Average Error: 39.2 → 0.7
Time: 26.9s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -5.453340308409901 \cdot 10^{-05}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 5.0711197596488575 \cdot 10^{-06}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -5.453340308409901 \cdot 10^{-05}:\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 5.0711197596488575 \cdot 10^{-06}:\\
\;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\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 r885164 = x;
        double r885165 = eps;
        double r885166 = r885164 + r885165;
        double r885167 = cos(r885166);
        double r885168 = cos(r885164);
        double r885169 = r885167 - r885168;
        return r885169;
}
↓
double f(double x, double eps) {
        double r885170 = eps;
        double r885171 = -5.453340308409901e-05;
        bool r885172 = r885170 <= r885171;
        double r885173 = x;
        double r885174 = cos(r885173);
        double r885175 = cos(r885170);
        double r885176 = r885174 * r885175;
        double r885177 = sin(r885173);
        double r885178 = sin(r885170);
        double r885179 = r885177 * r885178;
        double r885180 = r885176 - r885179;
        double r885181 = r885180 - r885174;
        double r885182 = 5.0711197596488575e-06;
        bool r885183 = r885170 <= r885182;
        double r885184 = -2.0;
        double r885185 = 2.0;
        double r885186 = r885170 / r885185;
        double r885187 = sin(r885186);
        double r885188 = r885184 * r885187;
        double r885189 = fma(r885185, r885173, r885170);
        double r885190 = r885189 / r885185;
        double r885191 = sin(r885190);
        double r885192 = r885188 * r885191;
        double r885193 = expm1(r885192);
        double r885194 = log1p(r885193);
        double r885195 = r885183 ? r885194 : r885181;
        double r885196 = r885172 ? r885181 : r885195;
        return r885196;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if eps < -5.453340308409901e-05 or 5.0711197596488575e-06 < eps

    1. Initial program 29.5

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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 -5.453340308409901e-05 < eps < 5.0711197596488575e-06

    1. Initial program 49.2

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

    3. Applied diff-cos37.8

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

    4. Simplified0.4

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

    6. Applied associate-*r*0.4

      \[\leadsto \color{blue}{\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)}\]
    7. Using strategy rm

    8. Applied log1p-expm1-u0.4

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -5.453340308409901 \cdot 10^{-05}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 5.0711197596488575 \cdot 10^{-06}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))