Average Error: 40.2 → 0.6
Time: 9.7s
Precision: binary64
\[\cos \left(x + \varepsilon\right) - \cos x \]
↓
\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ t_1 := -t_0\\ \mathbf{if}\;\varepsilon \leq -3.519909493265526 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(\cos x, \cos \varepsilon, t_1\right) + \mathsf{fma}\left(-\sin \varepsilon, \sin x, t_0\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 5.874348258429151 \cdot 10^{-5}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \mathsf{fma}\left(0.5, \varepsilon \cdot \cos x, \sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \cos x, t_1\right) - \cos x\\ \end{array} \]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
t_0 := \sin \varepsilon \cdot \sin x\\
t_1 := -t_0\\
\mathbf{if}\;\varepsilon \leq -3.519909493265526 \cdot 10^{-5}:\\
\;\;\;\;\left(\mathsf{fma}\left(\cos x, \cos \varepsilon, t_1\right) + \mathsf{fma}\left(-\sin \varepsilon, \sin x, t_0\right)\right) - \cos x\\

\mathbf{elif}\;\varepsilon \leq 5.874348258429151 \cdot 10^{-5}:\\
\;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \mathsf{fma}\left(0.5, \varepsilon \cdot \cos x, \sin x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \cos x, t_1\right) - \cos x\\


\end{array}

(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))
↓
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin eps) (sin x))) (t_1 (- t_0)))
   (if (<= eps -3.519909493265526e-5)
     (-
      (+ (fma (cos x) (cos eps) t_1) (fma (- (sin eps)) (sin x) t_0))
      (cos x))
     (if (<= eps 5.874348258429151e-5)
       (* -2.0 (* (sin (/ eps 2.0)) (fma 0.5 (* eps (cos x)) (sin x))))
       (- (fma (cos eps) (cos x) t_1) (cos x))))))
double code(double x, double eps) {
	return cos(x + eps) - cos(x);
}
↓
double code(double x, double eps) {
	double t_0 = sin(eps) * sin(x);
	double t_1 = -t_0;
	double tmp;
	if (eps <= -3.519909493265526e-5) {
		tmp = (fma(cos(x), cos(eps), t_1) + fma(-sin(eps), sin(x), t_0)) - cos(x);
	} else if (eps <= 5.874348258429151e-5) {
		tmp = -2.0 * (sin(eps / 2.0) * fma(0.5, (eps * cos(x)), sin(x)));
	} else {
		tmp = fma(cos(eps), cos(x), t_1) - cos(x);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if eps < -3.51990949326552626e-5

    1. Initial program 31.2

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

    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 prod-diff_binary640.9

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

    if -3.51990949326552626e-5 < eps < 5.87434825842915077e-5

    1. Initial program 49.5

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

    2. Applied diff-cos_binary6438.1

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

    3. Simplified0.5

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

    4. Taylor expanded in eps around 0 0.2

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

    5. Simplified0.2

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\mathsf{fma}\left(0.5, \varepsilon \cdot \cos x, \sin x\right)}\right) \]

    if 5.87434825842915077e-5 < eps

    1. Initial program 30.0

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

    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 prod-diff_binary640.9

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

    4. Simplified0.9

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

    5. Simplified0.9

      \[\leadsto \left(\mathsf{fma}\left(\cos \varepsilon, \cos x, -\sin \varepsilon \cdot \sin x\right) + \color{blue}{0}\right) - \cos x \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.519909493265526 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin \varepsilon \cdot \sin x\right) + \mathsf{fma}\left(-\sin \varepsilon, \sin x, \sin \varepsilon \cdot \sin x\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 5.874348258429151 \cdot 10^{-5}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \mathsf{fma}\left(0.5, \varepsilon \cdot \cos x, \sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \cos x, -\sin \varepsilon \cdot \sin x\right) - \cos x\\ \end{array} \]

Reproduce

herbie shell --seed 2021215 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))