Average Error: 40.0 → 16.0
Time: 6.7s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.19483323723045707450996508539042828545 \cdot 10^{-8}:\\ \;\;\;\;\frac{\left(-{\left(\cos x\right)}^{3}\right) + {\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right)}^{3}}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon - \cos x\right)\right) + \cos x \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 1.611471127494872569020696813935555459458 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \log \left(e^{\sin x \cdot \sin \varepsilon + \cos x}\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -8.19483323723045707450996508539042828545 \cdot 10^{-8}:\\
\;\;\;\;\frac{\left(-{\left(\cos x\right)}^{3}\right) + {\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right)}^{3}}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon - \cos x\right)\right) + \cos x \cdot \cos x}\\

\mathbf{elif}\;\varepsilon \le 1.611471127494872569020696813935555459458 \cdot 10^{-14}:\\
\;\;\;\;\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\\

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

\end{array}
double f(double x, double eps) {
        double r67936 = x;
        double r67937 = eps;
        double r67938 = r67936 + r67937;
        double r67939 = cos(r67938);
        double r67940 = cos(r67936);
        double r67941 = r67939 - r67940;
        return r67941;
}
↓
double f(double x, double eps) {
        double r67942 = eps;
        double r67943 = -8.194833237230457e-08;
        bool r67944 = r67942 <= r67943;
        double r67945 = x;
        double r67946 = cos(r67945);
        double r67947 = 3.0;
        double r67948 = pow(r67946, r67947);
        double r67949 = -r67948;
        double r67950 = cos(r67942);
        double r67951 = r67950 * r67946;
        double r67952 = sin(r67945);
        double r67953 = sin(r67942);
        double r67954 = r67952 * r67953;
        double r67955 = r67951 - r67954;
        double r67956 = pow(r67955, r67947);
        double r67957 = r67949 + r67956;
        double r67958 = r67946 * r67950;
        double r67959 = r67954 - r67946;
        double r67960 = r67958 - r67959;
        double r67961 = r67955 * r67960;
        double r67962 = r67946 * r67946;
        double r67963 = r67961 + r67962;
        double r67964 = r67957 / r67963;
        double r67965 = 1.6114711274948726e-14;
        bool r67966 = r67942 <= r67965;
        double r67967 = 0.041666666666666664;
        double r67968 = 4.0;
        double r67969 = pow(r67942, r67968);
        double r67970 = r67967 * r67969;
        double r67971 = r67945 * r67942;
        double r67972 = 0.5;
        double r67973 = 2.0;
        double r67974 = pow(r67942, r67973);
        double r67975 = r67972 * r67974;
        double r67976 = r67971 + r67975;
        double r67977 = r67970 - r67976;
        double r67978 = r67954 + r67946;
        double r67979 = exp(r67978);
        double r67980 = log(r67979);
        double r67981 = r67951 - r67980;
        double r67982 = r67966 ? r67977 : r67981;
        double r67983 = r67944 ? r67964 : r67982;
        return r67983;
}

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 < -8.194833237230457e-08

    1. Initial program 30.9

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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-cbrt-cube1.2

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

    6. Applied add-cbrt-cube1.3

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

    7. Applied cbrt-unprod1.2

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

    8. Simplified1.2

      \[\leadsto \left(\cos x \cdot \cos \varepsilon - \sqrt[3]{\color{blue}{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}}\right) - \cos x\]
    9. Using strategy rm

    10. Applied flip3--1.4

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

    11. Simplified1.4

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

    12. Simplified1.3

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

    if -8.194833237230457e-08 < eps < 1.6114711274948726e-14

    1. Initial program 49.6

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

    3. Applied cos-sum49.4

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

    5. Applied add-cbrt-cube49.4

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

    6. Applied add-cbrt-cube49.4

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

    7. Applied cbrt-unprod49.4

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

    8. Simplified49.4

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

    9. Taylor expanded around inf 49.4

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

    10. Taylor expanded around 0 31.9

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)}\]

    if 1.6114711274948726e-14 < eps

    1. Initial program 31.8

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

    3. Applied cos-sum1.7

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

    5. Applied add-cbrt-cube1.8

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

    6. Applied add-cbrt-cube1.8

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

    7. Applied cbrt-unprod1.8

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

    8. Simplified1.8

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

    9. Taylor expanded around inf 1.7

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

    11. Applied add-log-exp1.9

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

    12. Applied add-log-exp1.9

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

    13. Applied sum-log1.9

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

    14. Simplified1.8

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

  4. Final simplification16.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.19483323723045707450996508539042828545 \cdot 10^{-8}:\\ \;\;\;\;\frac{\left(-{\left(\cos x\right)}^{3}\right) + {\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right)}^{3}}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon - \cos x\right)\right) + \cos x \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 1.611471127494872569020696813935555459458 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \log \left(e^{\sin x \cdot \sin \varepsilon + \cos x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))