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

Average Error: 7.5 → 0.5

Time: 12.8s

Precision: binary64

Cost: 39880

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

Math

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

↓

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-300}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{{x}^{4} \cdot \left(\varepsilon \cdot 5\right)}{\varepsilon} \cdot \frac{\varepsilon \cdot 3}{3}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 (<= t_0 -1e-300)
     t_0
     (if (<= t_0 0.0)
       (* (/ (* (pow x 4.0) (* eps 5.0)) eps) (/ (* eps 3.0) 3.0))
       t_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 <= -1e-300) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = ((pow(x, 4.0) * (eps * 5.0)) / eps) * ((eps * 3.0) / 3.0);
	} else {
		tmp = t_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 <= (-1d-300)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = (((x ** 4.0d0) * (eps * 5.0d0)) / eps) * ((eps * 3.0d0) / 3.0d0)
    else
        tmp = t_0
    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 <= -1e-300) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = ((Math.pow(x, 4.0) * (eps * 5.0)) / eps) * ((eps * 3.0) / 3.0);
	} else {
		tmp = t_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 <= -1e-300:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = ((math.pow(x, 4.0) * (eps * 5.0)) / eps) * ((eps * 3.0) / 3.0)
	else:
		tmp = t_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 <= -1e-300)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64((x ^ 4.0) * Float64(eps * 5.0)) / eps) * Float64(Float64(eps * 3.0) / 3.0));
	else
		tmp = t_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 <= -1e-300)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = (((x ^ 4.0) * (eps * 5.0)) / eps) * ((eps * 3.0) / 3.0);
	else
		tmp = t_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[LessEqual[t$95$0, -1e-300], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] * N[(N[(eps * 3.0), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -1.00000000000000003e-300 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 1.4

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

   if -1.00000000000000003e-300 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 8.9

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

   2. Taylor expanded in x around inf 0.3

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

   3. Applied egg-rr 5.6

      \[\leadsto \color{blue}{\frac{{x}^{4} \cdot \left(\left(\varepsilon \cdot 5\right) \cdot \left(\varepsilon \cdot 3\right)\right)}{\varepsilon \cdot 3}} \]

   4. Applied egg-rr 0.3

      \[\leadsto \color{blue}{\frac{{x}^{4} \cdot \left(\varepsilon \cdot 5\right)}{\varepsilon} \cdot \frac{\varepsilon \cdot 3}{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-300}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\frac{{x}^{4} \cdot \left(\varepsilon \cdot 5\right)}{\varepsilon} \cdot \frac{\varepsilon \cdot 3}{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.7
Cost	7432

\[\begin{array}{l} t_0 := {x}^{3} \cdot \left(\varepsilon \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\\ \mathbf{if}\;x \leq -1.4775133797801709 \cdot 10^{-36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6.567601233948317 \cdot 10^{-77}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.7
Cost	7432

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4775133797801709 \cdot 10^{-36}:\\ \;\;\;\;{x}^{3} \cdot \left(\varepsilon \cdot \left(x \cdot 5\right) + \varepsilon \cdot \left(\varepsilon \cdot 10\right)\right)\\ \mathbf{elif}\;x \leq 6.567601233948317 \cdot 10^{-77}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{3} \cdot \left(\varepsilon \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\\ \end{array} \]

Alternative 3
Error	1.8
Cost	7176

\[\begin{array}{l} t_0 := 5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{if}\;x \leq -1.4775133797801709 \cdot 10^{-36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6.567601233948317 \cdot 10^{-77}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	1.8
Cost	7176

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4775133797801709 \cdot 10^{-36}:\\ \;\;\;\;\frac{{x}^{4}}{\frac{1}{\varepsilon \cdot 5}}\\ \mathbf{elif}\;x \leq 6.567601233948317 \cdot 10^{-77}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 5
Error	1.8
Cost	7176

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4775133797801709 \cdot 10^{-36}:\\ \;\;\;\;\varepsilon \cdot \frac{\varepsilon \cdot 5}{\frac{\varepsilon}{{x}^{4}}}\\ \mathbf{elif}\;x \leq 6.567601233948317 \cdot 10^{-77}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 6
Error	1.8
Cost	7048

\[\begin{array}{l} t_0 := {x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{if}\;x \leq -1.4775133797801709 \cdot 10^{-36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6.567601233948317 \cdot 10^{-77}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	1.8
Cost	7048

\[\begin{array}{l} t_0 := 5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{if}\;x \leq -1.4775133797801709 \cdot 10^{-36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6.567601233948317 \cdot 10^{-77}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	4.5
Cost	6792

\[\begin{array}{l} t_0 := \varepsilon \cdot \left(\varepsilon \cdot 15\right)\\ \mathbf{if}\;x \leq -1.4775133797801709 \cdot 10^{-36}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \frac{x \cdot x}{\varepsilon \cdot 3}\right) \cdot t_0\\ \mathbf{elif}\;x \leq 6.567601233948317 \cdot 10^{-77}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{0.3333333333333333}{\varepsilon}\right)\right)\\ \end{array} \]

Alternative 9
Error	13.9
Cost	1088

\[\left(\varepsilon \cdot \left(\varepsilon \cdot 15\right)\right) \cdot \left(\frac{x \cdot x}{\varepsilon} \cdot \frac{x \cdot x}{3}\right) \]

Alternative 10
Error	13.9
Cost	1088

\[\left(\varepsilon \cdot \left(\varepsilon \cdot 15\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{0.3333333333333333}{\varepsilon}\right)\right) \]

Alternative 11
Error	13.9
Cost	1088

\[\left(\left(x \cdot x\right) \cdot \frac{x \cdot x}{\varepsilon \cdot 3}\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 15\right)\right) \]

Reproduce

herbie shell --seed 2022317 
(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)))