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

Average Error: 88.6% → 97.8%

Time: 15.3s

Precision: binary64

Cost: 27337.00

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{-45} \lor \neg \left(x \leq 7.4 \cdot 10^{-65}\right):\\ \;\;\;\;\mathsf{fma}\left(5 \cdot \varepsilon, {x}^{4}, \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 10\right), x \cdot x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} + x \cdot \left(5 \cdot {\varepsilon}^{4}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.32e-45) (not (<= x 7.4e-65)))
   (fma
    (* 5.0 eps)
    (pow x 4.0)
    (fma
     (* (* eps eps) (* eps 10.0))
     (* x x)
     (* (* (* eps eps) 10.0) (pow x 3.0))))
   (+ (pow eps 5.0) (* x (* 5.0 (pow eps 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 tmp;
	if ((x <= -1.32e-45) || !(x <= 7.4e-65)) {
		tmp = fma((5.0 * eps), pow(x, 4.0), fma(((eps * eps) * (eps * 10.0)), (x * x), (((eps * eps) * 10.0) * pow(x, 3.0))));
	} else {
		tmp = pow(eps, 5.0) + (x * (5.0 * pow(eps, 4.0)));
	}
	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 <= -1.32e-45) || !(x <= 7.4e-65))
		tmp = fma(Float64(5.0 * eps), (x ^ 4.0), fma(Float64(Float64(eps * eps) * Float64(eps * 10.0)), Float64(x * x), Float64(Float64(Float64(eps * eps) * 10.0) * (x ^ 3.0))));
	else
		tmp = Float64((eps ^ 5.0) + Float64(x * Float64(5.0 * (eps ^ 4.0))));
	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, -1.32e-45], N[Not[LessEqual[x, 7.4e-65]], $MachinePrecision]], N[(N[(5.0 * eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision] + N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * 10.0), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[eps, 5.0], $MachinePrecision] + N[(x * N[(5.0 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.32000000000000005e-45 or 7.4e-65 < x

   1. Initial program 44.3

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

   2. Taylor expanded in x around inf 91.3

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

   3. Simplified 91.3

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

      [Start] 91.3	\[ \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} + \left(\left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]
      fma-def [=>] 91.3	\[ \color{blue}{\mathsf{fma}\left(4 \cdot \varepsilon + \varepsilon, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right)} \]
      distribute-lft1-in [=>] 91.3	\[ \mathsf{fma}\left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]
      metadata-eval [=>] 91.3	\[ \mathsf{fma}\left(\color{blue}{5} \cdot \varepsilon, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]
      +-commutative [=>] 91.3	\[ \mathsf{fma}\left(5 \cdot \varepsilon, {x}^{4}, \color{blue}{\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}}\right) \]
      fma-def [=>] 91.3	\[ \mathsf{fma}\left(5 \cdot \varepsilon, {x}^{4}, \color{blue}{\mathsf{fma}\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}, {x}^{2}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right)}\right) \]
      fma-def [=>] 91.3	\[ \mathsf{fma}\left(5 \cdot \varepsilon, {x}^{4}, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}, \varepsilon, 4 \cdot {\varepsilon}^{3}\right)}, {x}^{2}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right)\right) \]
      distribute-rgt-out [=>] 91.3	\[ \mathsf{fma}\left(5 \cdot \varepsilon, {x}^{4}, \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{{\varepsilon}^{2} \cdot \left(2 + 4\right)}, \varepsilon, 4 \cdot {\varepsilon}^{3}\right), {x}^{2}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right)\right) \]
      unpow2 [=>] 91.3	\[ \mathsf{fma}\left(5 \cdot \varepsilon, {x}^{4}, \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(2 + 4\right), \varepsilon, 4 \cdot {\varepsilon}^{3}\right), {x}^{2}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right)\right) \]
      metadata-eval [=>] 91.3	\[ \mathsf{fma}\left(5 \cdot \varepsilon, {x}^{4}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{6}, \varepsilon, 4 \cdot {\varepsilon}^{3}\right), {x}^{2}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right)\right) \]
      unpow2 [=>] 91.3	\[ \mathsf{fma}\left(5 \cdot \varepsilon, {x}^{4}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot 6, \varepsilon, 4 \cdot {\varepsilon}^{3}\right), \color{blue}{x \cdot x}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right)\right) \]

   4. Applied egg-rr 91.3

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

   5. Applied egg-rr 91.3

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

   if -1.32000000000000005e-45 < x < 7.4e-65

   1. Initial program 99.6

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

   2. Taylor expanded in x around 0 99.5

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

   3. Simplified 99.5

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

      [Start] 99.5	\[ {\varepsilon}^{5} + \left(4 \cdot {\varepsilon}^{4} + {\varepsilon}^{4}\right) \cdot x \]
      +-commutative [<=] 99.5	\[ {\varepsilon}^{5} + \color{blue}{\left({\varepsilon}^{4} + 4 \cdot {\varepsilon}^{4}\right)} \cdot x \]
      *-commutative [=>] 99.5	\[ {\varepsilon}^{5} + \color{blue}{x \cdot \left({\varepsilon}^{4} + 4 \cdot {\varepsilon}^{4}\right)} \]
      distribute-rgt1-in [=>] 99.5	\[ {\varepsilon}^{5} + x \cdot \color{blue}{\left(\left(4 + 1\right) \cdot {\varepsilon}^{4}\right)} \]
      metadata-eval [=>] 99.5	\[ {\varepsilon}^{5} + x \cdot \left(\color{blue}{5} \cdot {\varepsilon}^{4}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 97.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{-45} \lor \neg \left(x \leq 7.4 \cdot 10^{-65}\right):\\ \;\;\;\;\mathsf{fma}\left(5 \cdot \varepsilon, {x}^{4}, \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 10\right), x \cdot x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5} + x \cdot \left(5 \cdot {\varepsilon}^{4}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	99.4%
Cost	39881.00

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-313} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right) + 5 \cdot \left(x \cdot \left(x \cdot \varepsilon\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	97.4%
Cost	6792.00

\[\begin{array}{l} t_0 := x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right)\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{-41}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(t_0 + \varepsilon \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{-65}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(t_0 + 5 \cdot \left(x \cdot \left(x \cdot \varepsilon\right)\right)\right)\\ \end{array} \]

Alternative 3
Error	82.6%
Cost	1216.00

\[\left(x \cdot x\right) \cdot \left(10 \cdot \left(\varepsilon \cdot \left(x \cdot \varepsilon\right)\right) + \varepsilon \cdot \left(x \cdot \left(x \cdot 5\right)\right)\right) \]

Alternative 4
Error	82.6%
Cost	1216.00

\[\left(x \cdot x\right) \cdot \left(x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right) + 5 \cdot \left(x \cdot \left(x \cdot \varepsilon\right)\right)\right) \]

Reproduce

herbie shell --seed 2023093 
(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)))