Average Error: 36.5 → 0.6
Time: 28.9s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.0006966600469417059:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 8.159405127878992 \cdot 10^{-23}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -0.0006966600469417059:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 8.159405127878992 \cdot 10^{-23}:\\
\;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right) \cdot 2\\

\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 r2523047 = x;
        double r2523048 = eps;
        double r2523049 = r2523047 + r2523048;
        double r2523050 = sin(r2523049);
        double r2523051 = sin(r2523047);
        double r2523052 = r2523050 - r2523051;
        return r2523052;
}
↓
double f(double x, double eps) {
        double r2523053 = eps;
        double r2523054 = -0.0006966600469417059;
        bool r2523055 = r2523053 <= r2523054;
        double r2523056 = x;
        double r2523057 = sin(r2523056);
        double r2523058 = cos(r2523053);
        double r2523059 = r2523057 * r2523058;
        double r2523060 = cos(r2523056);
        double r2523061 = sin(r2523053);
        double r2523062 = r2523060 * r2523061;
        double r2523063 = r2523059 + r2523062;
        double r2523064 = r2523063 - r2523057;
        double r2523065 = 8.159405127878992e-23;
        bool r2523066 = r2523053 <= r2523065;
        double r2523067 = 2.0;
        double r2523068 = r2523053 / r2523067;
        double r2523069 = sin(r2523068);
        double r2523070 = fma(r2523067, r2523056, r2523053);
        double r2523071 = r2523070 / r2523067;
        double r2523072 = cos(r2523071);
        double r2523073 = r2523069 * r2523072;
        double r2523074 = r2523073 * r2523067;
        double r2523075 = r2523062 - r2523057;
        double r2523076 = r2523075 + r2523059;
        double r2523077 = r2523066 ? r2523074 : r2523076;
        double r2523078 = r2523055 ? r2523064 : r2523077;
        return r2523078;
}

Error

Bits error versus x

Bits error versus eps

Target

Original36.5
Target14.8
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 3 regimes

  2. if eps < -0.0006966600469417059

    1. Initial program 28.8

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 -0.0006966600469417059 < eps < 8.159405127878992e-23

    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. Simplified0.3

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

    if 8.159405127878992e-23 < eps

    1. Initial program 29.6

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

    3. Applied sin-sum1.4

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

    4. Applied associate--l+1.4

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.0006966600469417059:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 8.159405127878992 \cdot 10^{-23}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 +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)))