Average Error: 36.9 → 0.8
Time: 21.4s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.505892634593631943814391258018581254419 \cdot 10^{-9}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 8.5558943769231499470661433859635760918 \cdot 10^{-30}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -8.505892634593631943814391258018581254419 \cdot 10^{-9}:\\
\;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\

\mathbf{elif}\;\varepsilon \le 8.5558943769231499470661433859635760918 \cdot 10^{-30}:\\
\;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{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 r78357 = x;
        double r78358 = eps;
        double r78359 = r78357 + r78358;
        double r78360 = sin(r78359);
        double r78361 = sin(r78357);
        double r78362 = r78360 - r78361;
        return r78362;
}
↓
double f(double x, double eps) {
        double r78363 = eps;
        double r78364 = -8.505892634593632e-09;
        bool r78365 = r78363 <= r78364;
        double r78366 = x;
        double r78367 = sin(r78366);
        double r78368 = cos(r78363);
        double r78369 = r78367 * r78368;
        double r78370 = cos(r78366);
        double r78371 = sin(r78363);
        double r78372 = r78370 * r78371;
        double r78373 = r78372 - r78367;
        double r78374 = r78369 + r78373;
        double r78375 = 8.55589437692315e-30;
        bool r78376 = r78363 <= r78375;
        double r78377 = 2.0;
        double r78378 = r78363 / r78377;
        double r78379 = sin(r78378);
        double r78380 = r78366 + r78363;
        double r78381 = r78380 + r78366;
        double r78382 = r78381 / r78377;
        double r78383 = cos(r78382);
        double r78384 = r78379 * r78383;
        double r78385 = r78377 * r78384;
        double r78386 = r78369 + r78372;
        double r78387 = r78386 - r78367;
        double r78388 = r78376 ? r78385 : r78387;
        double r78389 = r78365 ? r78374 : r78388;
        return r78389;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.9
Target14.8
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 < -8.505892634593632e-09

    1. Initial program 28.9

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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

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

    if -8.505892634593632e-09 < eps < 8.55589437692315e-30

    1. Initial program 45.9

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

    3. Applied diff-sin45.9

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

    4. Simplified0.2

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

    if 8.55589437692315e-30 < eps

    1. Initial program 29.3

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

    3. Applied sin-sum2.0

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.505892634593631943814391258018581254419 \cdot 10^{-9}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 8.5558943769231499470661433859635760918 \cdot 10^{-30}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

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