Average Error: 36.6 → 0.6
Time: 19.5s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.894373646089193 \cdot 10^{-09}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \le 2.1490204748102395 \cdot 10^{-17}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -7.894373646089193 \cdot 10^{-09}:\\
\;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\

\mathbf{elif}\;\varepsilon \le 2.1490204748102395 \cdot 10^{-17}:\\
\;\;\;\;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 r6956514 = x;
        double r6956515 = eps;
        double r6956516 = r6956514 + r6956515;
        double r6956517 = sin(r6956516);
        double r6956518 = sin(r6956514);
        double r6956519 = r6956517 - r6956518;
        return r6956519;
}
↓
double f(double x, double eps) {
        double r6956520 = eps;
        double r6956521 = -7.894373646089193e-09;
        bool r6956522 = r6956520 <= r6956521;
        double r6956523 = x;
        double r6956524 = cos(r6956523);
        double r6956525 = sin(r6956520);
        double r6956526 = r6956524 * r6956525;
        double r6956527 = sin(r6956523);
        double r6956528 = r6956526 - r6956527;
        double r6956529 = cos(r6956520);
        double r6956530 = r6956527 * r6956529;
        double r6956531 = r6956528 + r6956530;
        double r6956532 = 2.1490204748102395e-17;
        bool r6956533 = r6956520 <= r6956532;
        double r6956534 = 2.0;
        double r6956535 = r6956520 / r6956534;
        double r6956536 = sin(r6956535);
        double r6956537 = r6956523 + r6956520;
        double r6956538 = r6956537 + r6956523;
        double r6956539 = r6956538 / r6956534;
        double r6956540 = cos(r6956539);
        double r6956541 = r6956536 * r6956540;
        double r6956542 = r6956534 * r6956541;
        double r6956543 = r6956533 ? r6956542 : r6956531;
        double r6956544 = r6956522 ? r6956531 : r6956543;
        return r6956544;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.6
Target15.3
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 < -7.894373646089193e-09 or 2.1490204748102395e-17 < eps

    1. Initial program 29.8

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

    3. Applied sin-sum0.8

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

    4. Applied associate--l+0.8

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

    if -7.894373646089193e-09 < eps < 2.1490204748102395e-17

    1. Initial program 44.3

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

    3. Applied diff-sin44.3

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

    4. Simplified0.2

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -7.894373646089193 \cdot 10^{-09}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \le 2.1490204748102395 \cdot 10^{-17}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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