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

Average Accuracy: 88.6% → 99.5%

Time: 8.3s

Precision: binary64

Cost: 46729

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

Math

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

↓

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-320} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5, \varepsilon \cdot {x}^{4}, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -10\right) \cdot \left(-{x}^{3}\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -5e-320) (not (<= t_0 0.0)))
     t_0
     (fma 5.0 (* eps (pow x 4.0)) (* (* (* eps eps) -10.0) (- (pow x 3.0)))))))

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 <= -5e-320) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = fma(5.0, (eps * pow(x, 4.0)), (((eps * eps) * -10.0) * -pow(x, 3.0)));
	}
	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 <= -5e-320) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = fma(5.0, Float64(eps * (x ^ 4.0)), Float64(Float64(Float64(eps * eps) * -10.0) * Float64(-(x ^ 3.0))));
	end
	return 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[Or[LessEqual[t$95$0, -5e-320], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(eps * eps), $MachinePrecision] * -10.0), $MachinePrecision] * (-N[Power[x, 3.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -4.99994e-320 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 98.0%

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

   if -4.99994e-320 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 86.3%

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

   2. Taylor expanded in x around -inf 99.9%

      \[\leadsto \color{blue}{-1 \cdot \left(\left(-1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + -4 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right) + \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(5, \varepsilon \cdot {x}^{4}, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -10\right) \cdot \left(-{x}^{3}\right)\right)} \]
      Proof

      [Start] 99.9	\[ -1 \cdot \left(\left(-1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + -4 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right) + \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} \]
      +-commutative [=>] 99.9	\[ \color{blue}{\left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} + -1 \cdot \left(\left(-1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + -4 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right)} \]
      mul-1-neg [=>] 99.9	\[ \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} + \color{blue}{\left(-\left(-1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + -4 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right)} \]
      unsub-neg [=>] 99.9	\[ \color{blue}{\left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} - \left(-1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + -4 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}} \]
      distribute-lft1-in [=>] 99.9	\[ \color{blue}{\left(\left(4 + 1\right) \cdot \varepsilon\right)} \cdot {x}^{4} - \left(-1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + -4 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} \]
      metadata-eval [=>] 99.9	\[ \left(\color{blue}{5} \cdot \varepsilon\right) \cdot {x}^{4} - \left(-1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + -4 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} \]
      associate-*l* [=>] 99.9	\[ \color{blue}{5 \cdot \left(\varepsilon \cdot {x}^{4}\right)} - \left(-1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + -4 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} \]
      *-commutative [<=] 99.9	\[ 5 \cdot \color{blue}{\left({x}^{4} \cdot \varepsilon\right)} - \left(-1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + -4 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} \]
      fma-neg [=>] 99.9	\[ \color{blue}{\mathsf{fma}\left(5, {x}^{4} \cdot \varepsilon, -\left(-1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + -4 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right)} \]
      *-commutative [=>] 99.9	\[ \mathsf{fma}\left(5, \color{blue}{\varepsilon \cdot {x}^{4}}, -\left(-1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + -4 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right) \]
      distribute-rgt-neg-in [=>] 99.9	\[ \mathsf{fma}\left(5, \varepsilon \cdot {x}^{4}, \color{blue}{\left(-1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + -4 \cdot {\varepsilon}^{2}\right) \cdot \left(-{x}^{3}\right)}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-320} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5, \varepsilon \cdot {x}^{4}, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -10\right) \cdot \left(-{x}^{3}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.5%
Cost	39881

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-320} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 2
Accuracy	97.8%
Cost	7433

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-51} \lor \neg \left(x \leq 1.9 \cdot 10^{-42}\right):\\ \;\;\;\;{x}^{3} \cdot \left(\varepsilon \cdot \left(x \cdot 5 + \varepsilon \cdot 10\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 3
Accuracy	97.7%
Cost	7049

\[\begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{-51} \lor \neg \left(x \leq 2.1 \cdot 10^{-42}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 4
Accuracy	97.7%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-51}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-41}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 5
Accuracy	87.7%
Cost	6528

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

Reproduce

herbie shell --seed 2023147 
(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)))