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

Average Error: 7.4 → 1.2

Time: 12.2s

Precision: binary64

Cost: 20488

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

Math

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

↓

\[\begin{array}{l} t_0 := {\varepsilon}^{2} \cdot \left({x}^{3} \cdot 10\right) + \left(\varepsilon + \varepsilon \cdot 4\right) \cdot {x}^{4}\\ \mathbf{if}\;x \leq -3.75 \cdot 10^{-56}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-64}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (+
          (* (pow eps 2.0) (* (pow x 3.0) 10.0))
          (* (+ eps (* eps 4.0)) (pow x 4.0)))))
   (if (<= x -3.75e-56) t_0 (if (<= x 8.6e-64) (pow eps 5.0) 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(eps, 2.0) * (pow(x, 3.0) * 10.0)) + ((eps + (eps * 4.0)) * pow(x, 4.0));
	double tmp;
	if (x <= -3.75e-56) {
		tmp = t_0;
	} else if (x <= 8.6e-64) {
		tmp = pow(eps, 5.0);
	} 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 = ((eps ** 2.0d0) * ((x ** 3.0d0) * 10.0d0)) + ((eps + (eps * 4.0d0)) * (x ** 4.0d0))
    if (x <= (-3.75d-56)) then
        tmp = t_0
    else if (x <= 8.6d-64) then
        tmp = eps ** 5.0d0
    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(eps, 2.0) * (Math.pow(x, 3.0) * 10.0)) + ((eps + (eps * 4.0)) * Math.pow(x, 4.0));
	double tmp;
	if (x <= -3.75e-56) {
		tmp = t_0;
	} else if (x <= 8.6e-64) {
		tmp = Math.pow(eps, 5.0);
	} 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(eps, 2.0) * (math.pow(x, 3.0) * 10.0)) + ((eps + (eps * 4.0)) * math.pow(x, 4.0))
	tmp = 0
	if x <= -3.75e-56:
		tmp = t_0
	elif x <= 8.6e-64:
		tmp = math.pow(eps, 5.0)
	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((eps ^ 2.0) * Float64((x ^ 3.0) * 10.0)) + Float64(Float64(eps + Float64(eps * 4.0)) * (x ^ 4.0)))
	tmp = 0.0
	if (x <= -3.75e-56)
		tmp = t_0;
	elseif (x <= 8.6e-64)
		tmp = eps ^ 5.0;
	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 = ((eps ^ 2.0) * ((x ^ 3.0) * 10.0)) + ((eps + (eps * 4.0)) * (x ^ 4.0));
	tmp = 0.0;
	if (x <= -3.75e-56)
		tmp = t_0;
	elseif (x <= 8.6e-64)
		tmp = eps ^ 5.0;
	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[(N[Power[eps, 2.0], $MachinePrecision] * N[(N[Power[x, 3.0], $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] + N[(N[(eps + N[(eps * 4.0), $MachinePrecision]), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.75e-56], t$95$0, If[LessEqual[x, 8.6e-64], N[Power[eps, 5.0], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.75000000000000021e-56 or 8.59999999999999947e-64 < x

   1. Initial program 34.4

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

   2. Taylor expanded in x around inf 5.5

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

   3. Simplified 5.5

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left({x}^{3} \cdot 10\right) + \left(\varepsilon + \varepsilon \cdot 4\right) \cdot {x}^{4}} \]
      Proof

      [Start] 5.5	\[ \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} \]
      rational.json-simplify-1 [=>] 5.5	\[ \color{blue}{\left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4}} \]
      rational.json-simplify-2 [=>] 5.5	\[ \color{blue}{{x}^{3} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)} + \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} \]
      rational.json-simplify-2 [=>] 5.5	\[ {x}^{3} \cdot \left(\color{blue}{{\varepsilon}^{2} \cdot 2} + 8 \cdot {\varepsilon}^{2}\right) + \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} \]
      rational.json-simplify-47 [=>] 5.5	\[ {x}^{3} \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(8 + 2\right)\right)} + \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} \]
      rational.json-simplify-43 [=>] 5.5	\[ \color{blue}{{\varepsilon}^{2} \cdot \left(\left(8 + 2\right) \cdot {x}^{3}\right)} + \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} \]
      rational.json-simplify-2 [=>] 5.5	\[ {\varepsilon}^{2} \cdot \color{blue}{\left({x}^{3} \cdot \left(8 + 2\right)\right)} + \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} \]
      metadata-eval [=>] 5.5	\[ {\varepsilon}^{2} \cdot \left({x}^{3} \cdot \color{blue}{10}\right) + \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} \]
      rational.json-simplify-1 [=>] 5.5	\[ {\varepsilon}^{2} \cdot \left({x}^{3} \cdot 10\right) + \color{blue}{\left(\varepsilon + 4 \cdot \varepsilon\right)} \cdot {x}^{4} \]
      rational.json-simplify-2 [=>] 5.5	\[ {\varepsilon}^{2} \cdot \left({x}^{3} \cdot 10\right) + \left(\varepsilon + \color{blue}{\varepsilon \cdot 4}\right) \cdot {x}^{4} \]

   if -3.75000000000000021e-56 < x < 8.59999999999999947e-64

   1. Initial program 0.0

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

   2. Taylor expanded in x around 0 0.1

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.75 \cdot 10^{-56}:\\ \;\;\;\;{\varepsilon}^{2} \cdot \left({x}^{3} \cdot 10\right) + \left(\varepsilon + \varepsilon \cdot 4\right) \cdot {x}^{4}\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-64}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{2} \cdot \left({x}^{3} \cdot 10\right) + \left(\varepsilon + \varepsilon \cdot 4\right) \cdot {x}^{4}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.3
Cost	20360

\[\begin{array}{l} t_0 := {\varepsilon}^{2} \cdot \left({x}^{3} \cdot 10\right) + 5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{if}\;x \leq -1.15 \cdot 10^{-56}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-64}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.4
Cost	7048

\[\begin{array}{l} t_0 := 5 \cdot \left({x}^{4} \cdot \varepsilon\right)\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-64}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	1.4
Cost	7048

\[\begin{array}{l} t_0 := \varepsilon \cdot \left({x}^{4} \cdot 5\right)\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-64}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	1.4
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -9.4 \cdot 10^{-57}:\\ \;\;\;\;{x}^{4} \cdot \left(5 \cdot \varepsilon\right)\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-64}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot 5\right)\\ \end{array} \]

Alternative 5
Error	8.0
Cost	6528

\[{\varepsilon}^{5} \]

Reproduce

herbie shell --seed 2023077 
(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)))