Average Error: 39.9 → 16.4
Time: 10.4s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -9.09194571234337580020110294837711693374 \cdot 10^{-27}:\\ \;\;\;\;\left(\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \cos \varepsilon\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 3.672352851463477013948306603677469717173 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(e^{\cos x \cdot \cos \varepsilon}\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -9.09194571234337580020110294837711693374 \cdot 10^{-27}:\\
\;\;\;\;\left(\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \cos \varepsilon\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 3.672352851463477013948306603677469717173 \cdot 10^{-9}:\\
\;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\

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

\end{array}
double f(double x, double eps) {
        double r57927 = x;
        double r57928 = eps;
        double r57929 = r57927 + r57928;
        double r57930 = cos(r57929);
        double r57931 = cos(r57927);
        double r57932 = r57930 - r57931;
        return r57932;
}
↓
double f(double x, double eps) {
        double r57933 = eps;
        double r57934 = -9.091945712343376e-27;
        bool r57935 = r57933 <= r57934;
        double r57936 = x;
        double r57937 = cos(r57936);
        double r57938 = cbrt(r57937);
        double r57939 = r57938 * r57938;
        double r57940 = cos(r57933);
        double r57941 = r57938 * r57940;
        double r57942 = r57939 * r57941;
        double r57943 = sin(r57936);
        double r57944 = sin(r57933);
        double r57945 = r57943 * r57944;
        double r57946 = r57942 - r57945;
        double r57947 = r57946 - r57937;
        double r57948 = 3.672352851463477e-09;
        bool r57949 = r57933 <= r57948;
        double r57950 = 0.16666666666666666;
        double r57951 = 3.0;
        double r57952 = pow(r57936, r57951);
        double r57953 = r57950 * r57952;
        double r57954 = r57953 - r57936;
        double r57955 = 0.5;
        double r57956 = r57933 * r57955;
        double r57957 = r57954 - r57956;
        double r57958 = r57933 * r57957;
        double r57959 = r57937 * r57940;
        double r57960 = exp(r57959);
        double r57961 = log(r57960);
        double r57962 = r57961 - r57945;
        double r57963 = r57962 - r57937;
        double r57964 = r57949 ? r57958 : r57963;
        double r57965 = r57935 ? r57947 : r57964;
        return r57965;
}

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 < -9.091945712343376e-27

    1. Initial program 32.2

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

    3. Applied cos-sum4.0

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

    5. Applied add-cube-cbrt4.3

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \sqrt[3]{\cos x}\right)} \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\]

    6. Applied associate-*l*4.3

      \[\leadsto \left(\color{blue}{\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \cos \varepsilon\right)} - \sin x \cdot \sin \varepsilon\right) - \cos x\]

    if -9.091945712343376e-27 < eps < 3.672352851463477e-09

    1. Initial program 49.5

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

    2. Taylor expanded around 0 31.5

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

    3. Simplified31.5

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

    if 3.672352851463477e-09 < eps

    1. Initial program 30.4

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

    3. Applied cos-sum1.2

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

    5. Applied add-log-exp1.5

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

  4. Final simplification16.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -9.09194571234337580020110294837711693374 \cdot 10^{-27}:\\ \;\;\;\;\left(\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \cos \varepsilon\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 3.672352851463477013948306603677469717173 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(e^{\cos x \cdot \cos \varepsilon}\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))