Average Error: 39.6 → 15.8
Time: 6.5s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.13315499271887297 \cdot 10^{-10} \lor \neg \left(\varepsilon \le 3.28455979101293219 \cdot 10^{-7}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \cos \varepsilon\right)\right) - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.13315499271887297 \cdot 10^{-10} \lor \neg \left(\varepsilon \le 3.28455979101293219 \cdot 10^{-7}\right):\\
\;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \cos \varepsilon\right)\right) - \mathsf{fma}\left(\sin x, \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 r51949 = x;
        double r51950 = eps;
        double r51951 = r51949 + r51950;
        double r51952 = cos(r51951);
        double r51953 = cos(r51949);
        double r51954 = r51952 - r51953;
        return r51954;
}
↓
double f(double x, double eps) {
        double r51955 = eps;
        double r51956 = -1.133154992718873e-10;
        bool r51957 = r51955 <= r51956;
        double r51958 = 3.284559791012932e-07;
        bool r51959 = r51955 <= r51958;
        double r51960 = !r51959;
        bool r51961 = r51957 || r51960;
        double r51962 = x;
        double r51963 = cos(r51962);
        double r51964 = cos(r51955);
        double r51965 = r51963 * r51964;
        double r51966 = expm1(r51965);
        double r51967 = log1p(r51966);
        double r51968 = sin(r51962);
        double r51969 = sin(r51955);
        double r51970 = fma(r51968, r51969, r51963);
        double r51971 = r51967 - r51970;
        double r51972 = 0.16666666666666666;
        double r51973 = 3.0;
        double r51974 = pow(r51962, r51973);
        double r51975 = r51972 * r51974;
        double r51976 = r51975 - r51962;
        double r51977 = 0.5;
        double r51978 = r51955 * r51977;
        double r51979 = r51976 - r51978;
        double r51980 = r51955 * r51979;
        double r51981 = r51961 ? r51971 : r51980;
        return r51981;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.133154992718873e-10 or 3.284559791012932e-07 < eps

    1. Initial program 30.5

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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. Applied associate--l-1.1

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

    5. Simplified1.1

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

    7. Applied log1p-expm1-u1.2

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \cos \varepsilon\right)\right)} - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\]

    if -1.133154992718873e-10 < eps < 3.284559791012932e-07

    1. Initial program 49.3

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

    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

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification15.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.13315499271887297 \cdot 10^{-10} \lor \neg \left(\varepsilon \le 3.28455979101293219 \cdot 10^{-7}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \cos \varepsilon\right)\right) - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))