Average Error: 39.5 → 16.8
Time: 7.6s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -9.8390408000405122 \cdot 10^{-17}:\\ \;\;\;\;\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \sqrt[3]{{\left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}^{3}} + \cos x \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 3.23942241978548316 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \left(\sqrt[3]{\sin x} \cdot \sin \varepsilon\right)\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -9.8390408000405122 \cdot 10^{-17}:\\
\;\;\;\;\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \sqrt[3]{{\left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}^{3}} + \cos x \cdot \cos x}\\

\mathbf{elif}\;\varepsilon \le 3.23942241978548316 \cdot 10^{-7}:\\
\;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\

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

\end{array}
double f(double x, double eps) {
        double r62670 = x;
        double r62671 = eps;
        double r62672 = r62670 + r62671;
        double r62673 = cos(r62672);
        double r62674 = cos(r62670);
        double r62675 = r62673 - r62674;
        return r62675;
}
↓
double f(double x, double eps) {
        double r62676 = eps;
        double r62677 = -9.839040800040512e-17;
        bool r62678 = r62676 <= r62677;
        double r62679 = x;
        double r62680 = cos(r62679);
        double r62681 = cos(r62676);
        double r62682 = r62680 * r62681;
        double r62683 = sin(r62679);
        double r62684 = sin(r62676);
        double r62685 = r62683 * r62684;
        double r62686 = r62682 - r62685;
        double r62687 = 3.0;
        double r62688 = pow(r62686, r62687);
        double r62689 = pow(r62680, r62687);
        double r62690 = r62688 - r62689;
        double r62691 = r62681 * r62680;
        double r62692 = r62691 - r62685;
        double r62693 = r62686 + r62680;
        double r62694 = pow(r62693, r62687);
        double r62695 = cbrt(r62694);
        double r62696 = r62692 * r62695;
        double r62697 = r62680 * r62680;
        double r62698 = r62696 + r62697;
        double r62699 = r62690 / r62698;
        double r62700 = 3.239422419785483e-07;
        bool r62701 = r62676 <= r62700;
        double r62702 = 0.16666666666666666;
        double r62703 = pow(r62679, r62687);
        double r62704 = r62702 * r62703;
        double r62705 = r62704 - r62679;
        double r62706 = 0.5;
        double r62707 = r62676 * r62706;
        double r62708 = r62705 - r62707;
        double r62709 = r62676 * r62708;
        double r62710 = cbrt(r62683);
        double r62711 = r62710 * r62710;
        double r62712 = r62710 * r62684;
        double r62713 = r62711 * r62712;
        double r62714 = r62682 - r62713;
        double r62715 = r62714 - r62680;
        double r62716 = r62701 ? r62709 : r62715;
        double r62717 = r62678 ? r62699 : r62716;
        return r62717;
}

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 < -9.839040800040512e-17

    1. Initial program 30.7

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

    3. Applied cos-sum2.2

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

    5. Applied flip3--2.3

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

    6. Simplified2.3

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

    8. Applied add-cbrt-cube2.4

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

    9. Simplified2.4

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

    if -9.839040800040512e-17 < eps < 3.239422419785483e-07

    1. Initial program 49.1

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

    2. Taylor expanded around 0 32.2

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

    3. Simplified32.2

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

    if 3.239422419785483e-07 < eps

    1. Initial program 29.8

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

    5. Applied add-cube-cbrt1.2

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

    6. Applied associate-*l*1.2

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

  4. Final simplification16.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -9.8390408000405122 \cdot 10^{-17}:\\ \;\;\;\;\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \sqrt[3]{{\left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}^{3}} + \cos x \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 3.23942241978548316 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \left(\sqrt[3]{\sin x} \cdot \sin \varepsilon\right)\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))