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

Average Error: 7.4 → 1.3

Time: 16.6s

Precision: binary64

Cost: 33668

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

Math

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

↓

\[\begin{array}{l} t_0 := \left({\varepsilon}^{2} \cdot 10\right) \cdot {x}^{3}\\ \mathbf{if}\;x \leq -1.38 \cdot 10^{-43}:\\ \;\;\;\;\left({\varepsilon}^{3} \cdot 10\right) \cdot {x}^{2} + \left(t_0 + \left(\varepsilon + \varepsilon \cdot 4\right) \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-60}:\\ \;\;\;\;{\varepsilon}^{5} + {\varepsilon}^{4} \cdot \left(x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + 5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (pow eps 2.0) 10.0) (pow x 3.0))))
   (if (<= x -1.38e-43)
     (+
      (* (* (pow eps 3.0) 10.0) (pow x 2.0))
      (+ t_0 (* (+ eps (* eps 4.0)) (pow x 4.0))))
     (if (<= x 9.8e-60)
       (+ (pow eps 5.0) (* (pow eps 4.0) (* x 5.0)))
       (+ t_0 (* 5.0 (* eps (pow x 4.0))))))))

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) * 10.0) * pow(x, 3.0);
	double tmp;
	if (x <= -1.38e-43) {
		tmp = ((pow(eps, 3.0) * 10.0) * pow(x, 2.0)) + (t_0 + ((eps + (eps * 4.0)) * pow(x, 4.0)));
	} else if (x <= 9.8e-60) {
		tmp = pow(eps, 5.0) + (pow(eps, 4.0) * (x * 5.0));
	} else {
		tmp = t_0 + (5.0 * (eps * pow(x, 4.0)));
	}
	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) * 10.0d0) * (x ** 3.0d0)
    if (x <= (-1.38d-43)) then
        tmp = (((eps ** 3.0d0) * 10.0d0) * (x ** 2.0d0)) + (t_0 + ((eps + (eps * 4.0d0)) * (x ** 4.0d0)))
    else if (x <= 9.8d-60) then
        tmp = (eps ** 5.0d0) + ((eps ** 4.0d0) * (x * 5.0d0))
    else
        tmp = t_0 + (5.0d0 * (eps * (x ** 4.0d0)))
    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(eps, 2.0) * 10.0) * Math.pow(x, 3.0);
	double tmp;
	if (x <= -1.38e-43) {
		tmp = ((Math.pow(eps, 3.0) * 10.0) * Math.pow(x, 2.0)) + (t_0 + ((eps + (eps * 4.0)) * Math.pow(x, 4.0)));
	} else if (x <= 9.8e-60) {
		tmp = Math.pow(eps, 5.0) + (Math.pow(eps, 4.0) * (x * 5.0));
	} else {
		tmp = t_0 + (5.0 * (eps * Math.pow(x, 4.0)));
	}
	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) * 10.0) * math.pow(x, 3.0)
	tmp = 0
	if x <= -1.38e-43:
		tmp = ((math.pow(eps, 3.0) * 10.0) * math.pow(x, 2.0)) + (t_0 + ((eps + (eps * 4.0)) * math.pow(x, 4.0)))
	elif x <= 9.8e-60:
		tmp = math.pow(eps, 5.0) + (math.pow(eps, 4.0) * (x * 5.0))
	else:
		tmp = t_0 + (5.0 * (eps * math.pow(x, 4.0)))
	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) * 10.0) * (x ^ 3.0))
	tmp = 0.0
	if (x <= -1.38e-43)
		tmp = Float64(Float64(Float64((eps ^ 3.0) * 10.0) * (x ^ 2.0)) + Float64(t_0 + Float64(Float64(eps + Float64(eps * 4.0)) * (x ^ 4.0))));
	elseif (x <= 9.8e-60)
		tmp = Float64((eps ^ 5.0) + Float64((eps ^ 4.0) * Float64(x * 5.0)));
	else
		tmp = Float64(t_0 + Float64(5.0 * Float64(eps * (x ^ 4.0))));
	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) * 10.0) * (x ^ 3.0);
	tmp = 0.0;
	if (x <= -1.38e-43)
		tmp = (((eps ^ 3.0) * 10.0) * (x ^ 2.0)) + (t_0 + ((eps + (eps * 4.0)) * (x ^ 4.0)));
	elseif (x <= 9.8e-60)
		tmp = (eps ^ 5.0) + ((eps ^ 4.0) * (x * 5.0));
	else
		tmp = t_0 + (5.0 * (eps * (x ^ 4.0)));
	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] * 10.0), $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.38e-43], N[(N[(N[(N[Power[eps, 3.0], $MachinePrecision] * 10.0), $MachinePrecision] * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 + N[(N[(eps + N[(eps * 4.0), $MachinePrecision]), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.8e-60], N[(N[Power[eps, 5.0], $MachinePrecision] + N[(N[Power[eps, 4.0], $MachinePrecision] * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3800000000000001e-43

   1. Initial program 42.0

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

   2. Taylor expanded in x around inf 4.5

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

   3. Simplified 4.5

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \left({\varepsilon}^{2} \cdot 6\right) + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2} + \left(\left({\varepsilon}^{2} \cdot 10\right) \cdot {x}^{3} + \left(\varepsilon + \varepsilon \cdot 4\right) \cdot {x}^{4}\right)} \]
      Proof

      [Start] 4.5	\[ \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} + \left(\left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]
      rational.json-simplify-1 [=>] 4.5	\[ \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} + \color{blue}{\left(\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right)} \]
      rational.json-simplify-41 [=>] 4.5	\[ \color{blue}{\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2} + \left(\left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4}\right)} \]
      rational.json-simplify-2 [=>] 4.5	\[ \left(\color{blue}{\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)} + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2} + \left(\left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4}\right) \]
      rational.json-simplify-2 [=>] 4.5	\[ \left(\varepsilon \cdot \left(\color{blue}{{\varepsilon}^{2} \cdot 2} + 4 \cdot {\varepsilon}^{2}\right) + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2} + \left(\left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4}\right) \]
      rational.json-simplify-51 [=>] 4.5	\[ \left(\varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(4 + 2\right)\right)} + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2} + \left(\left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4}\right) \]
      metadata-eval [=>] 4.5	\[ \left(\varepsilon \cdot \left({\varepsilon}^{2} \cdot \color{blue}{6}\right) + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2} + \left(\left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4}\right) \]

   4. Taylor expanded in eps around 0 4.5

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

   5. Simplified 4.5

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

      [Start] 4.5	\[ \left(10 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2} + \left(\left({\varepsilon}^{2} \cdot 10\right) \cdot {x}^{3} + \left(\varepsilon + \varepsilon \cdot 4\right) \cdot {x}^{4}\right) \]
      rational.json-simplify-2 [=>] 4.5	\[ \color{blue}{\left({\varepsilon}^{3} \cdot 10\right)} \cdot {x}^{2} + \left(\left({\varepsilon}^{2} \cdot 10\right) \cdot {x}^{3} + \left(\varepsilon + \varepsilon \cdot 4\right) \cdot {x}^{4}\right) \]

   if -1.3800000000000001e-43 < x < 9.79999999999999977e-60

   1. Initial program 0.4

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

   2. Taylor expanded in eps around inf 0.5

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

   3. Simplified 0.5

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

      [Start] 0.5	\[ {\varepsilon}^{4} \cdot \left(4 \cdot x + x\right) + {\varepsilon}^{5} \]
      rational.json-simplify-1 [=>] 0.5	\[ \color{blue}{{\varepsilon}^{5} + {\varepsilon}^{4} \cdot \left(4 \cdot x + x\right)} \]
      rational.json-simplify-1 [=>] 0.5	\[ {\varepsilon}^{5} + {\varepsilon}^{4} \cdot \color{blue}{\left(x + 4 \cdot x\right)} \]

   4. Taylor expanded in x around 0 0.5

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

   5. Simplified 0.5

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

      [Start] 0.5	\[ {\varepsilon}^{5} + 5 \cdot \left({\varepsilon}^{4} \cdot x\right) \]
      rational.json-simplify-43 [=>] 0.5	\[ {\varepsilon}^{5} + \color{blue}{{\varepsilon}^{4} \cdot \left(x \cdot 5\right)} \]

   if 9.79999999999999977e-60 < x

   1. Initial program 35.1

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

   2. Taylor expanded in x around inf 5.0

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

   3. Simplified 5.0

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

      [Start] 5.0	\[ \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} \]
      rational.json-simplify-1 [=>] 5.0	\[ \color{blue}{\left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4}} \]
      rational.json-simplify-2 [=>] 5.0	\[ \left(\color{blue}{{\varepsilon}^{2} \cdot 2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} \]
      rational.json-simplify-51 [=>] 5.0	\[ \color{blue}{\left({\varepsilon}^{2} \cdot \left(8 + 2\right)\right)} \cdot {x}^{3} + \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} \]
      metadata-eval [=>] 5.0	\[ \left({\varepsilon}^{2} \cdot \color{blue}{10}\right) \cdot {x}^{3} + \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} \]
      rational.json-simplify-1 [=>] 5.0	\[ \left({\varepsilon}^{2} \cdot 10\right) \cdot {x}^{3} + \color{blue}{\left(\varepsilon + 4 \cdot \varepsilon\right)} \cdot {x}^{4} \]
      rational.json-simplify-2 [=>] 5.0	\[ \left({\varepsilon}^{2} \cdot 10\right) \cdot {x}^{3} + \left(\varepsilon + \color{blue}{\varepsilon \cdot 4}\right) \cdot {x}^{4} \]

   4. Taylor expanded in eps around 0 5.1

      \[\leadsto \left({\varepsilon}^{2} \cdot 10\right) \cdot {x}^{3} + \color{blue}{5 \cdot \left(\varepsilon \cdot {x}^{4}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.38 \cdot 10^{-43}:\\ \;\;\;\;\left({\varepsilon}^{3} \cdot 10\right) \cdot {x}^{2} + \left(\left({\varepsilon}^{2} \cdot 10\right) \cdot {x}^{3} + \left(\varepsilon + \varepsilon \cdot 4\right) \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-60}:\\ \;\;\;\;{\varepsilon}^{5} + {\varepsilon}^{4} \cdot \left(x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\varepsilon}^{2} \cdot 10\right) \cdot {x}^{3} + 5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.4
Cost	20360

\[\begin{array}{l} t_0 := \left({\varepsilon}^{2} \cdot 10\right) \cdot {x}^{3} + 5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-60}:\\ \;\;\;\;{\varepsilon}^{5} + {\varepsilon}^{4} \cdot \left(x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.4
Cost	20360

\[\begin{array}{l} t_0 := \left({\varepsilon}^{2} \cdot 10\right) \cdot {x}^{3}\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;t_0 + \left(\varepsilon + \varepsilon \cdot 4\right) \cdot {x}^{4}\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-60}:\\ \;\;\;\;{\varepsilon}^{5} + {\varepsilon}^{4} \cdot \left(x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + 5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 3
Error	1.5
Cost	13640

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-43}:\\ \;\;\;\;{x}^{4} \cdot \left(5 \cdot \varepsilon\right)\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-60}:\\ \;\;\;\;{\varepsilon}^{5} + {\varepsilon}^{4} \cdot \left(x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left({x}^{4} \cdot \varepsilon\right)\\ \end{array} \]

Alternative 4
Error	1.4
Cost	13512

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-42}:\\ \;\;\;\;{x}^{4} \cdot \left(5 \cdot \varepsilon\right)\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-60}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left({x}^{4} \cdot \varepsilon\right)\\ \end{array} \]

Alternative 5
Error	1.6
Cost	7048

\[\begin{array}{l} t_0 := 5 \cdot \left({x}^{4} \cdot \varepsilon\right)\\ \mathbf{if}\;x \leq -1.62 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-60}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	1.6
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-43}:\\ \;\;\;\;{x}^{4} \cdot \left(5 \cdot \varepsilon\right)\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-60}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left({x}^{4} \cdot \varepsilon\right)\\ \end{array} \]

Alternative 7
Error	8.3
Cost	6528

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

Reproduce

herbie shell --seed 2023074 
(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)))