Average Error: 37.4 → 0.4
Time: 26.5s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.7134465698925543 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 3.970291838078694 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -2.7134465698925543 \cdot 10^{-08}:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 3.970291838078694 \cdot 10^{-08}:\\
\;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right) \cdot 2\\

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

\end{array}
double f(double x, double eps) {
        double r3091423 = x;
        double r3091424 = eps;
        double r3091425 = r3091423 + r3091424;
        double r3091426 = sin(r3091425);
        double r3091427 = sin(r3091423);
        double r3091428 = r3091426 - r3091427;
        return r3091428;
}
↓
double f(double x, double eps) {
        double r3091429 = eps;
        double r3091430 = -2.7134465698925543e-08;
        bool r3091431 = r3091429 <= r3091430;
        double r3091432 = x;
        double r3091433 = sin(r3091432);
        double r3091434 = cos(r3091429);
        double r3091435 = r3091433 * r3091434;
        double r3091436 = cos(r3091432);
        double r3091437 = sin(r3091429);
        double r3091438 = r3091436 * r3091437;
        double r3091439 = r3091435 + r3091438;
        double r3091440 = r3091439 - r3091433;
        double r3091441 = 3.970291838078694e-08;
        bool r3091442 = r3091429 <= r3091441;
        double r3091443 = 2.0;
        double r3091444 = r3091429 / r3091443;
        double r3091445 = sin(r3091444);
        double r3091446 = fma(r3091443, r3091432, r3091429);
        double r3091447 = r3091446 / r3091443;
        double r3091448 = cos(r3091447);
        double r3091449 = r3091445 * r3091448;
        double r3091450 = r3091449 * r3091443;
        double r3091451 = r3091438 - r3091433;
        double r3091452 = r3091451 + r3091435;
        double r3091453 = r3091442 ? r3091450 : r3091452;
        double r3091454 = r3091431 ? r3091440 : r3091453;
        return r3091454;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.4
Target15.8
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 3 regimes

  2. if eps < -2.7134465698925543e-08

    1. Initial program 30.8

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 -2.7134465698925543e-08 < eps < 3.970291838078694e-08

    1. Initial program 44.6

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

    3. Applied diff-sin44.6

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

    4. Simplified0.3

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

    if 3.970291838078694e-08 < eps

    1. Initial program 30.7

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

    3. Applied sin-sum0.5

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

    4. Applied associate--l+0.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.7134465698925543 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 3.970291838078694 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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