Average Error: 36.9 → 0.8
Time: 25.0s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.505892634593631943814391258018581254419 \cdot 10^{-9}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 8.5558943769231499470661433859635760918 \cdot 10^{-30}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -8.505892634593631943814391258018581254419 \cdot 10^{-9}:\\
\;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\

\mathbf{elif}\;\varepsilon \le 8.5558943769231499470661433859635760918 \cdot 10^{-30}:\\
\;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\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 r49519 = x;
        double r49520 = eps;
        double r49521 = r49519 + r49520;
        double r49522 = sin(r49521);
        double r49523 = sin(r49519);
        double r49524 = r49522 - r49523;
        return r49524;
}
↓
double f(double x, double eps) {
        double r49525 = eps;
        double r49526 = -8.505892634593632e-09;
        bool r49527 = r49525 <= r49526;
        double r49528 = x;
        double r49529 = sin(r49528);
        double r49530 = cos(r49525);
        double r49531 = r49529 * r49530;
        double r49532 = cos(r49528);
        double r49533 = sin(r49525);
        double r49534 = r49532 * r49533;
        double r49535 = r49534 - r49529;
        double r49536 = r49531 + r49535;
        double r49537 = 8.55589437692315e-30;
        bool r49538 = r49525 <= r49537;
        double r49539 = 2.0;
        double r49540 = r49525 / r49539;
        double r49541 = sin(r49540);
        double r49542 = fma(r49539, r49528, r49525);
        double r49543 = r49542 / r49539;
        double r49544 = cos(r49543);
        double r49545 = r49541 * r49544;
        double r49546 = r49539 * r49545;
        double r49547 = r49531 + r49534;
        double r49548 = r49547 - r49529;
        double r49549 = r49538 ? r49546 : r49548;
        double r49550 = r49527 ? r49536 : r49549;
        return r49550;
}

Error

Bits error versus x

Bits error versus eps

Target

Original36.9
Target14.8
Herbie0.8
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -8.505892634593632e-09

    1. Initial program 28.9

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

    3. Applied sin-sum0.6

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

    4. Applied associate--l+0.6

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

    if -8.505892634593632e-09 < eps < 8.55589437692315e-30

    1. Initial program 45.9

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

    3. Applied diff-sin45.9

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

    4. Simplified0.2

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

    if 8.55589437692315e-30 < eps

    1. Initial program 29.3

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

    3. Applied sin-sum2.0

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.505892634593631943814391258018581254419 \cdot 10^{-9}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 8.5558943769231499470661433859635760918 \cdot 10^{-30}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +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)))