Average Error: 39.8 → 0.5
Time: 8.6s
Precision: binary64
\[\cos \left(x + \varepsilon\right) - \cos x \]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0027141351705622318 \lor \neg \left(\varepsilon \leq 0.0023954047667611917\right):\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, 0.041666666666666664 \cdot {\varepsilon}^{4}, \mathsf{fma}\left(\sin x, 0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon, \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) \cdot -0.5\right)\right)\\ \end{array} \]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -0.0027141351705622318 \lor \neg \left(\varepsilon \leq 0.0023954047667611917\right):\\
\;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin \varepsilon \cdot \sin x\right) - \cos x\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\cos x, 0.041666666666666664 \cdot {\varepsilon}^{4}, \mathsf{fma}\left(\sin x, 0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon, \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) \cdot -0.5\right)\right)\\


\end{array}

(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0027141351705622318) (not (<= eps 0.0023954047667611917)))
   (- (fma (cos x) (cos eps) (- (* (sin eps) (sin x)))) (cos x))
   (fma
    (cos x)
    (* 0.041666666666666664 (pow eps 4.0))
    (fma
     (sin x)
     (- (* 0.16666666666666666 (pow eps 3.0)) eps)
     (* (* eps (* eps (cos x))) -0.5)))))
double code(double x, double eps) {
	return cos(x + eps) - cos(x);
}
↓
double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0027141351705622318) || !(eps <= 0.0023954047667611917)) {
		tmp = fma(cos(x), cos(eps), -(sin(eps) * sin(x))) - cos(x);
	} else {
		tmp = fma(cos(x), (0.041666666666666664 * pow(eps, 4.0)), fma(sin(x), ((0.16666666666666666 * pow(eps, 3.0)) - eps), ((eps * (eps * cos(x))) * -0.5)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if eps < -0.00271413517056223177 or 0.00239540476676119172 < eps

    1. Initial program 30.7

      \[\cos \left(x + \varepsilon\right) - \cos x \]
    2. Using strategy rm

    3. Applied cos-sum_binary640.8

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \]

    4. Applied fma-neg_binary640.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin x \cdot \sin \varepsilon\right)} - \cos x \]

    5. Simplified0.8

      \[\leadsto \mathsf{fma}\left(\cos x, \cos \varepsilon, \color{blue}{-\sin \varepsilon \cdot \sin x}\right) - \cos x \]

    if -0.00271413517056223177 < eps < 0.00239540476676119172

    1. Initial program 49.2

      \[\cos \left(x + \varepsilon\right) - \cos x \]
    2. Using strategy rm

    3. Applied diff-cos_binary6438.1

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

    4. Applied associate-*r*_binary6438.1

      \[\leadsto \color{blue}{\left(-2 \cdot \sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right)\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)} \]

    5. Simplified0.6

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot -2\right)} \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \]

    6. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right) - \left(0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \varepsilon \cdot \sin x\right)} \]

    7. Simplified0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, 0.041666666666666664 \cdot {\varepsilon}^{4}, \mathsf{fma}\left(\sin x, 0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon, \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) \cdot -0.5\right)\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0027141351705622318 \lor \neg \left(\varepsilon \leq 0.0023954047667611917\right):\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, 0.041666666666666664 \cdot {\varepsilon}^{4}, \mathsf{fma}\left(\sin x, 0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon, \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) \cdot -0.5\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021211 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))