2sin (example 3.3)

Average Error: 36.7 → 0.5

Time: 15.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.22306452923435388:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 6.765190492196658 \cdot 10^{-13}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

C

double f(double x, double eps) {
        double r1485908 = x;
        double r1485909 = eps;
        double r1485910 = r1485908 + r1485909;
        double r1485911 = sin(r1485910);
        double r1485912 = sin(r1485908);
        double r1485913 = r1485911 - r1485912;
        return r1485913;
}

↓

double f(double x, double eps) {
        double r1485914 = eps;
        double r1485915 = -0.22306452923435388;
        bool r1485916 = r1485914 <= r1485915;
        double r1485917 = x;
        double r1485918 = sin(r1485917);
        double r1485919 = cos(r1485914);
        double r1485920 = r1485918 * r1485919;
        double r1485921 = cos(r1485917);
        double r1485922 = sin(r1485914);
        double r1485923 = r1485921 * r1485922;
        double r1485924 = r1485920 + r1485923;
        double r1485925 = r1485924 - r1485918;
        double r1485926 = 6.765190492196658e-13;
        bool r1485927 = r1485914 <= r1485926;
        double r1485928 = 2.0;
        double r1485929 = r1485914 / r1485928;
        double r1485930 = sin(r1485929);
        double r1485931 = fma(r1485928, r1485917, r1485914);
        double r1485932 = r1485931 / r1485928;
        double r1485933 = cos(r1485932);
        double r1485934 = r1485930 * r1485933;
        double r1485935 = r1485934 * r1485928;
        double r1485936 = r1485927 ? r1485935 : r1485925;
        double r1485937 = r1485916 ? r1485925 : r1485936;
        return r1485937;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.7
Target	14.5
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 < -0.22306452923435388 or 6.765190492196658e-13 < eps

   1. Initial program 29.3

      \[\sin \left(x + \varepsilon\right) - \sin x\]
   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 -0.22306452923435388 < eps < 6.765190492196658e-13

   1. Initial program 44.2

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

   3. Applied diff-sin 44.2

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

   4. Simplified 0.4

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.22306452923435388:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 6.765190492196658 \cdot 10^{-13}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +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)))