2sin (example 3.3)

Average Error: 37.1 → 0.4

Time: 22.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.778095247816633 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.9528216764299977 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

C

double f(double x, double eps) {
        double r3365678 = x;
        double r3365679 = eps;
        double r3365680 = r3365678 + r3365679;
        double r3365681 = sin(r3365680);
        double r3365682 = sin(r3365678);
        double r3365683 = r3365681 - r3365682;
        return r3365683;
}

↓

double f(double x, double eps) {
        double r3365684 = eps;
        double r3365685 = -8.778095247816633e-09;
        bool r3365686 = r3365684 <= r3365685;
        double r3365687 = x;
        double r3365688 = sin(r3365687);
        double r3365689 = cos(r3365684);
        double r3365690 = r3365688 * r3365689;
        double r3365691 = cos(r3365687);
        double r3365692 = sin(r3365684);
        double r3365693 = r3365691 * r3365692;
        double r3365694 = r3365690 + r3365693;
        double r3365695 = r3365694 - r3365688;
        double r3365696 = 1.9528216764299977e-08;
        bool r3365697 = r3365684 <= r3365696;
        double r3365698 = 2.0;
        double r3365699 = r3365684 / r3365698;
        double r3365700 = sin(r3365699);
        double r3365701 = r3365687 + r3365684;
        double r3365702 = r3365687 + r3365701;
        double r3365703 = r3365702 / r3365698;
        double r3365704 = cos(r3365703);
        double r3365705 = r3365700 * r3365704;
        double r3365706 = r3365698 * r3365705;
        double r3365707 = r3365697 ? r3365706 : r3365695;
        double r3365708 = r3365686 ? r3365695 : r3365707;
        return r3365708;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.4
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 < -8.778095247816633e-09 or 1.9528216764299977e-08 < eps

   1. Initial program 30.1

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

   3. Applied sin-sum 0.5

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

   if -8.778095247816633e-09 < eps < 1.9528216764299977e-08

   1. Initial program 44.6

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

   3. Applied diff-sin 44.6

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

   4. Simplified 0.3

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.778095247816633 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.9528216764299977 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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