Average Error: 36.9 → 0.5
Time: 22.2s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.096393406185498984369787505094642754422 \cdot 10^{-8}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.730605182535497089020295258590936143817 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.096393406185498984369787505094642754422 \cdot 10^{-8}:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 1.730605182535497089020295258590936143817 \cdot 10^{-9}:\\
\;\;\;\;2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\right)\right)\right)\\

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

\end{array}
double f(double x, double eps) {
        double r4531847 = x;
        double r4531848 = eps;
        double r4531849 = r4531847 + r4531848;
        double r4531850 = sin(r4531849);
        double r4531851 = sin(r4531847);
        double r4531852 = r4531850 - r4531851;
        return r4531852;
}
↓
double f(double x, double eps) {
        double r4531853 = eps;
        double r4531854 = -1.096393406185499e-08;
        bool r4531855 = r4531853 <= r4531854;
        double r4531856 = x;
        double r4531857 = sin(r4531856);
        double r4531858 = cos(r4531853);
        double r4531859 = r4531857 * r4531858;
        double r4531860 = cos(r4531856);
        double r4531861 = sin(r4531853);
        double r4531862 = r4531860 * r4531861;
        double r4531863 = r4531859 + r4531862;
        double r4531864 = r4531863 - r4531857;
        double r4531865 = 1.730605182535497e-09;
        bool r4531866 = r4531853 <= r4531865;
        double r4531867 = 2.0;
        double r4531868 = 0.5;
        double r4531869 = r4531868 * r4531853;
        double r4531870 = sin(r4531869);
        double r4531871 = fma(r4531868, r4531853, r4531856);
        double r4531872 = cos(r4531871);
        double r4531873 = log1p(r4531872);
        double r4531874 = expm1(r4531873);
        double r4531875 = r4531870 * r4531874;
        double r4531876 = r4531867 * r4531875;
        double r4531877 = r4531866 ? r4531876 : r4531864;
        double r4531878 = r4531855 ? r4531864 : r4531877;
        return r4531878;
}

Error

Bits error versus x

Bits error versus eps

Target

Original36.9
Target15.3
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.096393406185499e-08 or 1.730605182535497e-09 < eps

    1. Initial program 29.8

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

    3. Applied sin-sum0.5

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

    if -1.096393406185499e-08 < eps < 1.730605182535497e-09

    1. Initial program 44.7

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

    3. Applied diff-sin44.7

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

    4. Simplified0.3

      \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)}\]
    5. Using strategy rm

    6. Applied add-cbrt-cube0.5

      \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \color{blue}{\sqrt[3]{\left(\cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)}}\right)\]

    7. Simplified0.5

      \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sqrt[3]{\color{blue}{\left(\cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right) \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\right) \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)}}\right)\]
    8. Using strategy rm

    9. Applied expm1-log1p-u0.5

      \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\left(\cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right) \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\right) \cdot \cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)}\right)\right)}\right)\]

    10. Simplified0.4

      \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\right)}\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.096393406185498984369787505094642754422 \cdot 10^{-8}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.730605182535497089020295258590936143817 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +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)))