Average Error: 39.4 → 0.7
Time: 16.2s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.005924238612320757:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \le 8.193560063624595 \cdot 10^{-05}:\\ \;\;\;\;\sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -0.005924238612320757:\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\

\mathbf{elif}\;\varepsilon \le 8.193560063624595 \cdot 10^{-05}:\\
\;\;\;\;\sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\

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

\end{array}
double f(double x, double eps) {
        double r733254 = x;
        double r733255 = eps;
        double r733256 = r733254 + r733255;
        double r733257 = cos(r733256);
        double r733258 = cos(r733254);
        double r733259 = r733257 - r733258;
        return r733259;
}
↓
double f(double x, double eps) {
        double r733260 = eps;
        double r733261 = -0.005924238612320757;
        bool r733262 = r733260 <= r733261;
        double r733263 = x;
        double r733264 = cos(r733263);
        double r733265 = cos(r733260);
        double r733266 = r733264 * r733265;
        double r733267 = sin(r733263);
        double r733268 = sin(r733260);
        double r733269 = r733267 * r733268;
        double r733270 = r733264 + r733269;
        double r733271 = r733266 - r733270;
        double r733272 = 8.193560063624595e-05;
        bool r733273 = r733260 <= r733272;
        double r733274 = r733263 + r733260;
        double r733275 = r733263 + r733274;
        double r733276 = 2.0;
        double r733277 = r733275 / r733276;
        double r733278 = sin(r733277);
        double r733279 = -2.0;
        double r733280 = r733260 / r733276;
        double r733281 = sin(r733280);
        double r733282 = r733279 * r733281;
        double r733283 = r733278 * r733282;
        double r733284 = r733266 - r733269;
        double r733285 = r733284 - r733264;
        double r733286 = r733273 ? r733283 : r733285;
        double r733287 = r733262 ? r733271 : r733286;
        return r733287;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if eps < -0.005924238612320757

    1. Initial program 29.1

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    2. Using strategy rm

    3. Applied cos-sum0.8

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

    4. Applied associate--l-0.8

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

    if -0.005924238612320757 < eps < 8.193560063624595e-05

    1. Initial program 48.7

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    2. Using strategy rm

    3. Applied diff-cos37.5

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

    4. Simplified0.6

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

    6. Applied associate-*r*0.5

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

    if 8.193560063624595e-05 < eps

    1. Initial program 30.9

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    2. Using strategy rm

    3. Applied cos-sum1.0

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.005924238612320757:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \le 8.193560063624595 \cdot 10^{-05}:\\ \;\;\;\;\sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))