Average Error: 37.5 → 0.5
Time: 21.5s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.1998709236678226 \cdot 10^{-08}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \le 1.1087986248072222 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.1998709236678226 \cdot 10^{-08}:\\
\;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\

\mathbf{elif}\;\varepsilon \le 1.1087986248072222 \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(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\end{array}
double f(double x, double eps) {
        double r5929254 = x;
        double r5929255 = eps;
        double r5929256 = r5929254 + r5929255;
        double r5929257 = sin(r5929256);
        double r5929258 = sin(r5929254);
        double r5929259 = r5929257 - r5929258;
        return r5929259;
}
↓
double f(double x, double eps) {
        double r5929260 = eps;
        double r5929261 = -1.1998709236678226e-08;
        bool r5929262 = r5929260 <= r5929261;
        double r5929263 = x;
        double r5929264 = cos(r5929263);
        double r5929265 = sin(r5929260);
        double r5929266 = r5929264 * r5929265;
        double r5929267 = sin(r5929263);
        double r5929268 = r5929266 - r5929267;
        double r5929269 = cos(r5929260);
        double r5929270 = r5929267 * r5929269;
        double r5929271 = r5929268 + r5929270;
        double r5929272 = 1.1087986248072222e-08;
        bool r5929273 = r5929260 <= r5929272;
        double r5929274 = 2.0;
        double r5929275 = r5929260 / r5929274;
        double r5929276 = sin(r5929275);
        double r5929277 = fma(r5929274, r5929263, r5929260);
        double r5929278 = r5929277 / r5929274;
        double r5929279 = cos(r5929278);
        double r5929280 = r5929276 * r5929279;
        double r5929281 = r5929280 * r5929274;
        double r5929282 = r5929270 + r5929266;
        double r5929283 = r5929282 - r5929267;
        double r5929284 = r5929273 ? r5929281 : r5929283;
        double r5929285 = r5929262 ? r5929271 : r5929284;
        return r5929285;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.5
Target15.6
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -1.1998709236678226e-08

    1. Initial program 31.1

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 -1.1998709236678226e-08 < eps < 1.1087986248072222e-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

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

    if 1.1087986248072222e-08 < eps

    1. Initial program 30.9

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

    3. Applied sin-sum0.6

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.1998709236678226 \cdot 10^{-08}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \le 1.1087986248072222 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +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)))