Average Error: 39.6 → 0.5
Time: 12.1s
Precision: binary64
\[\cos \left(x + \varepsilon\right) - \cos x \]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0027738346999691818 \lor \neg \left(\varepsilon \leq 0.002845823794585604\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, \sin x \cdot {\varepsilon}^{3}, \mathsf{fma}\left(\cos x, \mathsf{fma}\left({\varepsilon}^{4}, 0.041666666666666664, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right), -\varepsilon \cdot \sin x\right)\right)\\ \end{array} \]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -0.0027738346999691818 \lor \neg \left(\varepsilon \leq 0.002845823794585604\right):\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\

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


\end{array}

(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0027738346999691818) (not (<= eps 0.002845823794585604)))
   (- (* (cos x) (cos eps)) (fma (sin eps) (sin x) (cos x)))
   (fma
    0.16666666666666666
    (* (sin x) (pow eps 3.0))
    (fma
     (cos x)
     (fma (pow eps 4.0) 0.041666666666666664 (* (* eps eps) -0.5))
     (- (* eps (sin x)))))))
double code(double x, double eps) {
	return cos(x + eps) - cos(x);
}
↓
double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0027738346999691818) || !(eps <= 0.002845823794585604)) {
		tmp = (cos(x) * cos(eps)) - fma(sin(eps), sin(x), cos(x));
	} else {
		tmp = fma(0.16666666666666666, (sin(x) * pow(eps, 3.0)), fma(cos(x), fma(pow(eps, 4.0), 0.041666666666666664, ((eps * eps) * -0.5)), -(eps * sin(x))));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if eps < -0.0027738346999691818 or 0.0028458237945856038 < eps

    1. Initial program 29.7

      \[\cos \left(x + \varepsilon\right) - \cos x \]

    2. Applied cos-sum_binary640.8

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

    3. Applied associate--l-_binary640.9

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

    4. Simplified0.8

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

    if -0.0027738346999691818 < eps < 0.0028458237945856038

    1. Initial program 49.9

      \[\cos \left(x + \varepsilon\right) - \cos x \]

    2. Applied diff-cos_binary6438.0

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

    3. Simplified0.7

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

    4. Applied log1p-expm1-u_binary640.7

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\right)\right)} \]

    5. Simplified0.7

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(-2 \cdot \left(\sin \left(\frac{\mathsf{fma}\left(x, 2, \varepsilon\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\right)}\right) \]

    6. Taylor expanded in eps around 0 0.2

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

    7. Simplified0.2

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0027738346999691818 \lor \neg \left(\varepsilon \leq 0.002845823794585604\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, \sin x \cdot {\varepsilon}^{3}, \mathsf{fma}\left(\cos x, \mathsf{fma}\left({\varepsilon}^{4}, 0.041666666666666664, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right), -\varepsilon \cdot \sin x\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021344 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))