Average Error: 39.7 → 0.5
Time: 9.2s
Precision: binary64
\[\cos \left(x + \varepsilon\right) - \cos x \]
↓
\[\begin{array}{l} t_0 := \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.004004932165133894:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0035389504100430845:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
t_0 := \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\
\mathbf{if}\;\varepsilon \leq -0.004004932165133894:\\
\;\;\;\;t_0\\

\mathbf{elif}\;\varepsilon \leq 0.0035389504100430845:\\
\;\;\;\;\cos x \cdot \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\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)) (* (sin x) (sin eps))) (cos x))))
   (if (<= eps -0.004004932165133894)
     t_0
     (if (<= eps 0.0035389504100430845)
       (+
        (*
         (cos x)
         (fma 0.041666666666666664 (pow eps 4.0) (* (* eps eps) -0.5)))
        (* (sin x) (- (* 0.16666666666666666 (pow eps 3.0)) eps)))
       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)) - (sin(x) * sin(eps))) - cos(x);
	double tmp;
	if (eps <= -0.004004932165133894) {
		tmp = t_0;
	} else if (eps <= 0.0035389504100430845) {
		tmp = (cos(x) * fma(0.041666666666666664, pow(eps, 4.0), ((eps * eps) * -0.5))) + (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) - eps));
	} 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 < -0.0040049321651338936 or 0.0035389504100430845 < eps

    1. Initial program 29.9

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

    2. Applied cos-sum_binary640.8

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

    if -0.0040049321651338936 < eps < 0.0035389504100430845

    1. Initial program 49.5

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

    2. Taylor expanded in eps around 0 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.004004932165133894:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0035389504100430845:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array} \]

Reproduce

herbie shell --seed 2022077 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))