Average Error: 38.9 → 0.5
Time: 23.5s
Precision: binary64
\[\cos \left(x + \varepsilon\right) - \cos x \]
↓
\[\begin{array}{l} t_0 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.002752665168387367:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{expm1}\left(\mathsf{log1p}\left(t_0\right)\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0026284641846169455:\\ \;\;\;\;\cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right) - \sin x \cdot \mathsf{fma}\left({\varepsilon}^{3}, -0.16666666666666666, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - t_0\right) - \cos x\\ \end{array} \]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
t_0 := \sin x \cdot \sin \varepsilon\\
\mathbf{if}\;\varepsilon \leq -0.002752665168387367:\\
\;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{expm1}\left(\mathsf{log1p}\left(t_0\right)\right)\right) - \cos x\\

\mathbf{elif}\;\varepsilon \leq 0.0026284641846169455:\\
\;\;\;\;\cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right) - \sin x \cdot \mathsf{fma}\left({\varepsilon}^{3}, -0.16666666666666666, \varepsilon\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - t_0\right) - \cos x\\


\end{array}

(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))
↓
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin x) (sin eps))))
   (if (<= eps -0.002752665168387367)
     (- (fma (cos x) (cos eps) (- (expm1 (log1p t_0)))) (cos x))
     (if (<= eps 0.0026284641846169455)
       (-
        (*
         (cos x)
         (+ (* 0.041666666666666664 (pow eps 4.0)) (* (* eps eps) -0.5)))
        (* (sin x) (fma (pow eps 3.0) -0.16666666666666666 eps)))
       (- (- (* (cos x) (cos eps)) t_0) (cos x))))))
double code(double x, double eps) {
	return cos((x + eps)) - cos(x);
}
↓
double code(double x, double eps) {
	double t_0 = sin(x) * sin(eps);
	double tmp;
	if (eps <= -0.002752665168387367) {
		tmp = fma(cos(x), cos(eps), -expm1(log1p(t_0))) - cos(x);
	} else if (eps <= 0.0026284641846169455) {
		tmp = (cos(x) * ((0.041666666666666664 * pow(eps, 4.0)) + ((eps * eps) * -0.5))) - (sin(x) * fma(pow(eps, 3.0), -0.16666666666666666, eps));
	} else {
		tmp = ((cos(x) * cos(eps)) - t_0) - cos(x);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if eps < -0.002752665168387367

    1. Initial program 29.4

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

    2. Applied egg-rr0.8

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

    3. Applied egg-rr0.8

      \[\leadsto \mathsf{fma}\left(\cos x, \cos \varepsilon, -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin x \cdot \sin \varepsilon\right)\right)}\right) - \cos x \]

    if -0.002752665168387367 < eps < 0.0026284641846169455

    1. Initial program 49.1

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

    2. Applied egg-rr48.2

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

    3. Taylor expanded in eps around 0 0.1

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

    4. Simplified0.2

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

    if 0.0026284641846169455 < eps

    1. Initial program 28.6

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

    2. Applied egg-rr0.8

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.002752665168387367:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{expm1}\left(\mathsf{log1p}\left(\sin x \cdot \sin \varepsilon\right)\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0026284641846169455:\\ \;\;\;\;\cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right) - \sin x \cdot \mathsf{fma}\left({\varepsilon}^{3}, -0.16666666666666666, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array} \]

Reproduce

herbie shell --seed 2022130 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))