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

Average Error: 7.7 → 0.4

Time: 12.9s

Precision: binary64

Cost: 39881

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

Math

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

↓

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{-310} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{{x}^{-4}}{5}}\\ \end{array} \]

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 -4e-310) (not (<= t_0 0.0)))
     t_0
     (/ eps (/ (pow x -4.0) 5.0)))))

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 <= -4e-310) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = eps / (pow(x, -4.0) / 5.0);
	}
	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 <= (-4d-310)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = eps / ((x ** (-4.0d0)) / 5.0d0)
    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 <= -4e-310) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = eps / (Math.pow(x, -4.0) / 5.0);
	}
	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 <= -4e-310) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = eps / (math.pow(x, -4.0) / 5.0)
	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 <= -4e-310) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(eps / Float64((x ^ -4.0) / 5.0));
	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 <= -4e-310) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = eps / ((x ^ -4.0) / 5.0);
	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, -4e-310], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(eps / N[(N[Power[x, -4.0], $MachinePrecision] / 5.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -3.999999999999988e-310 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 1.6

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

   if -3.999999999999988e-310 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 9.1

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

   2. Taylor expanded in eps around 0 0.2

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

   3. Applied egg-rr 56.5

      \[\leadsto \varepsilon \cdot \color{blue}{\frac{\left({x}^{8} \cdot 17\right) \cdot \left({x}^{8} \cdot 15\right)}{\left({x}^{4} \cdot 3\right) \cdot \left({x}^{8} \cdot 17\right)}} \]

   4. Simplified 0.2

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(\frac{{x}^{4}}{1} \cdot 5.666666666666667\right) \cdot 0.8823529411764706\right)} \]
      Proof

      [Start] 56.5	\[ \varepsilon \cdot \frac{\left({x}^{8} \cdot 17\right) \cdot \left({x}^{8} \cdot 15\right)}{\left({x}^{4} \cdot 3\right) \cdot \left({x}^{8} \cdot 17\right)} \]
      times-frac [=>] 52.8	\[ \varepsilon \cdot \color{blue}{\left(\frac{{x}^{8} \cdot 17}{{x}^{4} \cdot 3} \cdot \frac{{x}^{8} \cdot 15}{{x}^{8} \cdot 17}\right)} \]
      times-frac [=>] 52.8	\[ \varepsilon \cdot \left(\color{blue}{\left(\frac{{x}^{8}}{{x}^{4}} \cdot \frac{17}{3}\right)} \cdot \frac{{x}^{8} \cdot 15}{{x}^{8} \cdot 17}\right) \]
      metadata-eval [<=] 52.8	\[ \varepsilon \cdot \left(\left(\frac{{x}^{\color{blue}{\left(2 \cdot 4\right)}}}{{x}^{4}} \cdot \frac{17}{3}\right) \cdot \frac{{x}^{8} \cdot 15}{{x}^{8} \cdot 17}\right) \]
      pow-sqr [<=] 52.8	\[ \varepsilon \cdot \left(\left(\frac{\color{blue}{{x}^{4} \cdot {x}^{4}}}{{x}^{4}} \cdot \frac{17}{3}\right) \cdot \frac{{x}^{8} \cdot 15}{{x}^{8} \cdot 17}\right) \]
      associate-/l* [=>] 52.8	\[ \varepsilon \cdot \left(\left(\color{blue}{\frac{{x}^{4}}{\frac{{x}^{4}}{{x}^{4}}}} \cdot \frac{17}{3}\right) \cdot \frac{{x}^{8} \cdot 15}{{x}^{8} \cdot 17}\right) \]
      *-inverses [=>] 52.8	\[ \varepsilon \cdot \left(\left(\frac{{x}^{4}}{\color{blue}{1}} \cdot \frac{17}{3}\right) \cdot \frac{{x}^{8} \cdot 15}{{x}^{8} \cdot 17}\right) \]
      metadata-eval [=>] 52.8	\[ \varepsilon \cdot \left(\left(\frac{{x}^{4}}{1} \cdot \color{blue}{5.666666666666667}\right) \cdot \frac{{x}^{8} \cdot 15}{{x}^{8} \cdot 17}\right) \]
      times-frac [=>] 52.7	\[ \varepsilon \cdot \left(\left(\frac{{x}^{4}}{1} \cdot 5.666666666666667\right) \cdot \color{blue}{\left(\frac{{x}^{8}}{{x}^{8}} \cdot \frac{15}{17}\right)}\right) \]
      *-inverses [=>] 0.2	\[ \varepsilon \cdot \left(\left(\frac{{x}^{4}}{1} \cdot 5.666666666666667\right) \cdot \left(\color{blue}{1} \cdot \frac{15}{17}\right)\right) \]
      metadata-eval [=>] 0.2	\[ \varepsilon \cdot \left(\left(\frac{{x}^{4}}{1} \cdot 5.666666666666667\right) \cdot \left(1 \cdot \color{blue}{0.8823529411764706}\right)\right) \]
      metadata-eval [=>] 0.2	\[ \varepsilon \cdot \left(\left(\frac{{x}^{4}}{1} \cdot 5.666666666666667\right) \cdot \color{blue}{0.8823529411764706}\right) \]

   5. Applied egg-rr 9.2

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\varepsilon \cdot 5}{{x}^{-4}}\right)} + -1} \]

   6. Simplified 0.2

      \[\leadsto \color{blue}{\frac{\varepsilon}{\frac{{x}^{-4}}{5}}} \]
      Proof

      [Start] 9.2	\[ e^{\mathsf{log1p}\left(\frac{\varepsilon \cdot 5}{{x}^{-4}}\right)} + -1 \]
      metadata-eval [<=] 9.2	\[ e^{\mathsf{log1p}\left(\frac{\varepsilon \cdot 5}{{x}^{-4}}\right)} + \color{blue}{\left(-1\right)} \]
      sub-neg [<=] 9.2	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{\varepsilon \cdot 5}{{x}^{-4}}\right)} - 1} \]
      expm1-def [=>] 0.2	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\varepsilon \cdot 5}{{x}^{-4}}\right)\right)} \]
      expm1-log1p [=>] 0.2	\[ \color{blue}{\frac{\varepsilon \cdot 5}{{x}^{-4}}} \]
      associate-/l* [=>] 0.2	\[ \color{blue}{\frac{\varepsilon}{\frac{{x}^{-4}}{5}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -4 \cdot 10^{-310} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{{x}^{-4}}{5}}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.5
Cost	6792

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-41}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right)\right)\right) + x \cdot \left(x \cdot \left(\varepsilon \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-46}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(10 \cdot \left(\varepsilon \cdot \left(x \cdot \varepsilon\right)\right) + \left(x \cdot x\right) \cdot \left(\varepsilon \cdot 5\right)\right)\\ \end{array} \]

Alternative 2
Error	11.1
Cost	1472

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

Alternative 3
Error	11.1
Cost	1216

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

Alternative 4
Error	11.1
Cost	1216

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

Alternative 5
Error	11.3
Cost	960

\[\varepsilon \cdot \left(\left(\frac{x}{\frac{1}{x}} \cdot \left(\left(x \cdot x\right) \cdot 5.666666666666667\right)\right) \cdot 0.8823529411764706\right) \]

Alternative 6
Error	11.3
Cost	832

\[\varepsilon \cdot \left(0.8823529411764706 \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 5.666666666666667\right)\right)\right)\right) \]

Alternative 7
Error	11.3
Cost	704

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

Reproduce

herbie shell --seed 2023010 
(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)))