2sin (example 3.3)

Average Error: 37.4 → 0.5

Time: 10.7s

Precision: 64

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

Math

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

↓

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

C

double f(double x, double eps) {
        double r139189 = x;
        double r139190 = eps;
        double r139191 = r139189 + r139190;
        double r139192 = sin(r139191);
        double r139193 = sin(r139189);
        double r139194 = r139192 - r139193;
        return r139194;
}

↓

double f(double x, double eps) {
        double r139195 = eps;
        double r139196 = -1.1161292204334507e-07;
        bool r139197 = r139195 <= r139196;
        double r139198 = 1.174838876346265e-08;
        bool r139199 = r139195 <= r139198;
        double r139200 = !r139199;
        bool r139201 = r139197 || r139200;
        double r139202 = x;
        double r139203 = sin(r139202);
        double r139204 = cos(r139195);
        double r139205 = r139203 * r139204;
        double r139206 = cos(r139202);
        double r139207 = sin(r139195);
        double r139208 = r139206 * r139207;
        double r139209 = r139205 + r139208;
        double r139210 = r139209 - r139203;
        double r139211 = r139202 + r139195;
        double r139212 = r139211 + r139202;
        double r139213 = 2.0;
        double r139214 = r139212 / r139213;
        double r139215 = cos(r139214);
        double r139216 = r139195 / r139213;
        double r139217 = sin(r139216);
        double r139218 = r139215 * r139217;
        double r139219 = r139218 * r139213;
        double r139220 = r139201 ? r139210 : r139219;
        return r139220;
}

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

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

4. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020047 +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)))