2sin (example 3.3)

Average Error: 36.6 → 0.4

Time: 12.9s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -6.83922365147160498477619820149701890255 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 7.263195155747593821712365094725644999729 \cdot 10^{-12}\right):\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r96495 = x;
        double r96496 = eps;
        double r96497 = r96495 + r96496;
        double r96498 = sin(r96497);
        double r96499 = sin(r96495);
        double r96500 = r96498 - r96499;
        return r96500;
}

↓

double f(double x, double eps) {
        double r96501 = eps;
        double r96502 = -6.839223651471605e-09;
        bool r96503 = r96501 <= r96502;
        double r96504 = 7.263195155747594e-12;
        bool r96505 = r96501 <= r96504;
        double r96506 = !r96505;
        bool r96507 = r96503 || r96506;
        double r96508 = x;
        double r96509 = sin(r96508);
        double r96510 = cos(r96501);
        double r96511 = r96509 * r96510;
        double r96512 = cos(r96508);
        double r96513 = sin(r96501);
        double r96514 = r96512 * r96513;
        double r96515 = r96514 - r96509;
        double r96516 = r96511 + r96515;
        double r96517 = 2.0;
        double r96518 = r96501 / r96517;
        double r96519 = sin(r96518);
        double r96520 = r96508 + r96501;
        double r96521 = r96520 + r96508;
        double r96522 = r96521 / r96517;
        double r96523 = cos(r96522);
        double r96524 = r96519 * r96523;
        double r96525 = r96517 * r96524;
        double r96526 = r96507 ? r96516 : r96525;
        return r96526;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.6
Target	14.8
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 < -6.839223651471605e-09 or 7.263195155747594e-12 < eps

   1. Initial program 29.8

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

   3. Applied sin-sum 0.6

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

   4. Applied associate--l+ 0.6

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

   if -6.839223651471605e-09 < eps < 7.263195155747594e-12

   1. Initial program 43.7

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

   3. Applied diff-sin 43.7

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

   4. Simplified 0.2

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6.83922365147160498477619820149701890255 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 7.263195155747593821712365094725644999729 \cdot 10^{-12}\right):\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(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)))