Average Error: 39.8 → 16.8
Time: 9.5s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.979810964893114292557903597185422883762 \cdot 10^{-22}:\\ \;\;\;\;\log \left(e^{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon}\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 1.075814197720993254357452900693559172396 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\cos \varepsilon \cdot \cos x - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}^{3}}\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -2.979810964893114292557903597185422883762 \cdot 10^{-22}:\\
\;\;\;\;\log \left(e^{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon}\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 1.075814197720993254357452900693559172396 \cdot 10^{-6}:\\
\;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\

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

\end{array}
double f(double x, double eps) {
        double r173091 = x;
        double r173092 = eps;
        double r173093 = r173091 + r173092;
        double r173094 = cos(r173093);
        double r173095 = cos(r173091);
        double r173096 = r173094 - r173095;
        return r173096;
}
↓
double f(double x, double eps) {
        double r173097 = eps;
        double r173098 = -2.9798109648931143e-22;
        bool r173099 = r173097 <= r173098;
        double r173100 = x;
        double r173101 = cos(r173100);
        double r173102 = cos(r173097);
        double r173103 = r173101 * r173102;
        double r173104 = sin(r173100);
        double r173105 = sin(r173097);
        double r173106 = r173104 * r173105;
        double r173107 = r173103 - r173106;
        double r173108 = exp(r173107);
        double r173109 = log(r173108);
        double r173110 = r173109 - r173101;
        double r173111 = 1.0758141977209933e-06;
        bool r173112 = r173097 <= r173111;
        double r173113 = 0.16666666666666666;
        double r173114 = 3.0;
        double r173115 = pow(r173100, r173114);
        double r173116 = r173113 * r173115;
        double r173117 = r173116 - r173100;
        double r173118 = 0.5;
        double r173119 = r173097 * r173118;
        double r173120 = r173117 - r173119;
        double r173121 = r173097 * r173120;
        double r173122 = r173102 * r173101;
        double r173123 = r173106 + r173101;
        double r173124 = r173122 - r173123;
        double r173125 = pow(r173124, r173114);
        double r173126 = cbrt(r173125);
        double r173127 = r173112 ? r173121 : r173126;
        double r173128 = r173099 ? r173110 : r173127;
        return r173128;
}

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 < -2.9798109648931143e-22

    1. Initial program 32.0

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

    3. Applied cos-sum3.7

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

    5. Applied add-log-exp3.8

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

    6. Applied add-log-exp3.9

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

    7. Applied diff-log3.9

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

    8. Simplified3.8

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

    if -2.9798109648931143e-22 < eps < 1.0758141977209933e-06

    1. Initial program 49.0

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

    2. Taylor expanded around 0 32.7

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

    3. Simplified32.7

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

    if 1.0758141977209933e-06 < eps

    1. Initial program 31.2

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

    3. Applied cos-sum1.0

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

    4. Applied associate--l-1.1

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

    6. Applied add-cbrt-cube1.3

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

    7. Simplified1.3

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

  4. Final simplification16.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.979810964893114292557903597185422883762 \cdot 10^{-22}:\\ \;\;\;\;\log \left(e^{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon}\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 1.075814197720993254357452900693559172396 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\cos \varepsilon \cdot \cos x - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))