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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.41467470001176498 \cdot 10^{-4}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 6.808328837524803 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(\sin \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot -2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;\varepsilon \le -2.41467470001176498 \cdot 10^{-4}:\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 6.808328837524803 \cdot 10^{-5}:\\
\;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(\sin \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot -2\right)\right)\right)\\

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

\end{array}

double f(double x, double eps) {
        double r46990 = x;
        double r46991 = eps;
        double r46992 = r46990 + r46991;
        double r46993 = cos(r46992);
        double r46994 = cos(r46990);
        double r46995 = r46993 - r46994;
        return r46995;
}

↓

double f(double x, double eps) {
        double r46996 = eps;
        double r46997 = -0.0002414674700011765;
        bool r46998 = r46996 <= r46997;
        double r46999 = x;
        double r47000 = cos(r46999);
        double r47001 = cos(r46996);
        double r47002 = r47000 * r47001;
        double r47003 = sin(r46999);
        double r47004 = sin(r46996);
        double r47005 = r47003 * r47004;
        double r47006 = r47002 - r47005;
        double r47007 = r47006 - r47000;
        double r47008 = 6.808328837524803e-05;
        bool r47009 = r46996 <= r47008;
        double r47010 = 0.5;
        double r47011 = r47010 * r46996;
        double r47012 = sin(r47011);
        double r47013 = 2.0;
        double r47014 = fma(r47013, r46999, r46996);
        double r47015 = r47014 * r47010;
        double r47016 = sin(r47015);
        double r47017 = -2.0;
        double r47018 = r47016 * r47017;
        double r47019 = r47012 * r47018;
        double r47020 = expm1(r47019);
        double r47021 = log1p(r47020);
        double r47022 = fma(r47003, r47004, r47000);
        double r47023 = r47002 - r47022;
        double r47024 = r47009 ? r47021 : r47023;
        double r47025 = r46998 ? r47007 : r47024;
        return r47025;
}

Derivation

Split input into 3 regimes
if eps < -0.0002414674700011765
1. Initial program 31.4
  \[\cos \left(x + \varepsilon\right) - \cos x\]
2. Using strategy rm
3. Applied cos-sum0.9
  \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
if -0.0002414674700011765 < eps < 6.808328837524803e-05
1. Initial program 48.9
  \[\cos \left(x + \varepsilon\right) - \cos x\]
2. Using strategy rm
3. Applied diff-cos37.9
  \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
4. Simplified0.5
  \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
5. Using strategy rm
6. Applied associate-*r*0.5
  \[\leadsto \color{blue}{\left(-2 \cdot \sin \left(\frac{0 + \varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)}\]
7. Simplified0.5
  \[\leadsto \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)} \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\]
8. Using strategy rm
9. Applied log1p-expm1-u0.5
  \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)}\]
10. Simplified0.5
  \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(\sin \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot -2\right)\right)}\right)\]
if 6.808328837524803e-05 < eps
1. Initial program 29.6
  \[\cos \left(x + \varepsilon\right) - \cos x\]
2. Using strategy rm
3. Applied cos-sum0.9
  \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
4. Applied associate--l-0.9
  \[\leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
5. Simplified0.9
  \[\leadsto \cos x \cdot \cos \varepsilon - \color{blue}{\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)}\]
Recombined 3 regimes into one program.
Final simplification0.7
\[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.41467470001176498 \cdot 10^{-4}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 6.808328837524803 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(\sin \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot -2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \end{array}\]

Falling back on racket egraph because egg-herbie package not installed (more)

Error

Derivation

`if eps < -0.0002414674700011765`

`if -0.0002414674700011765 < eps < 6.808328837524803e-05`

`if 6.808328837524803e-05 < eps`

Reproduce