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

Average Error: 7.0 → 0.3

Time: 8.7s

Precision: binary64

Cost: 39880

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

Math

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

↓

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-323}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(5 \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 -2e-323)
     t_0
     (if (<= t_0 0.0) (* (pow x 4.0) (* 5.0 eps)) 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 <= -2e-323) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = pow(x, 4.0) * (5.0 * eps);
	} 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 <= (-2d-323)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = (x ** 4.0d0) * (5.0d0 * eps)
    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 <= -2e-323) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = Math.pow(x, 4.0) * (5.0 * eps);
	} 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 <= -2e-323:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = math.pow(x, 4.0) * (5.0 * eps)
	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 <= -2e-323)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64((x ^ 4.0) * Float64(5.0 * eps));
	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 <= -2e-323)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = (x ^ 4.0) * (5.0 * eps);
	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, -2e-323], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[Power[x, 4.0], $MachinePrecision] * N[(5.0 * eps), $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)) < -1.97626e-323 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 -1.97626e-323 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 8.4

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

   2. Taylor expanded in eps around 0 0.0

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

   3. Simplified 0.0

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

      [Start] 0.0	\[ \varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right) \]
      rational_best-simplify-1 [=>] 0.0	\[ \varepsilon \cdot \color{blue}{\left({x}^{4} + 4 \cdot {x}^{4}\right)} \]

   4. Taylor expanded in eps around 0 0.0

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

   5. Simplified 0.1

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

      [Start] 0.0	\[ \varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right) \]
      rational_best-simplify-1 [=>] 0.0	\[ \varepsilon \cdot \color{blue}{\left({x}^{4} + 4 \cdot {x}^{4}\right)} \]
      rational_best-simplify-63 [<=] 0.0	\[ \varepsilon \cdot \color{blue}{\left(4 \cdot {x}^{4} - \left(-{x}^{4}\right)\right)} \]
      rational_best-simplify-3 [=>] 0.0	\[ \varepsilon \cdot \left(\color{blue}{{x}^{4} \cdot 4} - \left(-{x}^{4}\right)\right) \]
      rational_best-simplify-18 [=>] 0.0	\[ \varepsilon \cdot \left({x}^{4} \cdot 4 - \color{blue}{{x}^{4} \cdot -1}\right) \]
      rational_best-simplify-110 [=>] 0.0	\[ \varepsilon \cdot \color{blue}{\left({x}^{4} \cdot \left(4 - -1\right)\right)} \]
      metadata-eval [=>] 0.0	\[ \varepsilon \cdot \left({x}^{4} \cdot \color{blue}{5}\right) \]
      rational_best-simplify-113 [=>] 0.1	\[ \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5\right)} \]
      rational_best-simplify-3 [<=] 0.1	\[ {x}^{4} \cdot \color{blue}{\left(5 \cdot \varepsilon\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-323}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(5 \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.5
Cost	7048

\[\begin{array}{l} t_0 := 5 \cdot \left({x}^{4} \cdot \varepsilon\right)\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-49}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.5
Cost	7048

\[\begin{array}{l} t_0 := \varepsilon \cdot \left({x}^{4} \cdot 5\right)\\ \mathbf{if}\;x \leq -2.8 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-47}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	1.4
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-53}:\\ \;\;\;\;{x}^{4} \cdot \left(5 \cdot \varepsilon\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-47}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot 5\right)\\ \end{array} \]

Alternative 4
Error	7.7
Cost	6528

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

Reproduce

herbie shell --seed 2023104 
(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)))