ENA, Section 1.4, Exercise 4b, n=5

Average Error: 7.4 → 0.4

Time: 9.2s

Precision: binary64

Cost: 39881

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

Math

\[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

↓

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-321} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right) + x \cdot \left(5 \cdot \left(x \cdot \varepsilon\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (pow (+ x eps) 5.0) (pow x 5.0)))

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -5e-321) (not (<= t_0 0.0)))
     t_0
     (* (* x x) (+ (* x (* eps (* eps 10.0))) (* x (* 5.0 (* x eps))))))))

C

double code(double x, double eps) {
	return pow((x + eps), 5.0) - pow(x, 5.0);
}

↓

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -5e-321) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (x * x) * ((x * (eps * (eps * 10.0))) + (x * (5.0 * (x * eps))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
end function

↓

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 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if ((t_0 <= (-5d-321)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = (x * x) * ((x * (eps * (eps * 10.0d0))) + (x * (5.0d0 * (x * eps))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	return Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
}

↓

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if ((t_0 <= -5e-321) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (x * x) * ((x * (eps * (eps * 10.0))) + (x * (5.0 * (x * eps))));
	}
	return tmp;
}

Python

def code(x, eps):
	return math.pow((x + eps), 5.0) - math.pow(x, 5.0)

↓

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if (t_0 <= -5e-321) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = (x * x) * ((x * (eps * (eps * 10.0))) + (x * (5.0 * (x * eps))))
	return tmp

Julia

function code(x, eps)
	return Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
end

↓

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -5e-321) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(x * x) * Float64(Float64(x * Float64(eps * Float64(eps * 10.0))) + Float64(x * Float64(5.0 * Float64(x * eps)))));
	end
	return tmp
end

MATLAB

function tmp = code(x, eps)
	tmp = ((x + eps) ^ 5.0) - (x ^ 5.0);
end

↓

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if ((t_0 <= -5e-321) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = (x * x) * ((x * (eps * (eps * 10.0))) + (x * (5.0 * (x * eps))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]

↓

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-321], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(N[(x * x), $MachinePrecision] * N[(N[(x * N[(eps * N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(5.0 * N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -4.99994e-321 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 1.8

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   if -4.99994e-321 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 8.7

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in x around inf 0.1

      \[\leadsto \color{blue}{\left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}} \]

   3. Simplified 0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right) \cdot {x}^{3}\right)} \]
      Proof

      [Start] 0.1	\[ \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} \]
      fma-def [=>] 0.1	\[ \color{blue}{\mathsf{fma}\left(4 \cdot \varepsilon + \varepsilon, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right)} \]
      distribute-lft1-in [=>] 0.1	\[ \mathsf{fma}\left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right) \]
      metadata-eval [=>] 0.1	\[ \mathsf{fma}\left(\color{blue}{5} \cdot \varepsilon, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right) \]
      *-commutative [=>] 0.1	\[ \mathsf{fma}\left(\color{blue}{\varepsilon \cdot 5}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right) \]
      distribute-rgt-out [=>] 0.1	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left({\varepsilon}^{2} \cdot \left(2 + 8\right)\right)} \cdot {x}^{3}\right) \]
      unpow2 [=>] 0.1	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(2 + 8\right)\right) \cdot {x}^{3}\right) \]
      metadata-eval [=>] 0.1	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{10}\right) \cdot {x}^{3}\right) \]
      associate-*l* [=>] 0.1	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right)} \cdot {x}^{3}\right) \]

   4. Applied egg-rr 0.1

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right) \cdot x + \left(\varepsilon \cdot 5\right) \cdot \left(x \cdot x\right)\right)} \]

   5. Taylor expanded in eps around 0 0.1

      \[\leadsto \left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right) \cdot x + \color{blue}{5 \cdot \left(\varepsilon \cdot {x}^{2}\right)}\right) \]

   6. Simplified 0.1

      \[\leadsto \left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right) \cdot x + \color{blue}{x \cdot \left(5 \cdot \left(\varepsilon \cdot x\right)\right)}\right) \]
      Proof

      [Start] 0.1	\[ \left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right) \cdot x + 5 \cdot \left(\varepsilon \cdot {x}^{2}\right)\right) \]
      associate-*r* [=>] 0.1	\[ \left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right) \cdot x + \color{blue}{\left(5 \cdot \varepsilon\right) \cdot {x}^{2}}\right) \]
      *-commutative [<=] 0.1	\[ \left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right) \cdot x + \color{blue}{\left(\varepsilon \cdot 5\right)} \cdot {x}^{2}\right) \]
      unpow2 [=>] 0.1	\[ \left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right) \cdot x + \left(\varepsilon \cdot 5\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \]
      *-commutative [=>] 0.1	\[ \left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right) \cdot x + \color{blue}{\left(x \cdot x\right) \cdot \left(\varepsilon \cdot 5\right)}\right) \]
      associate-*l* [=>] 0.1	\[ \left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right) \cdot x + \color{blue}{x \cdot \left(x \cdot \left(\varepsilon \cdot 5\right)\right)}\right) \]
      *-commutative [<=] 0.1	\[ \left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right) \cdot x + x \cdot \color{blue}{\left(\left(\varepsilon \cdot 5\right) \cdot x\right)}\right) \]
      *-commutative [=>] 0.1	\[ \left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right) \cdot x + x \cdot \left(\color{blue}{\left(5 \cdot \varepsilon\right)} \cdot x\right)\right) \]
      associate-*l* [=>] 0.1	\[ \left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right) \cdot x + x \cdot \color{blue}{\left(5 \cdot \left(\varepsilon \cdot x\right)\right)}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-321} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right) + x \cdot \left(5 \cdot \left(x \cdot \varepsilon\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.5
Cost	6793

\[\begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{-53} \lor \neg \left(x \leq 1.5 \cdot 10^{-67}\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right) + x \cdot \left(5 \cdot \left(x \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 2
Error	11.0
Cost	1216

\[\left(x \cdot x\right) \cdot \left(x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right) + x \cdot \left(5 \cdot \left(x \cdot \varepsilon\right)\right)\right) \]

Reproduce

herbie shell --seed 2023031 
(FPCore (x eps)
  :name "ENA, Section 1.4, Exercise 4b, n=5"
  :precision binary64
  :pre (and (and (<= -1000000000.0 x) (<= x 1000000000.0)) (and (<= -1.0 eps) (<= eps 1.0)))
  (- (pow (+ x eps) 5.0) (pow x 5.0)))