Average Error: 7.0 → 0.6
Time: 5.0s
Precision: binary64
\[\left(-1000000000 \leq x \land x \leq 1000000000\right) \land \left(-1 \leq \varepsilon \land \varepsilon \leq 1\right)\]
\[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
↓
\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t_0 - {x}^{5}\\ \mathbf{if}\;t_1 \leq -3.769278611 \cdot 10^{-314}:\\ \;\;\;\;10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right) + \left({\varepsilon}^{5} + \left(5 \cdot \left(x \cdot {\varepsilon}^{4}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)\right)\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + 10 \cdot \left(x \cdot \left(x \cdot {\varepsilon}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, {x}^{5}, t_0\right)\\ \end{array} \]
{\left(x + \varepsilon\right)}^{5} - {x}^{5}
↓
\begin{array}{l}
t_0 := {\left(x + \varepsilon\right)}^{5}\\
t_1 := t_0 - {x}^{5}\\
\mathbf{if}\;t_1 \leq -3.769278611 \cdot 10^{-314}:\\
\;\;\;\;10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right) + \left({\varepsilon}^{5} + \left(5 \cdot \left(x \cdot {\varepsilon}^{4}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)\right)\right)\\

\mathbf{elif}\;t_1 \leq 0:\\
\;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + 10 \cdot \left(x \cdot \left(x \cdot {\varepsilon}^{2}\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(-1, {x}^{5}, t_0\right)\\


\end{array}

(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)) (t_1 (- t_0 (pow x 5.0))))
   (if (<= t_1 -3.769278611e-314)
     (+
      (* 10.0 (* (pow eps 3.0) (pow x 2.0)))
      (+
       (pow eps 5.0)
       (+ (* 5.0 (* x (pow eps 4.0))) (* 10.0 (* (pow eps 2.0) (pow x 3.0))))))
     (if (<= t_1 0.0)
       (* eps (+ (* 5.0 (pow x 4.0)) (* 10.0 (* x (* x (pow eps 2.0))))))
       (fma -1.0 (pow x 5.0) t_0)))))
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);
	double t_1 = t_0 - pow(x, 5.0);
	double tmp;
	if (t_1 <= -3.769278611e-314) {
		tmp = (10.0 * (pow(eps, 3.0) * pow(x, 2.0))) + (pow(eps, 5.0) + ((5.0 * (x * pow(eps, 4.0))) + (10.0 * (pow(eps, 2.0) * pow(x, 3.0)))));
	} else if (t_1 <= 0.0) {
		tmp = eps * ((5.0 * pow(x, 4.0)) + (10.0 * (x * (x * pow(eps, 2.0)))));
	} else {
		tmp = fma(-1.0, pow(x, 5.0), t_0);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -3.769278611e-314

    1. Initial program 1.4

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

    2. Taylor expanded in x around 0 3.8

      \[\leadsto \color{blue}{10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right) + \left({\varepsilon}^{5} + \left(5 \cdot \left({\varepsilon}^{4} \cdot x\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)\right)\right)} \]

    if -3.769278611e-314 < (-.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 x around inf 0.1

      \[\leadsto \color{blue}{5 \cdot \left(\varepsilon \cdot {x}^{4}\right) + \left(10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)\right)} \]

    3. Simplified0.1

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

    4. Taylor expanded in x around 0 0.1

      \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + 10 \cdot \left(x \cdot \color{blue}{\left({\varepsilon}^{2} \cdot x\right)}\right)\right) \]

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

    1. Initial program 1.2

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

    2. Applied egg-rr1.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, {x}^{5}, {\left(x + \varepsilon\right)}^{5}\right)} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -3.769278611 \cdot 10^{-314}:\\ \;\;\;\;10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right) + \left({\varepsilon}^{5} + \left(5 \cdot \left(x \cdot {\varepsilon}^{4}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + 10 \cdot \left(x \cdot \left(x \cdot {\varepsilon}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, {x}^{5}, {\left(x + \varepsilon\right)}^{5}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022130 
(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)))