Average Error: 39.7 → 0.6
Time: 9.7s
Precision: binary64
\[\cos \left(x + \varepsilon\right) - \cos x \]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.8611454772111663 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 3.026368651951747 \cdot 10^{-5}\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\varepsilon, \sin x, 0.5 \cdot \left(\cos x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -2.8611454772111663 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 3.026368651951747 \cdot 10^{-5}\right):\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\

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


\end{array}

(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -2.8611454772111663e-5) (not (<= eps 3.026368651951747e-5)))
   (- (* (cos x) (cos eps)) (fma (sin eps) (sin x) (cos x)))
   (- (fma eps (sin x) (* 0.5 (* (cos x) (* eps eps)))))))
double code(double x, double eps) {
	return cos(x + eps) - cos(x);
}
↓
double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.8611454772111663e-5) || !(eps <= 3.026368651951747e-5)) {
		tmp = (cos(x) * cos(eps)) - fma(sin(eps), sin(x), cos(x));
	} else {
		tmp = -fma(eps, sin(x), (0.5 * (cos(x) * (eps * eps))));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if eps < -2.86114547721116629e-5 or 3.02636865195174714e-5 < 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 -2.86114547721116629e-5 < eps < 3.02636865195174714e-5

    1. Initial program 49.1

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.8611454772111663 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 3.026368651951747 \cdot 10^{-5}\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\varepsilon, \sin x, 0.5 \cdot \left(\cos x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021210 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))