2sin (example 3.3)

Average Error: 36.6 → 0.6

Time: 19.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.307134048656964 \cdot 10^{-05}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.2025547292869671 \cdot 10^{-20}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

C

double f(double x, double eps) {
        double r7350246 = x;
        double r7350247 = eps;
        double r7350248 = r7350246 + r7350247;
        double r7350249 = sin(r7350248);
        double r7350250 = sin(r7350246);
        double r7350251 = r7350249 - r7350250;
        return r7350251;
}

↓

double f(double x, double eps) {
        double r7350252 = eps;
        double r7350253 = -2.307134048656964e-05;
        bool r7350254 = r7350252 <= r7350253;
        double r7350255 = x;
        double r7350256 = sin(r7350255);
        double r7350257 = cos(r7350252);
        double r7350258 = r7350256 * r7350257;
        double r7350259 = cos(r7350255);
        double r7350260 = sin(r7350252);
        double r7350261 = r7350259 * r7350260;
        double r7350262 = r7350258 + r7350261;
        double r7350263 = r7350262 - r7350256;
        double r7350264 = 1.2025547292869671e-20;
        bool r7350265 = r7350252 <= r7350264;
        double r7350266 = 2.0;
        double r7350267 = r7350252 / r7350266;
        double r7350268 = sin(r7350267);
        double r7350269 = r7350255 + r7350252;
        double r7350270 = r7350269 + r7350255;
        double r7350271 = r7350270 / r7350266;
        double r7350272 = cos(r7350271);
        double r7350273 = r7350268 * r7350272;
        double r7350274 = r7350266 * r7350273;
        double r7350275 = r7350265 ? r7350274 : r7350263;
        double r7350276 = r7350254 ? r7350263 : r7350275;
        return r7350276;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.6
Target	15.1
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if eps < -2.307134048656964e-05 or 1.2025547292869671e-20 < eps

   1. Initial program 29.4

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

   3. Applied sin-sum 0.8

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

   if -2.307134048656964e-05 < eps < 1.2025547292869671e-20

   1. Initial program 44.5

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

   3. Applied diff-sin 44.5

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

   4. Simplified 0.3

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.307134048656964 \cdot 10^{-05}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.2025547292869671 \cdot 10^{-20}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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