2sin (example 3.3)

Average Error: 37.0 → 0.5

Time: 19.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.082748610758504 \cdot 10^{-07}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 2.4662243898819633 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

C

double f(double x, double eps) {
        double r6644284 = x;
        double r6644285 = eps;
        double r6644286 = r6644284 + r6644285;
        double r6644287 = sin(r6644286);
        double r6644288 = sin(r6644284);
        double r6644289 = r6644287 - r6644288;
        return r6644289;
}

↓

double f(double x, double eps) {
        double r6644290 = eps;
        double r6644291 = -2.082748610758504e-07;
        bool r6644292 = r6644290 <= r6644291;
        double r6644293 = x;
        double r6644294 = sin(r6644293);
        double r6644295 = cos(r6644290);
        double r6644296 = r6644294 * r6644295;
        double r6644297 = cos(r6644293);
        double r6644298 = sin(r6644290);
        double r6644299 = r6644297 * r6644298;
        double r6644300 = r6644296 + r6644299;
        double r6644301 = r6644300 - r6644294;
        double r6644302 = 2.4662243898819633e-08;
        bool r6644303 = r6644290 <= r6644302;
        double r6644304 = 2.0;
        double r6644305 = r6644290 / r6644304;
        double r6644306 = sin(r6644305);
        double r6644307 = r6644293 + r6644290;
        double r6644308 = r6644307 + r6644293;
        double r6644309 = r6644308 / r6644304;
        double r6644310 = cos(r6644309);
        double r6644311 = r6644306 * r6644310;
        double r6644312 = r6644304 * r6644311;
        double r6644313 = r6644303 ? r6644312 : r6644301;
        double r6644314 = r6644292 ? r6644301 : r6644313;
        return r6644314;
}

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 2 regimes

2. if eps < -2.082748610758504e-07 or 2.4662243898819633e-08 < eps

   1. Initial program 29.8

      \[\sin \left(x + \varepsilon\right) - \sin x\]
   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 -2.082748610758504e-07 < eps < 2.4662243898819633e-08

   1. Initial program 44.7

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

   3. Applied diff-sin 44.7

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

   4. Simplified 0.4

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.082748610758504 \cdot 10^{-07}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 2.4662243898819633 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

  (- (sin (+ x eps)) (sin x)))