Average Error: 39.7 → 16.1
Time: 21.3s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.705040344446819412541570780417199415524 \cdot 10^{-8} \lor \neg \left(\varepsilon \le 4.159942496741691009894952628723681058577 \cdot 10^{-13}\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.705040344446819412541570780417199415524 \cdot 10^{-8} \lor \neg \left(\varepsilon \le 4.159942496741691009894952628723681058577 \cdot 10^{-13}\right):\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\

\mathbf{else}:\\
\;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\

\end{array}
double f(double x, double eps) {
        double r61242 = x;
        double r61243 = eps;
        double r61244 = r61242 + r61243;
        double r61245 = cos(r61244);
        double r61246 = cos(r61242);
        double r61247 = r61245 - r61246;
        return r61247;
}
↓
double f(double x, double eps) {
        double r61248 = eps;
        double r61249 = -1.7050403444468194e-08;
        bool r61250 = r61248 <= r61249;
        double r61251 = 4.159942496741691e-13;
        bool r61252 = r61248 <= r61251;
        double r61253 = !r61252;
        bool r61254 = r61250 || r61253;
        double r61255 = x;
        double r61256 = cos(r61255);
        double r61257 = cos(r61248);
        double r61258 = r61256 * r61257;
        double r61259 = sin(r61255);
        double r61260 = sin(r61248);
        double r61261 = r61259 * r61260;
        double r61262 = r61261 + r61256;
        double r61263 = r61258 - r61262;
        double r61264 = 0.16666666666666666;
        double r61265 = 3.0;
        double r61266 = pow(r61255, r61265);
        double r61267 = r61264 * r61266;
        double r61268 = r61267 - r61255;
        double r61269 = 0.5;
        double r61270 = r61248 * r61269;
        double r61271 = r61268 - r61270;
        double r61272 = r61248 * r61271;
        double r61273 = r61254 ? r61263 : r61272;
        return r61273;
}

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 < -1.7050403444468194e-08

    1. Initial program 30.3

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

    3. Applied cos-sum1.1

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
    4. Using strategy rm

    5. Applied add-log-exp1.2

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

    if -1.7050403444468194e-08 < eps < 4.159942496741691e-13

    1. Initial program 49.2

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

    2. Taylor expanded around 0 31.7

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({x}^{3} \cdot \varepsilon\right) - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)}\]

    3. Simplified31.7

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)}\]

    if 4.159942496741691e-13 < eps

    1. Initial program 31.3

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

    3. Applied cos-sum1.9

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
    4. Using strategy rm

    5. Applied add-log-exp2.0

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

    6. Applied add-log-exp2.1

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

    7. Applied add-log-exp2.2

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

    8. Applied diff-log2.3

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

    9. Applied diff-log2.4

      \[\leadsto \color{blue}{\log \left(\frac{\frac{e^{\cos x \cdot \cos \varepsilon}}{e^{\sin x \cdot \sin \varepsilon}}}{e^{\cos x}}\right)}\]

    10. Simplified2.0

      \[\leadsto \log \color{blue}{\left(e^{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x}\right)}\]
    11. Using strategy rm

    12. Applied associate--l-2.0

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

  4. Final simplification16.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.705040344446819412541570780417199415524 \cdot 10^{-8} \lor \neg \left(\varepsilon \le 4.159942496741691009894952628723681058577 \cdot 10^{-13}\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019298 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))