2sin (example 3.3)

Average Error: 37.0 → 0.5

Time: 14.0s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -6.6823151206119754 \cdot 10^{-9}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 7.4364775183871708 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

C

double f(double x, double eps) {
        double r115188 = x;
        double r115189 = eps;
        double r115190 = r115188 + r115189;
        double r115191 = sin(r115190);
        double r115192 = sin(r115188);
        double r115193 = r115191 - r115192;
        return r115193;
}

↓

double f(double x, double eps) {
        double r115194 = eps;
        double r115195 = -6.6823151206119754e-09;
        bool r115196 = r115194 <= r115195;
        double r115197 = x;
        double r115198 = sin(r115197);
        double r115199 = cos(r115194);
        double r115200 = r115198 * r115199;
        double r115201 = cos(r115197);
        double r115202 = sin(r115194);
        double r115203 = r115201 * r115202;
        double r115204 = r115203 - r115198;
        double r115205 = r115200 + r115204;
        double r115206 = 7.436477518387171e-09;
        bool r115207 = r115194 <= r115206;
        double r115208 = 2.0;
        double r115209 = r115197 + r115194;
        double r115210 = r115209 + r115197;
        double r115211 = r115210 / r115208;
        double r115212 = cos(r115211);
        double r115213 = r115194 / r115208;
        double r115214 = sin(r115213);
        double r115215 = r115212 * r115214;
        double r115216 = r115208 * r115215;
        double r115217 = r115200 + r115203;
        double r115218 = r115217 - r115198;
        double r115219 = r115207 ? r115216 : r115218;
        double r115220 = r115196 ? r115205 : r115219;
        return r115220;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.0
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 3 regimes

2. if eps < -6.6823151206119754e-09

   1. Initial program 30.1

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

   3. Applied sin-sum 0.6

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

   4. Applied associate--l+ 0.7

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

   if -6.6823151206119754e-09 < eps < 7.436477518387171e-09

   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

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

   if 7.436477518387171e-09 < eps

   1. Initial program 29.2

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

   3. Applied sin-sum 0.6

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6.6823151206119754 \cdot 10^{-9}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 7.4364775183871708 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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)))