Average Error: 7.2 → 1.2
Time: 4.5s
Precision: binary64
\[\left(-1000000000 \leq x \land x \leq 1000000000\right) \land \left(-1 \leq \varepsilon \land \varepsilon \leq 1\right)\]
\[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
↓
\[\begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(\left(10 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x + \varepsilon\right)\right)\\ \mathbf{if}\;x \leq -6.091962211134749 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 5 \cdot {x}^{4}, t_0\right)\\ \mathbf{elif}\;x \leq 1.1002273010802258 \cdot 10^{-50}:\\ \;\;\;\;{\varepsilon}^{5} + \mathsf{fma}\left(5, x \cdot {\varepsilon}^{4}, t_0\right)\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)\\ \end{array} \]
{\left(x + \varepsilon\right)}^{5} - {x}^{5}
↓
\begin{array}{l}
t_0 := \left(x \cdot x\right) \cdot \left(\left(10 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x + \varepsilon\right)\right)\\
\mathbf{if}\;x \leq -6.091962211134749 \cdot 10^{-61}:\\
\;\;\;\;\mathsf{fma}\left(\varepsilon, 5 \cdot {x}^{4}, t_0\right)\\

\mathbf{elif}\;x \leq 1.1002273010802258 \cdot 10^{-50}:\\
\;\;\;\;{\varepsilon}^{5} + \mathsf{fma}\left(5, x \cdot {\varepsilon}^{4}, t_0\right)\\

\mathbf{else}:\\
\;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\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 (* (* x x) (* (* 10.0 (* eps eps)) (+ x eps)))))
   (if (<= x -6.091962211134749e-61)
     (fma eps (* 5.0 (pow x 4.0)) t_0)
     (if (<= x 1.1002273010802258e-50)
       (+ (pow eps 5.0) (fma 5.0 (* x (pow eps 4.0)) t_0))
       (+
        (* 5.0 (* eps (pow x 4.0)))
        (* 10.0 (* (pow eps 2.0) (pow x 3.0))))))))
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 = (x * x) * ((10.0 * (eps * eps)) * (x + eps));
	double tmp;
	if (x <= -6.091962211134749e-61) {
		tmp = fma(eps, (5.0 * pow(x, 4.0)), t_0);
	} else if (x <= 1.1002273010802258e-50) {
		tmp = pow(eps, 5.0) + fma(5.0, (x * pow(eps, 4.0)), t_0);
	} else {
		tmp = (5.0 * (eps * pow(x, 4.0))) + (10.0 * (pow(eps, 2.0) * pow(x, 3.0)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if x < -6.0919622111347492e-61

    1. Initial program 33.2

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

    2. Taylor expanded in x around inf 5.7

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

    3. Simplified5.6

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

    if -6.0919622111347492e-61 < x < 1.10022730108022582e-50

    1. Initial program 0.1

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

    2. Taylor expanded in x around 0 0.1

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

    3. Simplified0.1

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

    if 1.10022730108022582e-50 < x

    1. Initial program 38.1

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

    2. Taylor expanded in x around inf 5.5

      \[\leadsto \color{blue}{5 \cdot \left(\varepsilon \cdot {x}^{4}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.091962211134749 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 5 \cdot {x}^{4}, \left(x \cdot x\right) \cdot \left(\left(10 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x + \varepsilon\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.1002273010802258 \cdot 10^{-50}:\\ \;\;\;\;{\varepsilon}^{5} + \mathsf{fma}\left(5, x \cdot {\varepsilon}^{4}, \left(x \cdot x\right) \cdot \left(\left(10 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x + \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(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)))