Average Error: 37.1 → 0.4
Time: 10.3s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.78183908127777317 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 1.19054048034770088 \cdot 10^{-8}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{1}{2} \cdot \mathsf{fma}\left(x, 2, \varepsilon\right)\right)\right)\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -8.78183908127777317 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 1.19054048034770088 \cdot 10^{-8}\right):\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{else}:\\
\;\;\;\;2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{1}{2} \cdot \mathsf{fma}\left(x, 2, \varepsilon\right)\right)\right)\\

\end{array}
double f(double x, double eps) {
        double r111918 = x;
        double r111919 = eps;
        double r111920 = r111918 + r111919;
        double r111921 = sin(r111920);
        double r111922 = sin(r111918);
        double r111923 = r111921 - r111922;
        return r111923;
}
↓
double f(double x, double eps) {
        double r111924 = eps;
        double r111925 = -8.781839081277773e-09;
        bool r111926 = r111924 <= r111925;
        double r111927 = 1.1905404803477009e-08;
        bool r111928 = r111924 <= r111927;
        double r111929 = !r111928;
        bool r111930 = r111926 || r111929;
        double r111931 = x;
        double r111932 = sin(r111931);
        double r111933 = cos(r111924);
        double r111934 = r111932 * r111933;
        double r111935 = cos(r111931);
        double r111936 = sin(r111924);
        double r111937 = r111935 * r111936;
        double r111938 = r111934 + r111937;
        double r111939 = r111938 - r111932;
        double r111940 = 2.0;
        double r111941 = 0.5;
        double r111942 = r111941 * r111924;
        double r111943 = sin(r111942);
        double r111944 = fma(r111931, r111940, r111924);
        double r111945 = r111941 * r111944;
        double r111946 = cos(r111945);
        double r111947 = r111943 * r111946;
        double r111948 = r111940 * r111947;
        double r111949 = r111930 ? r111939 : r111948;
        return r111949;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.1
Target15.2
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -8.781839081277773e-09 or 1.1905404803477009e-08 < eps

    1. Initial program 30.1

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 -8.781839081277773e-09 < eps < 1.1905404803477009e-08

    1. Initial program 44.4

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

    3. Applied diff-sin44.4

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

    4. Simplified0.3

      \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
    5. Using strategy rm

    6. Applied expm1-log1p-u0.4

      \[\leadsto 2 \cdot \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)}\right)\]

    7. Simplified0.4

      \[\leadsto 2 \cdot \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\cos \left(\frac{\mathsf{fma}\left(x, 2, \varepsilon\right)}{2}\right)\right)}\right)\right)\]

    8. Taylor expanded around inf 0.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.78183908127777317 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 1.19054048034770088 \cdot 10^{-8}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{1}{2} \cdot \mathsf{fma}\left(x, 2, \varepsilon\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

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