Average Error: 40.0 → 0.7
Time: 25.5s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -9.798283548084824 \cdot 10^{-06}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 0.00011275979133792468:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -9.798283548084824 \cdot 10^{-06}:\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 0.00011275979133792468:\\
\;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\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 r1334831 = x;
        double r1334832 = eps;
        double r1334833 = r1334831 + r1334832;
        double r1334834 = cos(r1334833);
        double r1334835 = cos(r1334831);
        double r1334836 = r1334834 - r1334835;
        return r1334836;
}
↓
double f(double x, double eps) {
        double r1334837 = eps;
        double r1334838 = -9.798283548084824e-06;
        bool r1334839 = r1334837 <= r1334838;
        double r1334840 = x;
        double r1334841 = cos(r1334840);
        double r1334842 = cos(r1334837);
        double r1334843 = r1334841 * r1334842;
        double r1334844 = sin(r1334840);
        double r1334845 = sin(r1334837);
        double r1334846 = r1334844 * r1334845;
        double r1334847 = r1334843 - r1334846;
        double r1334848 = r1334847 - r1334841;
        double r1334849 = 0.00011275979133792468;
        bool r1334850 = r1334837 <= r1334849;
        double r1334851 = 2.0;
        double r1334852 = fma(r1334851, r1334840, r1334837);
        double r1334853 = r1334852 / r1334851;
        double r1334854 = sin(r1334853);
        double r1334855 = log1p(r1334854);
        double r1334856 = expm1(r1334855);
        double r1334857 = -2.0;
        double r1334858 = r1334837 / r1334851;
        double r1334859 = sin(r1334858);
        double r1334860 = r1334857 * r1334859;
        double r1334861 = r1334856 * r1334860;
        double r1334862 = r1334850 ? r1334861 : r1334848;
        double r1334863 = r1334839 ? r1334848 : r1334862;
        return r1334863;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if eps < -9.798283548084824e-06 or 0.00011275979133792468 < eps

    1. Initial program 30.7

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

    3. Applied cos-sum1.0

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

    if -9.798283548084824e-06 < eps < 0.00011275979133792468

    1. Initial program 49.9

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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. Simplified0.4

      \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)}\]
    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 expm1-log1p-u0.5

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -9.798283548084824 \cdot 10^{-06}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 0.00011275979133792468:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))