Average Error: 39.7 → 0.5
Time: 8.9s
Precision: binary64
\[\cos \left(x + \varepsilon\right) - \cos x \]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0023808861058280564:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0027000306911697434:\\ \;\;\;\;\left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right) - \left(0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \varepsilon \cdot \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\right)\\ \end{array} \]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -0.0023808861058280564:\\
\;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{elif}\;\varepsilon \leq 0.0027000306911697434:\\
\;\;\;\;\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)\\

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


\end{array}

(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.0023808861058280564)
   (- (fma (cos x) (cos eps) (- (* (sin x) (sin eps)))) (cos x))
   (if (<= eps 0.0027000306911697434)
     (-
      (+
       (* 0.16666666666666666 (* (pow eps 3.0) (sin x)))
       (* 0.041666666666666664 (* (pow eps 4.0) (cos x))))
      (+ (* 0.5 (* (pow eps 2.0) (cos x))) (* eps (sin x))))
     (fma (cos eps) (cos x) (- (fma (sin eps) (sin x) (cos x)))))))
double code(double x, double eps) {
	return cos((x + eps)) - cos(x);
}
↓
double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0023808861058280564) {
		tmp = fma(cos(x), cos(eps), -(sin(x) * sin(eps))) - cos(x);
	} else if (eps <= 0.0027000306911697434) {
		tmp = ((0.16666666666666666 * (pow(eps, 3.0) * sin(x))) + (0.041666666666666664 * (pow(eps, 4.0) * cos(x)))) - ((0.5 * (pow(eps, 2.0) * cos(x))) + (eps * sin(x)));
	} else {
		tmp = fma(cos(eps), cos(x), -fma(sin(eps), sin(x), cos(x)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if eps < -0.00238088610582805641

    1. Initial program 30.6

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

    2. Applied egg-rr0.8

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

    if -0.00238088610582805641 < eps < 0.00270003069116974337

    1. Initial program 49.2

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

    2. Taylor expanded in eps around 0 0.1

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

    if 0.00270003069116974337 < eps

    1. Initial program 30.6

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

    2. Applied egg-rr0.9

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

    3. Taylor expanded in x around inf 0.9

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

    4. Simplified0.9

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

    5. Applied egg-rr0.8

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0023808861058280564:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0027000306911697434:\\ \;\;\;\;\left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right) - \left(0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \varepsilon \cdot \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022129 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))