2sin (example 3.3)

Average Error: 37.1 → 0.4

Time: 12.5s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.0651460236276763 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 3.1297217212623504 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot 2\\ \end{array}\]

C

double f(double x, double eps) {
        double r75948 = x;
        double r75949 = eps;
        double r75950 = r75948 + r75949;
        double r75951 = sin(r75950);
        double r75952 = sin(r75948);
        double r75953 = r75951 - r75952;
        return r75953;
}

↓

double f(double x, double eps) {
        double r75954 = eps;
        double r75955 = -8.065146023627676e-09;
        bool r75956 = r75954 <= r75955;
        double r75957 = 3.1297217212623504e-09;
        bool r75958 = r75954 <= r75957;
        double r75959 = !r75958;
        bool r75960 = r75956 || r75959;
        double r75961 = x;
        double r75962 = sin(r75961);
        double r75963 = cos(r75954);
        double r75964 = r75962 * r75963;
        double r75965 = cos(r75961);
        double r75966 = sin(r75954);
        double r75967 = r75965 * r75966;
        double r75968 = r75964 + r75967;
        double r75969 = r75968 - r75962;
        double r75970 = r75961 + r75954;
        double r75971 = r75970 + r75961;
        double r75972 = 2.0;
        double r75973 = r75971 / r75972;
        double r75974 = cos(r75973);
        double r75975 = r75954 / r75972;
        double r75976 = sin(r75975);
        double r75977 = r75974 * r75976;
        double r75978 = r75977 * r75972;
        double r75979 = r75960 ? r75969 : r75978;
        return r75979;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.1
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if eps < -8.065146023627676e-09 or 3.1297217212623504e-09 < eps

   1. Initial program 29.9

      \[\sin \left(x + \varepsilon\right) - \sin x\]
   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 -8.065146023627676e-09 < eps < 3.1297217212623504e-09

   1. Initial program 44.8

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

   3. Applied diff-sin 44.8

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

   4. Simplified 0.3

      \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.0651460236276763 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 3.1297217212623504 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot 2\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +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)))