Average Error: 36.9 → 15.6
Time: 20.0s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.39838213499975321573025536050478104761 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(-\sqrt[3]{\sin x}, \sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}, \sqrt[3]{\sin x} \cdot \left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right)\right) + \mathsf{fma}\left(\cos \varepsilon, \sin x, \cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 3.395662673764271314665716906433539720035 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(x \cdot \varepsilon, x, \varepsilon \cdot \left(x \cdot \varepsilon\right)\right), \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\sqrt[3]{\sin x}, \sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}, \sqrt[3]{\sin x} \cdot \left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right)\right) + \mathsf{fma}\left(\cos \varepsilon, \sin x, \cos x \cdot \sin \varepsilon - \sin x\right)\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.39838213499975321573025536050478104761 \cdot 10^{-69}:\\
\;\;\;\;\mathsf{fma}\left(-\sqrt[3]{\sin x}, \sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}, \sqrt[3]{\sin x} \cdot \left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right)\right) + \mathsf{fma}\left(\cos \varepsilon, \sin x, \cos x \cdot \sin \varepsilon - \sin x\right)\\

\mathbf{elif}\;\varepsilon \le 3.395662673764271314665716906433539720035 \cdot 10^{-46}:\\
\;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(x \cdot \varepsilon, x, \varepsilon \cdot \left(x \cdot \varepsilon\right)\right), \varepsilon\right)\\

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

\end{array}
double f(double x, double eps) {
        double r5320142 = x;
        double r5320143 = eps;
        double r5320144 = r5320142 + r5320143;
        double r5320145 = sin(r5320144);
        double r5320146 = sin(r5320142);
        double r5320147 = r5320145 - r5320146;
        return r5320147;
}
↓
double f(double x, double eps) {
        double r5320148 = eps;
        double r5320149 = -1.3983821349997532e-69;
        bool r5320150 = r5320148 <= r5320149;
        double r5320151 = x;
        double r5320152 = sin(r5320151);
        double r5320153 = cbrt(r5320152);
        double r5320154 = -r5320153;
        double r5320155 = r5320153 * r5320153;
        double r5320156 = r5320153 * r5320155;
        double r5320157 = fma(r5320154, r5320155, r5320156);
        double r5320158 = cos(r5320148);
        double r5320159 = cos(r5320151);
        double r5320160 = sin(r5320148);
        double r5320161 = r5320159 * r5320160;
        double r5320162 = r5320161 - r5320152;
        double r5320163 = fma(r5320158, r5320152, r5320162);
        double r5320164 = r5320157 + r5320163;
        double r5320165 = 3.3956626737642713e-46;
        bool r5320166 = r5320148 <= r5320165;
        double r5320167 = -0.5;
        double r5320168 = r5320151 * r5320148;
        double r5320169 = r5320148 * r5320168;
        double r5320170 = fma(r5320168, r5320151, r5320169);
        double r5320171 = fma(r5320167, r5320170, r5320148);
        double r5320172 = r5320166 ? r5320171 : r5320164;
        double r5320173 = r5320150 ? r5320164 : r5320172;
        return r5320173;
}

Error

Bits error versus x

Bits error versus eps

Target

Original36.9
Target15.1
Herbie15.6
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.3983821349997532e-69 or 3.3956626737642713e-46 < eps

    1. Initial program 30.3

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

    3. Applied sin-sum4.7

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

    4. Applied associate--l+4.8

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

    6. Applied add-cube-cbrt5.1

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

    7. Applied prod-diff5.1

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

    8. Applied associate-+r+5.1

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

    9. Simplified4.7

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

    if -1.3983821349997532e-69 < eps < 3.3956626737642713e-46

    1. Initial program 46.9

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

    3. Applied sin-sum46.9

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

    4. Taylor expanded around 0 32.2

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

    5. Simplified32.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\varepsilon \cdot x, x, \varepsilon \cdot \left(\varepsilon \cdot x\right)\right), \varepsilon\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification15.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.39838213499975321573025536050478104761 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(-\sqrt[3]{\sin x}, \sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}, \sqrt[3]{\sin x} \cdot \left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right)\right) + \mathsf{fma}\left(\cos \varepsilon, \sin x, \cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 3.395662673764271314665716906433539720035 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(x \cdot \varepsilon, x, \varepsilon \cdot \left(x \cdot \varepsilon\right)\right), \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\sqrt[3]{\sin x}, \sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}, \sqrt[3]{\sin x} \cdot \left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right)\right) + \mathsf{fma}\left(\cos \varepsilon, \sin x, \cos x \cdot \sin \varepsilon - \sin x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

  (- (sin (+ x eps)) (sin x)))