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

Average Accuracy: 88.8% → 98.1%

Time: 18.6s

Precision: binary64

Cost: 40457

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -4.95 \cdot 10^{-60} \lor \neg \left(x \leq 2.3 \cdot 10^{-49}\right):\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \mathsf{fma}\left(5 \cdot {\varepsilon}^{4}, x, \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 10\right), x \cdot x, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -4.95e-60) (not (<= x 2.3e-49)))
   (fma
    (* eps 5.0)
    (pow x 4.0)
    (fma
     (* 5.0 (pow eps 4.0))
     x
     (fma
      (* (* eps eps) (* eps 10.0))
      (* x x)
      (* (* eps eps) (* 10.0 (pow x 3.0))))))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	return pow((x + eps), 5.0) - pow(x, 5.0);
}

↓

double code(double x, double eps) {
	double tmp;
	if ((x <= -4.95e-60) || !(x <= 2.3e-49)) {
		tmp = fma((eps * 5.0), pow(x, 4.0), fma((5.0 * pow(eps, 4.0)), x, fma(((eps * eps) * (eps * 10.0)), (x * x), ((eps * eps) * (10.0 * pow(x, 3.0))))));
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	return Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
end

↓

function code(x, eps)
	tmp = 0.0
	if ((x <= -4.95e-60) || !(x <= 2.3e-49))
		tmp = fma(Float64(eps * 5.0), (x ^ 4.0), fma(Float64(5.0 * (eps ^ 4.0)), x, fma(Float64(Float64(eps * eps) * Float64(eps * 10.0)), Float64(x * x), Float64(Float64(eps * eps) * Float64(10.0 * (x ^ 3.0))))));
	else
		tmp = eps ^ 5.0;
	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_] := If[Or[LessEqual[x, -4.95e-60], N[Not[LessEqual[x, 2.3e-49]], $MachinePrecision]], N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision] + N[(N[(5.0 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision] * x + N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * 10.0), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * N[(10.0 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9499999999999997e-60 or 2.2999999999999999e-49 < x

   1. Initial program 44.1%

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

   2. Applied egg-rr 12.1%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(x + \varepsilon\right)}^{15}}} - {x}^{5} \]
      Proof

      [Start] 44.1	\[ {\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
      add-cbrt-cube [=>] 12.4	\[ \color{blue}{\sqrt[3]{\left({\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5}\right) \cdot {\left(x + \varepsilon\right)}^{5}}} - {x}^{5} \]
      unpow3 [<=] 12.4	\[ \sqrt[3]{\color{blue}{{\left({\left(x + \varepsilon\right)}^{5}\right)}^{3}}} - {x}^{5} \]
      pow-pow [=>] 12.1	\[ \sqrt[3]{\color{blue}{{\left(x + \varepsilon\right)}^{\left(5 \cdot 3\right)}}} - {x}^{5} \]
      metadata-eval [=>] 12.1	\[ \sqrt[3]{{\left(x + \varepsilon\right)}^{\color{blue}{15}}} - {x}^{5} \]

   3. Taylor expanded in x around inf 92.0%

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

   4. Simplified 92.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \mathsf{fma}\left(5 \cdot {\varepsilon}^{4}, x, \mathsf{fma}\left(4 \cdot {\varepsilon}^{3} + {\varepsilon}^{3} \cdot 6, x \cdot x, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3}\right)\right)\right)\right)} \]
      Proof

      [Start] 92.0	\[ \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} + \left(\left({\varepsilon}^{4} + 4 \cdot {\varepsilon}^{4}\right) \cdot x + \left(\left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right)\right) \]
      fma-def [=>] 92.0	\[ \color{blue}{\mathsf{fma}\left(4 \cdot \varepsilon + \varepsilon, {x}^{4}, \left({\varepsilon}^{4} + 4 \cdot {\varepsilon}^{4}\right) \cdot x + \left(\left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right)\right)} \]
      distribute-lft1-in [=>] 92.0	\[ \mathsf{fma}\left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon}, {x}^{4}, \left({\varepsilon}^{4} + 4 \cdot {\varepsilon}^{4}\right) \cdot x + \left(\left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right)\right) \]
      metadata-eval [=>] 92.0	\[ \mathsf{fma}\left(\color{blue}{5} \cdot \varepsilon, {x}^{4}, \left({\varepsilon}^{4} + 4 \cdot {\varepsilon}^{4}\right) \cdot x + \left(\left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right)\right) \]
      *-commutative [=>] 92.0	\[ \mathsf{fma}\left(\color{blue}{\varepsilon \cdot 5}, {x}^{4}, \left({\varepsilon}^{4} + 4 \cdot {\varepsilon}^{4}\right) \cdot x + \left(\left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right)\right) \]
      +-commutative [<=] 92.0	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(4 \cdot {\varepsilon}^{4} + {\varepsilon}^{4}\right)} \cdot x + \left(\left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right)\right) \]
      fma-def [=>] 92.0	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\mathsf{fma}\left(4 \cdot {\varepsilon}^{4} + {\varepsilon}^{4}, x, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right)}\right) \]

   5. Applied egg-rr 92.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \mathsf{fma}\left(5 \cdot {\varepsilon}^{4}, x, \mathsf{fma}\left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 6 + \varepsilon \cdot 4\right)}, x \cdot x, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3}\right)\right)\right)\right) \]
      Proof

      [Start] 92.0	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \mathsf{fma}\left(5 \cdot {\varepsilon}^{4}, x, \mathsf{fma}\left(4 \cdot {\varepsilon}^{3} + {\varepsilon}^{3} \cdot 6, x \cdot x, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3}\right)\right)\right)\right) \]
      +-commutative [=>] 92.0	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \mathsf{fma}\left(5 \cdot {\varepsilon}^{4}, x, \mathsf{fma}\left(\color{blue}{{\varepsilon}^{3} \cdot 6 + 4 \cdot {\varepsilon}^{3}}, x \cdot x, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3}\right)\right)\right)\right) \]
      unpow3 [=>] 92.0	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \mathsf{fma}\left(5 \cdot {\varepsilon}^{4}, x, \mathsf{fma}\left(\color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)} \cdot 6 + 4 \cdot {\varepsilon}^{3}, x \cdot x, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3}\right)\right)\right)\right) \]
      associate-*l* [=>] 92.0	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \mathsf{fma}\left(5 \cdot {\varepsilon}^{4}, x, \mathsf{fma}\left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 6\right)} + 4 \cdot {\varepsilon}^{3}, x \cdot x, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3}\right)\right)\right)\right) \]
      *-commutative [=>] 92.0	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \mathsf{fma}\left(5 \cdot {\varepsilon}^{4}, x, \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 6\right) + \color{blue}{{\varepsilon}^{3} \cdot 4}, x \cdot x, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3}\right)\right)\right)\right) \]
      unpow3 [=>] 92.0	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \mathsf{fma}\left(5 \cdot {\varepsilon}^{4}, x, \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 6\right) + \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)} \cdot 4, x \cdot x, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3}\right)\right)\right)\right) \]
      associate-*l* [=>] 92.0	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \mathsf{fma}\left(5 \cdot {\varepsilon}^{4}, x, \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 6\right) + \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 4\right)}, x \cdot x, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3}\right)\right)\right)\right) \]
      distribute-lft-out [=>] 92.0	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \mathsf{fma}\left(5 \cdot {\varepsilon}^{4}, x, \mathsf{fma}\left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 6 + \varepsilon \cdot 4\right)}, x \cdot x, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3}\right)\right)\right)\right) \]

   6. Applied egg-rr 92.0%

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

      [Start] 92.0	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \mathsf{fma}\left(5 \cdot {\varepsilon}^{4}, x, \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 6 + \varepsilon \cdot 4\right), x \cdot x, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3}\right)\right)\right)\right) \]
      distribute-lft-out [=>] 92.0	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \mathsf{fma}\left(5 \cdot {\varepsilon}^{4}, x, \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(6 + 4\right)\right)}, x \cdot x, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3}\right)\right)\right)\right) \]
      metadata-eval [=>] 92.0	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \mathsf{fma}\left(5 \cdot {\varepsilon}^{4}, x, \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \color{blue}{10}\right), x \cdot x, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3}\right)\right)\right)\right) \]

   if -4.9499999999999997e-60 < x < 2.2999999999999999e-49

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 99.6%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.95 \cdot 10^{-60} \lor \neg \left(x \leq 2.3 \cdot 10^{-49}\right):\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \mathsf{fma}\left(5 \cdot {\varepsilon}^{4}, x, \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 10\right), x \cdot x, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.0%
Cost	40265

\[\begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-61} \lor \neg \left(x \leq 1.95 \cdot 10^{-49}\right):\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \mathsf{fma}\left(4 \cdot {\varepsilon}^{3} + {\varepsilon}^{3} \cdot 6, x \cdot x, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 2
Accuracy	99.0%
Cost	39881

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-294} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]

Alternative 3
Accuracy	97.9%
Cost	20296

\[\begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \left(x \cdot x\right) \cdot \left(x \cdot 10\right), \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right)\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-49}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{3} \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	97.9%
Cost	14089

\[\begin{array}{l} \mathbf{if}\;x \leq -4.95 \cdot 10^{-60} \lor \neg \left(x \leq 2.7 \cdot 10^{-49}\right):\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \left(x \cdot x\right) \cdot \left(x \cdot 10\right), \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 5
Accuracy	97.7%
Cost	7049

\[\begin{array}{l} \mathbf{if}\;x \leq -3.65 \cdot 10^{-60} \lor \neg \left(x \leq 3.5 \cdot 10^{-49}\right):\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 6
Accuracy	97.7%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{-60}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-49}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 7
Accuracy	97.7%
Cost	6792

\[\begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{-60}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot 5\right)\right)\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-49}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	83.2%
Cost	704

\[\left(\varepsilon \cdot 5\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

Alternative 9
Accuracy	83.2%
Cost	704

\[\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot 5\right)\right) \]

Reproduce

herbie shell --seed 2023131 
(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)))