squares N/A

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. associate--l+ N/A

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

   7. +-inverses N/A

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

   8. +-rgt-identity N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\left(x + \varepsilon\right) + x\right)}\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(\varepsilon + x\right) + x\right)\right) \]

   11. associate-+l+ N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon + \color{blue}{\left(x + x\right)}\right)\right) \]

   12. --rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(\varepsilon - 0\right) + \left(\color{blue}{x} + x\right)\right)\right) \]

   13. associate-+l- N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon - \color{blue}{\left(0 - \left(x + x\right)\right)}\right)\right) \]

   14. neg-sub0 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon - \left(\mathsf{neg}\left(\left(x + x\right)\right)\right)\right)\right) \]

   15. --lowering--.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \color{blue}{\left(\mathsf{neg}\left(\left(x + x\right)\right)\right)}\right)\right) \]

   16. count-2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \left(\mathsf{neg}\left(2 \cdot x\right)\right)\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \left(\mathsf{neg}\left(x \cdot 2\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

   20. metadata-eval 100.0%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, -2\right)\right)\right) \]

3. Simplified 100.0%

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

   1. sub-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. distribute-lft-neg-out N/A

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

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

      \[\leadsto \mathsf{+.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \color{blue}{\left(\mathsf{neg}\left(\left(x \cdot -2\right) \cdot \varepsilon\right)\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\mathsf{neg}\left(\color{blue}{\left(x \cdot -2\right) \cdot \varepsilon}\right)\right)\right) \]

   6. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\left(\mathsf{neg}\left(x \cdot -2\right)\right) \cdot \color{blue}{\varepsilon}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\left(\mathsf{neg}\left(-2 \cdot x\right)\right) \cdot \varepsilon\right)\right) \]

   8. distribute-lft-neg-in N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot x\right) \cdot \varepsilon\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\left(2 \cdot x\right) \cdot \varepsilon\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(2 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{*.f64}\left(2, \color{blue}{\left(x \cdot \varepsilon\right)}\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{*.f64}\left(2, \left(\varepsilon \cdot \color{blue}{x}\right)\right)\right) \]

   13. *-lowering-*.f64 100.0%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\varepsilon, \color{blue}{x}\right)\right)\right) \]

6. Applied egg-rr 100.0%

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. fma-define N/A

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

   4. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\left(2 \cdot \varepsilon\right), \color{blue}{x}, \left(\varepsilon \cdot \varepsilon\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma.f64}\left(\left(\varepsilon \cdot 2\right), x, \left(\varepsilon \cdot \varepsilon\right)\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\varepsilon, 2\right), x, \left(\varepsilon \cdot \varepsilon\right)\right) \]

   7. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\varepsilon, 2\right), x, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right) \]

8. Applied egg-rr 100.0%

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

Alternative 2: 89.9% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(2 \cdot x\right)\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-130}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-120}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* eps (* 2.0 x))))
   (if (<= x -3.4e-130) t_0 (if (<= x 7.5e-120) (* eps eps) t_0))))

C

double code(double x, double eps) {
	double t_0 = eps * (2.0 * x);
	double tmp;
	if (x <= -3.4e-130) {
		tmp = t_0;
	} else if (x <= 7.5e-120) {
		tmp = eps * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = eps * (2.0d0 * x)
    if (x <= (-3.4d-130)) then
        tmp = t_0
    else if (x <= 7.5d-120) then
        tmp = eps * eps
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = eps * (2.0 * x);
	double tmp;
	if (x <= -3.4e-130) {
		tmp = t_0;
	} else if (x <= 7.5e-120) {
		tmp = eps * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = eps * (2.0 * x)
	tmp = 0
	if x <= -3.4e-130:
		tmp = t_0
	elif x <= 7.5e-120:
		tmp = eps * eps
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(2.0 * x))
	tmp = 0.0
	if (x <= -3.4e-130)
		tmp = t_0;
	elseif (x <= 7.5e-120)
		tmp = Float64(eps * eps);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = eps * (2.0 * x);
	tmp = 0.0;
	if (x <= -3.4e-130)
		tmp = t_0;
	elseif (x <= 7.5e-120)
		tmp = eps * eps;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(2.0 * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.4e-130], t$95$0, If[LessEqual[x, 7.5e-120], N[(eps * eps), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.40000000000000005e-130 or 7.5000000000000004e-120 < x

   1. Initial program 39.9%

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. difference-of-squares N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. +-inverses N/A

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

      8. +-rgt-identity N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\left(x + \varepsilon\right) + x\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(\varepsilon + x\right) + x\right)\right) \]

      11. associate-+l+ N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon + \color{blue}{\left(x + x\right)}\right)\right) \]

      12. --rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(\varepsilon - 0\right) + \left(\color{blue}{x} + x\right)\right)\right) \]

      13. associate-+l- N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon - \color{blue}{\left(0 - \left(x + x\right)\right)}\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon - \left(\mathsf{neg}\left(\left(x + x\right)\right)\right)\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \color{blue}{\left(\mathsf{neg}\left(\left(x + x\right)\right)\right)}\right)\right) \]

      16. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \left(\mathsf{neg}\left(2 \cdot x\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \left(\mathsf{neg}\left(x \cdot 2\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      20. metadata-eval 100.0%

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, -2\right)\right)\right) \]

   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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(2 \cdot x\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 83.9%

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(2, \color{blue}{x}\right)\right) \]

   7. Simplified 83.9%

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

   if -3.40000000000000005e-130 < x < 7.5000000000000004e-120

   1. Initial program 98.8%

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. difference-of-squares N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. +-inverses N/A

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

      8. +-rgt-identity N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\left(x + \varepsilon\right) + x\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(\varepsilon + x\right) + x\right)\right) \]

      11. associate-+l+ N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon + \color{blue}{\left(x + x\right)}\right)\right) \]

      12. --rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(\varepsilon - 0\right) + \left(\color{blue}{x} + x\right)\right)\right) \]

      13. associate-+l- N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon - \color{blue}{\left(0 - \left(x + x\right)\right)}\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon - \left(\mathsf{neg}\left(\left(x + x\right)\right)\right)\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \color{blue}{\left(\mathsf{neg}\left(\left(x + x\right)\right)\right)}\right)\right) \]

      16. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \left(\mathsf{neg}\left(2 \cdot x\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \left(\mathsf{neg}\left(x \cdot 2\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      20. metadata-eval 100.0%

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, -2\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 98.0%

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\varepsilon}\right) \]

   7. Simplified 98.0%

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

Alternative 3: 100.0% 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 72.3%

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

   1. unpow2 N/A

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

   2. unpow2 N/A

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

   3. difference-of-squares N/A

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. associate--l+ N/A

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

   7. +-inverses N/A

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

   8. +-rgt-identity N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\left(x + \varepsilon\right) + x\right)}\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(\varepsilon + x\right) + x\right)\right) \]

   11. associate-+l+ N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon + \color{blue}{\left(x + x\right)}\right)\right) \]

   12. --rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(\varepsilon - 0\right) + \left(\color{blue}{x} + x\right)\right)\right) \]

   13. associate-+l- N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon - \color{blue}{\left(0 - \left(x + x\right)\right)}\right)\right) \]

   14. neg-sub0 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon - \left(\mathsf{neg}\left(\left(x + x\right)\right)\right)\right)\right) \]

   15. --lowering--.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \color{blue}{\left(\mathsf{neg}\left(\left(x + x\right)\right)\right)}\right)\right) \]

   16. count-2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \left(\mathsf{neg}\left(2 \cdot x\right)\right)\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \left(\mathsf{neg}\left(x \cdot 2\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

   20. metadata-eval 100.0%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, -2\right)\right)\right) \]

3. Simplified 100.0%

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

Alternative 4: 72.9% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.3%

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

   1. unpow2 N/A

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

   2. unpow2 N/A

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

   3. difference-of-squares N/A

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. associate--l+ N/A

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

   7. +-inverses N/A

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

   8. +-rgt-identity N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\left(x + \varepsilon\right) + x\right)}\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(\varepsilon + x\right) + x\right)\right) \]

   11. associate-+l+ N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon + \color{blue}{\left(x + x\right)}\right)\right) \]

   12. --rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(\varepsilon - 0\right) + \left(\color{blue}{x} + x\right)\right)\right) \]

   13. associate-+l- N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon - \color{blue}{\left(0 - \left(x + x\right)\right)}\right)\right) \]

   14. neg-sub0 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon - \left(\mathsf{neg}\left(\left(x + x\right)\right)\right)\right)\right) \]

   15. --lowering--.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \color{blue}{\left(\mathsf{neg}\left(\left(x + x\right)\right)\right)}\right)\right) \]

   16. count-2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \left(\mathsf{neg}\left(2 \cdot x\right)\right)\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \left(\mathsf{neg}\left(x \cdot 2\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

   20. metadata-eval 100.0%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, -2\right)\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in eps around inf

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 69.8%

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\varepsilon}\right) \]

7. Simplified 69.8%

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

Reproduce

herbie shell --seed 2024152 
(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)))