Average Error: 40.3 → 0.5
Time: 12.4s
Precision: binary64
\[\cos \left(x + \varepsilon\right) - \cos x \]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00021764184323813543 \lor \neg \left(\varepsilon \leq 0.00018112974332094064\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \mathsf{fma}\left(\sin x \cdot \left(\varepsilon \cdot \varepsilon\right), -0.125, \mathsf{fma}\left(0.5, \varepsilon \cdot \cos x, \sin x\right)\right)\right)\\ \end{array} \]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -0.00021764184323813543 \lor \neg \left(\varepsilon \leq 0.00018112974332094064\right):\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\

\mathbf{else}:\\
\;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \mathsf{fma}\left(\sin x \cdot \left(\varepsilon \cdot \varepsilon\right), -0.125, \mathsf{fma}\left(0.5, \varepsilon \cdot \cos x, \sin x\right)\right)\right)\\


\end{array}

(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00021764184323813543)
         (not (<= eps 0.00018112974332094064)))
   (- (* (cos x) (cos eps)) (fma (sin eps) (sin x) (cos x)))
   (*
    -2.0
    (*
     (sin (/ eps 2.0))
     (fma (* (sin x) (* eps eps)) -0.125 (fma 0.5 (* eps (cos x)) (sin x)))))))
double code(double x, double eps) {
	return cos(x + eps) - cos(x);
}
↓
double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00021764184323813543) || !(eps <= 0.00018112974332094064)) {
		tmp = (cos(x) * cos(eps)) - fma(sin(eps), sin(x), cos(x));
	} else {
		tmp = -2.0 * (sin(eps / 2.0) * fma((sin(x) * (eps * eps)), -0.125, fma(0.5, (eps * cos(x)), sin(x))));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if eps < -2.1764184323813543e-4 or 1.8112974332094064e-4 < eps

    1. Initial program 30.8

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

    2. Applied cos-sum_binary640.9

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

    3. Applied associate--l-_binary640.9

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

    4. Simplified0.9

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

    if -2.1764184323813543e-4 < eps < 1.8112974332094064e-4

    1. Initial program 49.6

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

    2. Applied diff-cos_binary6437.6

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

    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. Taylor expanded in eps around 0 0.2

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\left(\left(0.5 \cdot \left(\varepsilon \cdot \cos x\right) + \sin x\right) - 0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)}\right) \]

    5. Simplified0.2

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\mathsf{fma}\left(\sin x \cdot \left(\varepsilon \cdot \varepsilon\right), -0.125, \mathsf{fma}\left(0.5, \varepsilon \cdot \cos x, \sin x\right)\right)}\right) \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00021764184323813543 \lor \neg \left(\varepsilon \leq 0.00018112974332094064\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \mathsf{fma}\left(\sin x \cdot \left(\varepsilon \cdot \varepsilon\right), -0.125, \mathsf{fma}\left(0.5, \varepsilon \cdot \cos x, \sin x\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021224 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))