2sin (example 3.3)

Average Error: 37.4 → 0.9

Time: 25.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.01398185726995361012614527140840436914004 \lor \neg \left(\varepsilon \le 5.173054498508759236112133194996942845024 \cdot 10^{-30}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r112303 = x;
        double r112304 = eps;
        double r112305 = r112303 + r112304;
        double r112306 = sin(r112305);
        double r112307 = sin(r112303);
        double r112308 = r112306 - r112307;
        return r112308;
}

↓

double f(double x, double eps) {
        double r112309 = eps;
        double r112310 = -0.01398185726995361;
        bool r112311 = r112309 <= r112310;
        double r112312 = 5.173054498508759e-30;
        bool r112313 = r112309 <= r112312;
        double r112314 = !r112313;
        bool r112315 = r112311 || r112314;
        double r112316 = x;
        double r112317 = sin(r112316);
        double r112318 = cos(r112309);
        double r112319 = r112317 * r112318;
        double r112320 = cos(r112316);
        double r112321 = sin(r112309);
        double r112322 = r112320 * r112321;
        double r112323 = r112319 + r112322;
        double r112324 = r112323 - r112317;
        double r112325 = 2.0;
        double r112326 = r112309 / r112325;
        double r112327 = sin(r112326);
        double r112328 = fma(r112325, r112316, r112309);
        double r112329 = r112328 / r112325;
        double r112330 = cos(r112329);
        double r112331 = r112327 * r112330;
        double r112332 = r112325 * r112331;
        double r112333 = r112315 ? r112324 : r112332;
        return r112333;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.4
Target	15.3
Herbie	0.9

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

Derivation

1. Split input into 2 regimes

2. if eps < -0.01398185726995361 or 5.173054498508759e-30 < eps

   1. Initial program 30.3

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

   3. Applied sin-sum 1.4

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

   if -0.01398185726995361 < eps < 5.173054498508759e-30

   1. Initial program 45.4

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

   3. Applied diff-sin 45.4

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

   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 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.01398185726995361012614527140840436914004 \lor \neg \left(\varepsilon \le 5.173054498508759236112133194996942845024 \cdot 10^{-30}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +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)))