Average Error: 37.6 → 0.5
Time: 30.1s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.370205872151545 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.1744534610942515 \cdot 10^{-09}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -8.370205872151545 \cdot 10^{-09}:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 1.1744534610942515 \cdot 10^{-09}:\\
\;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)\right)\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 r18567202 = x;
        double r18567203 = eps;
        double r18567204 = r18567202 + r18567203;
        double r18567205 = sin(r18567204);
        double r18567206 = sin(r18567202);
        double r18567207 = r18567205 - r18567206;
        return r18567207;
}
↓
double f(double x, double eps) {
        double r18567208 = eps;
        double r18567209 = -8.370205872151545e-09;
        bool r18567210 = r18567208 <= r18567209;
        double r18567211 = x;
        double r18567212 = sin(r18567211);
        double r18567213 = cos(r18567208);
        double r18567214 = r18567212 * r18567213;
        double r18567215 = cos(r18567211);
        double r18567216 = sin(r18567208);
        double r18567217 = r18567215 * r18567216;
        double r18567218 = r18567214 + r18567217;
        double r18567219 = r18567218 - r18567212;
        double r18567220 = 1.1744534610942515e-09;
        bool r18567221 = r18567208 <= r18567220;
        double r18567222 = 2.0;
        double r18567223 = r18567208 / r18567222;
        double r18567224 = sin(r18567223);
        double r18567225 = r18567211 + r18567208;
        double r18567226 = r18567225 + r18567211;
        double r18567227 = r18567226 / r18567222;
        double r18567228 = cos(r18567227);
        double r18567229 = expm1(r18567228);
        double r18567230 = log1p(r18567229);
        double r18567231 = r18567224 * r18567230;
        double r18567232 = r18567222 * r18567231;
        double r18567233 = r18567221 ? r18567232 : r18567219;
        double r18567234 = r18567210 ? r18567219 : r18567233;
        return r18567234;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.6
Target15.2
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 2 regimes

  2. if eps < -8.370205872151545e-09 or 1.1744534610942515e-09 < eps

    1. Initial program 30.4

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 -8.370205872151545e-09 < eps < 1.1744534610942515e-09

    1. Initial program 45.2

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

    3. Applied diff-sin45.2

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

    4. Simplified0.3

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

    6. Applied log1p-expm1-u0.3

      \[\leadsto 2 \cdot \left(\color{blue}{\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\right)\right)\right)} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.370205872151545 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.1744534610942515 \cdot 10^{-09}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +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)))