Average Error: 36.7 → 0.7
Time: 16.7s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -892737702.61047351360321044921875:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 3.394506424383313479258624062949965816127 \cdot 10^{-9}:\\ \;\;\;\;\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \sin \varepsilon, -\sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -892737702.61047351360321044921875:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 3.394506424383313479258624062949965816127 \cdot 10^{-9}:\\
\;\;\;\;\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot 2\\

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

\end{array}
double f(double x, double eps) {
        double r84320 = x;
        double r84321 = eps;
        double r84322 = r84320 + r84321;
        double r84323 = sin(r84322);
        double r84324 = sin(r84320);
        double r84325 = r84323 - r84324;
        return r84325;
}
↓
double f(double x, double eps) {
        double r84326 = eps;
        double r84327 = -892737702.6104735;
        bool r84328 = r84326 <= r84327;
        double r84329 = x;
        double r84330 = sin(r84329);
        double r84331 = cos(r84326);
        double r84332 = r84330 * r84331;
        double r84333 = cos(r84329);
        double r84334 = sin(r84326);
        double r84335 = r84333 * r84334;
        double r84336 = r84332 + r84335;
        double r84337 = r84336 - r84330;
        double r84338 = 3.3945064243833135e-09;
        bool r84339 = r84326 <= r84338;
        double r84340 = 2.0;
        double r84341 = fma(r84340, r84329, r84326);
        double r84342 = r84341 / r84340;
        double r84343 = cos(r84342);
        double r84344 = expm1(r84343);
        double r84345 = log1p(r84344);
        double r84346 = r84326 / r84340;
        double r84347 = sin(r84346);
        double r84348 = r84345 * r84347;
        double r84349 = r84348 * r84340;
        double r84350 = -r84330;
        double r84351 = fma(r84333, r84334, r84350);
        double r84352 = r84351 + r84332;
        double r84353 = r84339 ? r84349 : r84352;
        double r84354 = r84328 ? r84337 : r84353;
        return r84354;
}

Error

Bits error versus x

Bits error versus eps

Target

Original36.7
Target14.9
Herbie0.7
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -892737702.6104735

    1. Initial program 29.6

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

    3. Applied sin-sum0.4

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

    if -892737702.6104735 < eps < 3.3945064243833135e-09

    1. Initial program 43.6

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

    3. Applied diff-sin43.6

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

    4. Simplified0.8

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

    6. Applied log1p-expm1-u0.9

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

    7. Simplified0.9

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

    9. Applied *-un-lft-identity0.9

      \[\leadsto 2 \cdot \color{blue}{\left(1 \cdot \left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\right)\right)\right)}\]

    if 3.3945064243833135e-09 < eps

    1. Initial program 29.9

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

    3. Applied sin-sum0.6

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

    4. Applied associate--l+0.6

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

    5. Simplified0.6

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -892737702.61047351360321044921875:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 3.394506424383313479258624062949965816127 \cdot 10^{-9}:\\ \;\;\;\;\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \sin \varepsilon, -\sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

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