Average Error: 39.6 → 16.2
Time: 7.4s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.1193854390859995 \cdot 10^{-28}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3} + {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x - \sin x \cdot \sin \varepsilon\right) + \left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}\\ \mathbf{elif}\;\varepsilon \le 4.5634137760449539 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\cos x \cdot \cos \varepsilon}\right) - \left(\sqrt[3]{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}} + \cos x\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -2.1193854390859995 \cdot 10^{-28}:\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3} + {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x - \sin x \cdot \sin \varepsilon\right) + \left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}\\

\mathbf{elif}\;\varepsilon \le 4.5634137760449539 \cdot 10^{-6}:\\
\;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(e^{\cos x \cdot \cos \varepsilon}\right) - \left(\sqrt[3]{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}} + \cos x\right)\\

\end{array}
double f(double x, double eps) {
        double r39962 = x;
        double r39963 = eps;
        double r39964 = r39962 + r39963;
        double r39965 = cos(r39964);
        double r39966 = cos(r39962);
        double r39967 = r39965 - r39966;
        return r39967;
}
↓
double f(double x, double eps) {
        double r39968 = eps;
        double r39969 = -2.1193854390859995e-28;
        bool r39970 = r39968 <= r39969;
        double r39971 = x;
        double r39972 = cos(r39971);
        double r39973 = cos(r39968);
        double r39974 = r39972 * r39973;
        double r39975 = sin(r39971);
        double r39976 = sin(r39968);
        double r39977 = r39975 * r39976;
        double r39978 = 3.0;
        double r39979 = pow(r39977, r39978);
        double r39980 = pow(r39972, r39978);
        double r39981 = r39979 + r39980;
        double r39982 = r39972 - r39977;
        double r39983 = r39972 * r39982;
        double r39984 = r39977 * r39977;
        double r39985 = r39983 + r39984;
        double r39986 = r39981 / r39985;
        double r39987 = r39974 - r39986;
        double r39988 = 4.563413776044954e-06;
        bool r39989 = r39968 <= r39988;
        double r39990 = 0.16666666666666666;
        double r39991 = pow(r39971, r39978);
        double r39992 = r39990 * r39991;
        double r39993 = r39992 - r39971;
        double r39994 = 0.5;
        double r39995 = r39968 * r39994;
        double r39996 = r39993 - r39995;
        double r39997 = r39968 * r39996;
        double r39998 = exp(r39974);
        double r39999 = log(r39998);
        double r40000 = cbrt(r39979);
        double r40001 = r40000 + r39972;
        double r40002 = r39999 - r40001;
        double r40003 = r39989 ? r39997 : r40002;
        double r40004 = r39970 ? r39987 : r40003;
        return r40004;
}

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 < -2.1193854390859995e-28

    1. Initial program 32.6

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

    3. Applied cos-sum4.6

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

    4. Applied associate--l-4.7

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

    6. Applied flip3-+4.8

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

    7. Simplified4.8

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

    if -2.1193854390859995e-28 < eps < 4.563413776044954e-06

    1. Initial program 49.1

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

    2. Taylor expanded around 0 31.0

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

    3. Simplified31.0

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

    if 4.563413776044954e-06 < eps

    1. Initial program 29.8

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

    3. Applied cos-sum0.9

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

    4. Applied associate--l-0.9

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

    6. Applied add-log-exp1.1

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

    8. Applied add-cbrt-cube1.2

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

    9. Applied add-cbrt-cube1.2

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

    10. Applied cbrt-unprod1.2

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

    11. Simplified1.2

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

  4. Final simplification16.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.1193854390859995 \cdot 10^{-28}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3} + {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x - \sin x \cdot \sin \varepsilon\right) + \left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}\\ \mathbf{elif}\;\varepsilon \le 4.5634137760449539 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\cos x \cdot \cos \varepsilon}\right) - \left(\sqrt[3]{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}} + \cos x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))