Average Error: 36.6 → 0.6
Time: 23.3s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.307134048656964 \cdot 10^{-05}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.2025547292869671 \cdot 10^{-20}:\\ \;\;\;\;2 \cdot (e^{\log_* (1 + \sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right))} - 1)^*\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -2.307134048656964 \cdot 10^{-05}:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 1.2025547292869671 \cdot 10^{-20}:\\
\;\;\;\;2 \cdot (e^{\log_* (1 + \sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right))} - 1)^*\\

\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 r14162456 = x;
        double r14162457 = eps;
        double r14162458 = r14162456 + r14162457;
        double r14162459 = sin(r14162458);
        double r14162460 = sin(r14162456);
        double r14162461 = r14162459 - r14162460;
        return r14162461;
}
↓
double f(double x, double eps) {
        double r14162462 = eps;
        double r14162463 = -2.307134048656964e-05;
        bool r14162464 = r14162462 <= r14162463;
        double r14162465 = x;
        double r14162466 = sin(r14162465);
        double r14162467 = cos(r14162462);
        double r14162468 = r14162466 * r14162467;
        double r14162469 = cos(r14162465);
        double r14162470 = sin(r14162462);
        double r14162471 = r14162469 * r14162470;
        double r14162472 = r14162468 + r14162471;
        double r14162473 = r14162472 - r14162466;
        double r14162474 = 1.2025547292869671e-20;
        bool r14162475 = r14162462 <= r14162474;
        double r14162476 = 2.0;
        double r14162477 = r14162462 / r14162476;
        double r14162478 = sin(r14162477);
        double r14162479 = r14162465 + r14162462;
        double r14162480 = r14162479 + r14162465;
        double r14162481 = r14162480 / r14162476;
        double r14162482 = cos(r14162481);
        double r14162483 = r14162478 * r14162482;
        double r14162484 = log1p(r14162483);
        double r14162485 = expm1(r14162484);
        double r14162486 = r14162476 * r14162485;
        double r14162487 = r14162475 ? r14162486 : r14162473;
        double r14162488 = r14162464 ? r14162473 : r14162487;
        return r14162488;
}

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
Target15.1
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 < -2.307134048656964e-05 or 1.2025547292869671e-20 < eps

    1. Initial program 29.4

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

    3. Applied sin-sum0.8

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

    if -2.307134048656964e-05 < eps < 1.2025547292869671e-20

    1. Initial program 44.5

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

    3. Applied diff-sin44.5

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

    4. Simplified0.3

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

    6. Applied expm1-log1p-u0.3

      \[\leadsto 2 \cdot \color{blue}{(e^{\log_* (1 + \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right))} - 1)^*}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.307134048656964 \cdot 10^{-05}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.2025547292869671 \cdot 10^{-20}:\\ \;\;\;\;2 \cdot (e^{\log_* (1 + \sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right))} - 1)^*\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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