Average Error: 39.4 → 0.5
Time: 8.8s
Precision: binary64
\[\cos \left(x + \varepsilon\right) - \cos x \]
↓
\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.0034224773079427465:\\ \;\;\;\;t_0 - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.003535501464615805:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\sin x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.125 + 1\right) + \cos x \cdot \left(\varepsilon \cdot 0.5 + {\varepsilon}^{3} \cdot -0.020833333333333332\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \left(\cos x + \sin \varepsilon \cdot \sin x\right)\\ \end{array} \]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
t_0 := \cos x \cdot \cos \varepsilon\\
\mathbf{if}\;\varepsilon \leq -0.0034224773079427465:\\
\;\;\;\;t_0 - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\

\mathbf{elif}\;\varepsilon \leq 0.003535501464615805:\\
\;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\sin x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.125 + 1\right) + \cos x \cdot \left(\varepsilon \cdot 0.5 + {\varepsilon}^{3} \cdot -0.020833333333333332\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;t_0 - \left(\cos x + \sin \varepsilon \cdot \sin x\right)\\


\end{array}

(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))
↓
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (cos x) (cos eps))))
   (if (<= eps -0.0034224773079427465)
     (- t_0 (fma (sin eps) (sin x) (cos x)))
     (if (<= eps 0.003535501464615805)
       (*
        -2.0
        (*
         (sin (/ eps 2.0))
         (+
          (* (sin x) (+ (* (* eps eps) -0.125) 1.0))
          (*
           (cos x)
           (+ (* eps 0.5) (* (pow eps 3.0) -0.020833333333333332))))))
       (- t_0 (+ (cos x) (* (sin eps) (sin x))))))))
double code(double x, double eps) {
	return cos(x + eps) - cos(x);
}
↓
double code(double x, double eps) {
	double t_0 = cos(x) * cos(eps);
	double tmp;
	if (eps <= -0.0034224773079427465) {
		tmp = t_0 - fma(sin(eps), sin(x), cos(x));
	} else if (eps <= 0.003535501464615805) {
		tmp = -2.0 * (sin(eps / 2.0) * ((sin(x) * (((eps * eps) * -0.125) + 1.0)) + (cos(x) * ((eps * 0.5) + (pow(eps, 3.0) * -0.020833333333333332)))));
	} else {
		tmp = t_0 - (cos(x) + (sin(eps) * sin(x)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if eps < -0.00342247730794274655

    1. Initial program 29.3

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

    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-_binary640.8

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

    4. Simplified0.8

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

    if -0.00342247730794274655 < eps < 0.003535501464615805

    1. Initial program 48.9

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

    2. Applied diff-cos_binary6436.9

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

    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

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\left(\left(0.5 \cdot \left(\varepsilon \cdot \cos x\right) + \sin x\right) - \left(0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right)}\right) \]

    5. Simplified0.2

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.125 + 1\right) \cdot \sin x + \cos x \cdot \left(0.5 \cdot \varepsilon + {\varepsilon}^{3} \cdot -0.020833333333333332\right)\right)}\right) \]

    if 0.003535501464615805 < eps

    1. Initial program 30.8

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

    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-_binary640.8

      \[\leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0034224773079427465:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.003535501464615805:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\sin x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.125 + 1\right) + \cos x \cdot \left(\varepsilon \cdot 0.5 + {\varepsilon}^{3} \cdot -0.020833333333333332\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin \varepsilon \cdot \sin x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))