Average Error: 37.1 → 0.5
Time: 16.9s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.256929432906464 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.136872105742304 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -7.256929432906464 \cdot 10^{-09}:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 1.136872105742304 \cdot 10^{-08}:\\
\;\;\;\;2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\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 r4532359 = x;
        double r4532360 = eps;
        double r4532361 = r4532359 + r4532360;
        double r4532362 = sin(r4532361);
        double r4532363 = sin(r4532359);
        double r4532364 = r4532362 - r4532363;
        return r4532364;
}
↓
double f(double x, double eps) {
        double r4532365 = eps;
        double r4532366 = -7.256929432906464e-09;
        bool r4532367 = r4532365 <= r4532366;
        double r4532368 = x;
        double r4532369 = sin(r4532368);
        double r4532370 = cos(r4532365);
        double r4532371 = r4532369 * r4532370;
        double r4532372 = cos(r4532368);
        double r4532373 = sin(r4532365);
        double r4532374 = r4532372 * r4532373;
        double r4532375 = r4532371 + r4532374;
        double r4532376 = r4532375 - r4532369;
        double r4532377 = 1.136872105742304e-08;
        bool r4532378 = r4532365 <= r4532377;
        double r4532379 = 2.0;
        double r4532380 = r4532368 + r4532365;
        double r4532381 = r4532368 + r4532380;
        double r4532382 = r4532381 / r4532379;
        double r4532383 = cos(r4532382);
        double r4532384 = expm1(r4532383);
        double r4532385 = log1p(r4532384);
        double r4532386 = r4532365 / r4532379;
        double r4532387 = sin(r4532386);
        double r4532388 = r4532385 * r4532387;
        double r4532389 = r4532379 * r4532388;
        double r4532390 = r4532378 ? r4532389 : r4532376;
        double r4532391 = r4532367 ? r4532376 : r4532390;
        return r4532391;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
Target15.3
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 < -7.256929432906464e-09 or 1.136872105742304e-08 < eps

    1. Initial program 30.0

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 -7.256929432906464e-09 < eps < 1.136872105742304e-08

    1. Initial program 44.8

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

    3. Applied diff-sin44.8

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

    4. Simplified0.3

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

    6. Applied log1p-expm1-u0.4

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -7.256929432906464 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.136872105742304 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 +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)))