Average Error: 36.8 → 0.4
Time: 26.9s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -9.760584355965868752318115308952428677003 \cdot 10^{-5}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 9.433852813731427826173936825124993754699 \cdot 10^{-9}:\\ \;\;\;\;\left(2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -9.760584355965868752318115308952428677003 \cdot 10^{-5}:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 9.433852813731427826173936825124993754699 \cdot 10^{-9}:\\
\;\;\;\;\left(2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\\

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

\end{array}
double f(double x, double eps) {
        double r99544 = x;
        double r99545 = eps;
        double r99546 = r99544 + r99545;
        double r99547 = sin(r99546);
        double r99548 = sin(r99544);
        double r99549 = r99547 - r99548;
        return r99549;
}
↓
double f(double x, double eps) {
        double r99550 = eps;
        double r99551 = -9.760584355965869e-05;
        bool r99552 = r99550 <= r99551;
        double r99553 = x;
        double r99554 = sin(r99553);
        double r99555 = cos(r99550);
        double r99556 = r99554 * r99555;
        double r99557 = cos(r99553);
        double r99558 = sin(r99550);
        double r99559 = r99557 * r99558;
        double r99560 = r99556 + r99559;
        double r99561 = r99560 - r99554;
        double r99562 = 9.433852813731428e-09;
        bool r99563 = r99550 <= r99562;
        double r99564 = 2.0;
        double r99565 = r99550 / r99564;
        double r99566 = sin(r99565);
        double r99567 = r99564 * r99566;
        double r99568 = fma(r99564, r99553, r99550);
        double r99569 = r99568 / r99564;
        double r99570 = cos(r99569);
        double r99571 = r99567 * r99570;
        double r99572 = r99559 - r99554;
        double r99573 = r99556 + r99572;
        double r99574 = r99563 ? r99571 : r99573;
        double r99575 = r99552 ? r99561 : r99574;
        return r99575;
}

Error

Bits error versus x

Bits error versus eps

Target

Original36.8
Target14.8
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -9.760584355965869e-05

    1. Initial program 30.4

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

    3. Applied sin-sum0.4

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

    if -9.760584355965869e-05 < eps < 9.433852813731428e-09

    1. Initial program 44.6

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

    3. Applied diff-sin44.6

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

    4. Simplified0.3

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

    6. Applied associate-*r*0.3

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

    if 9.433852813731428e-09 < eps

    1. Initial program 28.2

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

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -9.760584355965868752318115308952428677003 \cdot 10^{-5}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 9.433852813731427826173936825124993754699 \cdot 10^{-9}:\\ \;\;\;\;\left(2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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)))