Average Error: 7.4 → 0.6
Time: 12.0s
Precision: binary64
Cost: 39880
\[\left(-1000000000 \leq x \land x \leq 1000000000\right) \land \left(-1 \leq \varepsilon \land \varepsilon \leq 1\right)\]
\[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
↓
\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-293}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
(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 -2e-293)
     t_0
     (if (<= t_0 0.0) (* (pow x 4.0) (* eps 5.0)) t_0))))
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 <= -2e-293) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = pow(x, 4.0) * (eps * 5.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

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 <= (-2d-293)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = (x ** 4.0d0) * (eps * 5.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

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 <= -2e-293) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = Math.pow(x, 4.0) * (eps * 5.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

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 <= -2e-293:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = math.pow(x, 4.0) * (eps * 5.0)
	else:
		tmp = t_0
	return tmp

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 <= -2e-293)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	else
		tmp = t_0;
	end
	return tmp
end

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 <= -2e-293)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = (x ^ 4.0) * (eps * 5.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

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, -2e-293], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

{\left(x + \varepsilon\right)}^{5} - {x}^{5}
↓
\begin{array}{l}
t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\
\mathbf{if}\;t_0 \leq -2 \cdot 10^{-293}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;t_0 \leq 0:\\
\;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -2.0000000000000001e-293 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

    1. Initial program 1.5

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

    if -2.0000000000000001e-293 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

    1. Initial program 8.8

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

    2. Taylor expanded in x around inf 0.4

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

    3. Applied egg-rr7.9

      \[\leadsto \color{blue}{\sqrt{\left(25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot {x}^{8}}} \]

    4. Applied egg-rr0.4

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-293}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \]

Alternatives

Alternative 1
Error1.2
Cost7688
\[\begin{array}{l} \mathbf{if}\;x \leq -9.544928685124598 \cdot 10^{-52}:\\ \;\;\;\;{x}^{3} \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 10 + x \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 5.30602998556533 \cdot 10^{-47}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x + x \cdot 4\right) + {\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
Alternative 2
Error1.3
Cost7300
\[\begin{array}{l} \mathbf{if}\;x \leq -9.544928685124598 \cdot 10^{-52}:\\ \;\;\;\;{x}^{3} \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 10 + x \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 5.30602998556533 \cdot 10^{-47}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
Alternative 3
Error1.3
Cost7176
\[\begin{array}{l} \mathbf{if}\;x \leq -9.544928685124598 \cdot 10^{-52}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 5.30602998556533 \cdot 10^{-47}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
Alternative 4
Error1.4
Cost7048
\[\begin{array}{l} t_0 := {x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{if}\;x \leq -9.544928685124598 \cdot 10^{-52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.30602998556533 \cdot 10^{-47}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 5
Error1.4
Cost7048
\[\begin{array}{l} \mathbf{if}\;x \leq -9.544928685124598 \cdot 10^{-52}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 5.30602998556533 \cdot 10^{-47}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]
Alternative 6
Error8.1
Cost832
\[\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon + x \cdot 5\right) \]
Alternative 7
Error8.2
Cost576
\[\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]

Error

Reproduce

herbie shell --seed 2022316 
(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)))