Average Error: 36.8 → 0.5
Time: 17.3s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.754166658325086627164164643222221684482 \cdot 10^{-5} \lor \neg \left(\varepsilon \le 5.156835808069300787421688310116496545277 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)\right)\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -2.754166658325086627164164643222221684482 \cdot 10^{-5} \lor \neg \left(\varepsilon \le 5.156835808069300787421688310116496545277 \cdot 10^{-9}\right):\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{else}:\\
\;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)\right)\\

\end{array}
double f(double x, double eps) {
        double r95388 = x;
        double r95389 = eps;
        double r95390 = r95388 + r95389;
        double r95391 = sin(r95390);
        double r95392 = sin(r95388);
        double r95393 = r95391 - r95392;
        return r95393;
}
↓
double f(double x, double eps) {
        double r95394 = eps;
        double r95395 = -2.7541666583250866e-05;
        bool r95396 = r95394 <= r95395;
        double r95397 = 5.156835808069301e-09;
        bool r95398 = r95394 <= r95397;
        double r95399 = !r95398;
        bool r95400 = r95396 || r95399;
        double r95401 = x;
        double r95402 = sin(r95401);
        double r95403 = cos(r95394);
        double r95404 = r95402 * r95403;
        double r95405 = cos(r95401);
        double r95406 = sin(r95394);
        double r95407 = r95405 * r95406;
        double r95408 = r95404 + r95407;
        double r95409 = r95408 - r95402;
        double r95410 = 2.0;
        double r95411 = r95394 / r95410;
        double r95412 = sin(r95411);
        double r95413 = r95401 + r95394;
        double r95414 = r95413 + r95401;
        double r95415 = r95414 / r95410;
        double r95416 = cos(r95415);
        double r95417 = log1p(r95416);
        double r95418 = expm1(r95417);
        double r95419 = r95412 * r95418;
        double r95420 = r95410 * r95419;
        double r95421 = r95400 ? r95409 : r95420;
        return r95421;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.8
Target14.8
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 < -2.7541666583250866e-05 or 5.156835808069301e-09 < eps

    1. Initial program 29.0

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 -2.7541666583250866e-05 < eps < 5.156835808069301e-09

    1. Initial program 45.0

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

    3. Applied diff-sin45.0

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

    4. Simplified0.4

      \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
    5. Using strategy rm

    6. Applied expm1-log1p-u0.4

      \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.754166658325086627164164643222221684482 \cdot 10^{-5} \lor \neg \left(\varepsilon \le 5.156835808069300787421688310116496545277 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(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)))