2sin (example 3.3)

Average Error: 37.1 → 0.4

Time: 27.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.008151886891035409199446348793571814894676 \lor \neg \left(\varepsilon \le 8.837665134522452590648921417169903147482 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r97543 = x;
        double r97544 = eps;
        double r97545 = r97543 + r97544;
        double r97546 = sin(r97545);
        double r97547 = sin(r97543);
        double r97548 = r97546 - r97547;
        return r97548;
}

↓

double f(double x, double eps) {
        double r97549 = eps;
        double r97550 = -0.00815188689103541;
        bool r97551 = r97549 <= r97550;
        double r97552 = 8.837665134522453e-09;
        bool r97553 = r97549 <= r97552;
        double r97554 = !r97553;
        bool r97555 = r97551 || r97554;
        double r97556 = x;
        double r97557 = sin(r97556);
        double r97558 = cos(r97549);
        double r97559 = r97557 * r97558;
        double r97560 = cos(r97556);
        double r97561 = sin(r97549);
        double r97562 = r97560 * r97561;
        double r97563 = r97559 + r97562;
        double r97564 = r97563 - r97557;
        double r97565 = 2.0;
        double r97566 = fma(r97565, r97556, r97549);
        double r97567 = r97566 / r97565;
        double r97568 = cos(r97567);
        double r97569 = r97549 / r97565;
        double r97570 = sin(r97569);
        double r97571 = r97568 * r97570;
        double r97572 = r97565 * r97571;
        double r97573 = r97555 ? r97564 : r97572;
        return r97573;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.2
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if eps < -0.00815188689103541 or 8.837665134522453e-09 < eps

   1. Initial program 30.3

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

   3. Applied sin-sum 0.4

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

   if -0.00815188689103541 < eps < 8.837665134522453e-09

   1. Initial program 44.1

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

   3. Applied diff-sin 44.1

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

   4. Simplified 0.4

      \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)}\]
   5. Using strategy rm

   6. Applied *-commutative 0.4

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.008151886891035409199446348793571814894676 \lor \neg \left(\varepsilon \le 8.837665134522452590648921417169903147482 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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)))