2sin (example 3.3)

Average Error: 37.1 → 0.5

Time: 19.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -3.523695819775611137291554397776849327784 \cdot 10^{-6} \lor \neg \left(\varepsilon \le 4.06104011737916238035658440629202189931 \cdot 10^{-11}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r143968 = x;
        double r143969 = eps;
        double r143970 = r143968 + r143969;
        double r143971 = sin(r143970);
        double r143972 = sin(r143968);
        double r143973 = r143971 - r143972;
        return r143973;
}

↓

double f(double x, double eps) {
        double r143974 = eps;
        double r143975 = -3.523695819775611e-06;
        bool r143976 = r143974 <= r143975;
        double r143977 = 4.0610401173791624e-11;
        bool r143978 = r143974 <= r143977;
        double r143979 = !r143978;
        bool r143980 = r143976 || r143979;
        double r143981 = x;
        double r143982 = sin(r143981);
        double r143983 = cos(r143974);
        double r143984 = r143982 * r143983;
        double r143985 = cos(r143981);
        double r143986 = sin(r143974);
        double r143987 = r143985 * r143986;
        double r143988 = r143984 + r143987;
        double r143989 = r143988 - r143982;
        double r143990 = 2.0;
        double r143991 = r143974 / r143990;
        double r143992 = sin(r143991);
        double r143993 = fma(r143990, r143981, r143974);
        double r143994 = r143993 / r143990;
        double r143995 = cos(r143994);
        double r143996 = r143992 * r143995;
        double r143997 = r143990 * r143996;
        double r143998 = r143980 ? r143989 : r143997;
        return r143998;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	14.9
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if eps < -3.523695819775611e-06 or 4.0610401173791624e-11 < eps

   1. Initial program 29.6

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

   3. Applied sin-sum 0.6

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

   if -3.523695819775611e-06 < eps < 4.0610401173791624e-11

   1. Initial program 45.0

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

   3. Applied diff-sin 45.0

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

   4. Simplified 0.3

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.523695819775611137291554397776849327784 \cdot 10^{-6} \lor \neg \left(\varepsilon \le 4.06104011737916238035658440629202189931 \cdot 10^{-11}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +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)))