Average Error: 36.9 → 0.6
Time: 16.0s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -6443.306290177518349082674831151962280273:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \le 5.463141894552383288993626720472102098469 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -6443.306290177518349082674831151962280273:\\
\;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\

\mathbf{elif}\;\varepsilon \le 5.463141894552383288993626720472102098469 \cdot 10^{-9}:\\
\;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\

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

\end{array}
double f(double x, double eps) {
        double r5499287 = x;
        double r5499288 = eps;
        double r5499289 = r5499287 + r5499288;
        double r5499290 = sin(r5499289);
        double r5499291 = sin(r5499287);
        double r5499292 = r5499290 - r5499291;
        return r5499292;
}
↓
double f(double x, double eps) {
        double r5499293 = eps;
        double r5499294 = -6443.306290177518;
        bool r5499295 = r5499293 <= r5499294;
        double r5499296 = x;
        double r5499297 = cos(r5499296);
        double r5499298 = sin(r5499293);
        double r5499299 = r5499297 * r5499298;
        double r5499300 = sin(r5499296);
        double r5499301 = r5499299 - r5499300;
        double r5499302 = cos(r5499293);
        double r5499303 = r5499300 * r5499302;
        double r5499304 = r5499301 + r5499303;
        double r5499305 = 5.463141894552383e-09;
        bool r5499306 = r5499293 <= r5499305;
        double r5499307 = 2.0;
        double r5499308 = r5499293 / r5499307;
        double r5499309 = sin(r5499308);
        double r5499310 = r5499296 + r5499293;
        double r5499311 = r5499310 + r5499296;
        double r5499312 = r5499311 / r5499307;
        double r5499313 = cos(r5499312);
        double r5499314 = r5499309 * r5499313;
        double r5499315 = r5499307 * r5499314;
        double r5499316 = r5499306 ? r5499315 : r5499304;
        double r5499317 = r5499295 ? r5499304 : r5499316;
        return r5499317;
}

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
Target15.1
Herbie0.6
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -6443.306290177518 or 5.463141894552383e-09 < eps

    1. Initial program 29.9

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

    3. Applied sin-sum0.5

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

    4. Applied associate--l+0.5

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

    if -6443.306290177518 < eps < 5.463141894552383e-09

    1. Initial program 44.2

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

    3. Applied diff-sin44.2

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

    4. Simplified0.7

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6443.306290177518349082674831151962280273:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \le 5.463141894552383288993626720472102098469 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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)))