Average Error: 39.2 → 0.5
Time: 8.2s
Precision: binary64
\[\cos \left(x + \varepsilon\right) - \cos x \]
↓
\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \mathbf{if}\;\varepsilon \leq -0.00016442173469881995:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.00013931906783766504:\\ \;\;\;\;\mathsf{fma}\left(\sin x, 0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon, \cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
t_0 := \cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\
\mathbf{if}\;\varepsilon \leq -0.00016442173469881995:\\
\;\;\;\;t_0\\

\mathbf{elif}\;\varepsilon \leq 0.00013931906783766504:\\
\;\;\;\;\mathsf{fma}\left(\sin x, 0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon, \cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right)\right)\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}

(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))
↓
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (* (cos x) (cos eps)) (fma (sin eps) (sin x) (cos x)))))
   (if (<= eps -0.00016442173469881995)
     t_0
     (if (<= eps 0.00013931906783766504)
       (fma
        (sin x)
        (- (* 0.16666666666666666 (pow eps 3.0)) eps)
        (* (cos x) (* (* eps eps) -0.5)))
       t_0))))
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)) - fma(sin(eps), sin(x), cos(x));
	double tmp;
	if (eps <= -0.00016442173469881995) {
		tmp = t_0;
	} else if (eps <= 0.00013931906783766504) {
		tmp = fma(sin(x), ((0.16666666666666666 * pow(eps, 3.0)) - eps), (cos(x) * ((eps * eps) * -0.5)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.6442173469881995e-4 or 1.39319067837665036e-4 < eps

    1. Initial program 30.1

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

    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 -1.6442173469881995e-4 < eps < 1.39319067837665036e-4

    1. Initial program 48.6

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

    2. Taylor expanded in eps around 0 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00016442173469881995:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00013931906783766504:\\ \;\;\;\;\mathsf{fma}\left(\sin x, 0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon, \cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022082 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))