Average Error: 36.9 → 0.5
Time: 17.0s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.576402784825059484581493698344273335366 \cdot 10^{-9}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 8.790268235921677084067261552426802984073 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -7.576402784825059484581493698344273335366 \cdot 10^{-9}:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 8.790268235921677084067261552426802984073 \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 r6482410 = x;
        double r6482411 = eps;
        double r6482412 = r6482410 + r6482411;
        double r6482413 = sin(r6482412);
        double r6482414 = sin(r6482410);
        double r6482415 = r6482413 - r6482414;
        return r6482415;
}
↓
double f(double x, double eps) {
        double r6482416 = eps;
        double r6482417 = -7.57640278482506e-09;
        bool r6482418 = r6482416 <= r6482417;
        double r6482419 = x;
        double r6482420 = sin(r6482419);
        double r6482421 = cos(r6482416);
        double r6482422 = r6482420 * r6482421;
        double r6482423 = cos(r6482419);
        double r6482424 = sin(r6482416);
        double r6482425 = r6482423 * r6482424;
        double r6482426 = r6482422 + r6482425;
        double r6482427 = r6482426 - r6482420;
        double r6482428 = 8.790268235921677e-09;
        bool r6482429 = r6482416 <= r6482428;
        double r6482430 = 2.0;
        double r6482431 = r6482416 / r6482430;
        double r6482432 = sin(r6482431);
        double r6482433 = r6482419 + r6482416;
        double r6482434 = r6482433 + r6482419;
        double r6482435 = r6482434 / r6482430;
        double r6482436 = cos(r6482435);
        double r6482437 = r6482432 * r6482436;
        double r6482438 = r6482430 * r6482437;
        double r6482439 = r6482425 - r6482420;
        double r6482440 = r6482439 + r6482422;
        double r6482441 = r6482429 ? r6482438 : r6482440;
        double r6482442 = r6482418 ? r6482427 : r6482441;
        return r6482442;
}

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
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -7.57640278482506e-09

    1. Initial program 30.0

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

    3. Applied sin-sum0.5

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

    if -7.57640278482506e-09 < eps < 8.790268235921677e-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. Simplified0.3

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

    if 8.790268235921677e-09 < eps

    1. Initial program 30.1

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

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -7.576402784825059484581493698344273335366 \cdot 10^{-9}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 8.790268235921677084067261552426802984073 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(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)))