2sin (example 3.3)

Average Error: 37.4 → 0.5

Time: 13.0s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.1161292204334507 \cdot 10^{-7} \lor \neg \left(\varepsilon \le 1.174838876346265 \cdot 10^{-8}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r129951 = x;
        double r129952 = eps;
        double r129953 = r129951 + r129952;
        double r129954 = sin(r129953);
        double r129955 = sin(r129951);
        double r129956 = r129954 - r129955;
        return r129956;
}

↓

double f(double x, double eps) {
        double r129957 = eps;
        double r129958 = -1.1161292204334507e-07;
        bool r129959 = r129957 <= r129958;
        double r129960 = 1.174838876346265e-08;
        bool r129961 = r129957 <= r129960;
        double r129962 = !r129961;
        bool r129963 = r129959 || r129962;
        double r129964 = x;
        double r129965 = sin(r129964);
        double r129966 = cos(r129957);
        double r129967 = r129965 * r129966;
        double r129968 = cos(r129964);
        double r129969 = sin(r129957);
        double r129970 = r129968 * r129969;
        double r129971 = r129967 + r129970;
        double r129972 = r129971 - r129965;
        double r129973 = 2.0;
        double r129974 = r129957 / r129973;
        double r129975 = sin(r129974);
        double r129976 = r129964 + r129957;
        double r129977 = r129976 + r129964;
        double r129978 = r129977 / r129973;
        double r129979 = cos(r129978);
        double r129980 = r129975 * r129979;
        double r129981 = r129973 * r129980;
        double r129982 = r129963 ? r129972 : r129981;
        return r129982;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.4
Target	15.1
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 < -1.1161292204334507e-07 or 1.174838876346265e-08 < eps

   1. Initial program 30.0

      \[\sin \left(x + \varepsilon\right) - \sin x\]
   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 -1.1161292204334507e-07 < eps < 1.174838876346265e-08

   1. Initial program 45.1

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

   3. Applied diff-sin 45.1

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

   4. Simplified 0.4

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.1161292204334507 \cdot 10^{-7} \lor \neg \left(\varepsilon \le 1.174838876346265 \cdot 10^{-8}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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)))