Average Error: 37.1 → 0.8
Time: 20.9s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.141151458641036 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 2.1464224965526296 \cdot 10^{-27}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -7.141151458641036 \cdot 10^{-09}:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 2.1464224965526296 \cdot 10^{-27}:\\
\;\;\;\;\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 r3983339 = x;
        double r3983340 = eps;
        double r3983341 = r3983339 + r3983340;
        double r3983342 = sin(r3983341);
        double r3983343 = sin(r3983339);
        double r3983344 = r3983342 - r3983343;
        return r3983344;
}
↓
double f(double x, double eps) {
        double r3983345 = eps;
        double r3983346 = -7.141151458641036e-09;
        bool r3983347 = r3983345 <= r3983346;
        double r3983348 = x;
        double r3983349 = sin(r3983348);
        double r3983350 = cos(r3983345);
        double r3983351 = r3983349 * r3983350;
        double r3983352 = cos(r3983348);
        double r3983353 = sin(r3983345);
        double r3983354 = r3983352 * r3983353;
        double r3983355 = r3983351 + r3983354;
        double r3983356 = r3983355 - r3983349;
        double r3983357 = 2.1464224965526296e-27;
        bool r3983358 = r3983345 <= r3983357;
        double r3983359 = 2.0;
        double r3983360 = r3983345 / r3983359;
        double r3983361 = sin(r3983360);
        double r3983362 = fma(r3983359, r3983348, r3983345);
        double r3983363 = r3983362 / r3983359;
        double r3983364 = cos(r3983363);
        double r3983365 = r3983361 * r3983364;
        double r3983366 = r3983365 * r3983359;
        double r3983367 = r3983354 - r3983349;
        double r3983368 = r3983367 + r3983351;
        double r3983369 = r3983358 ? r3983366 : r3983368;
        double r3983370 = r3983347 ? r3983356 : r3983369;
        return r3983370;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.1
Target14.4
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 < -7.141151458641036e-09

    1. Initial program 29.9

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

    3. Applied sin-sum0.6

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

    if -7.141151458641036e-09 < eps < 2.1464224965526296e-27

    1. Initial program 45.0

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

    3. Applied diff-sin45.0

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

    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 2.1464224965526296e-27 < eps

    1. Initial program 29.3

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

    3. Applied sin-sum2.0

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

    4. Applied associate--l+2.0

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -7.141151458641036 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 2.1464224965526296 \cdot 10^{-27}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +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)))