Average Error: 37.1 → 0.4
Time: 17.1s
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 r169694 = x;
        double r169695 = eps;
        double r169696 = r169694 + r169695;
        double r169697 = sin(r169696);
        double r169698 = sin(r169694);
        double r169699 = r169697 - r169698;
        return r169699;
}
↓
double f(double x, double eps) {
        double r169700 = eps;
        double r169701 = -8.781839081277773e-09;
        bool r169702 = r169700 <= r169701;
        double r169703 = 1.1905404803477009e-08;
        bool r169704 = r169700 <= r169703;
        double r169705 = !r169704;
        bool r169706 = r169702 || r169705;
        double r169707 = x;
        double r169708 = sin(r169707);
        double r169709 = cos(r169700);
        double r169710 = r169708 * r169709;
        double r169711 = cos(r169707);
        double r169712 = sin(r169700);
        double r169713 = r169711 * r169712;
        double r169714 = r169710 + r169713;
        double r169715 = r169714 - r169708;
        double r169716 = 2.0;
        double r169717 = 0.5;
        double r169718 = r169717 * r169700;
        double r169719 = sin(r169718);
        double r169720 = fma(r169707, r169716, r169700);
        double r169721 = r169717 * r169720;
        double r169722 = cos(r169721);
        double r169723 = r169719 * r169722;
        double r169724 = r169716 * r169723;
        double r169725 = r169706 ? r169715 : r169724;
        return r169725;
}

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)))