2cos (problem 3.3.5)

Average Error: 39.4 → 0.7

Time: 18.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.347365952010613 \cdot 10^{-07}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 2.2247927857601395 \cdot 10^{-06}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

C

double f(double x, double eps) {
        double r612671 = x;
        double r612672 = eps;
        double r612673 = r612671 + r612672;
        double r612674 = cos(r612673);
        double r612675 = cos(r612671);
        double r612676 = r612674 - r612675;
        return r612676;
}

↓

double f(double x, double eps) {
        double r612677 = eps;
        double r612678 = -1.347365952010613e-07;
        bool r612679 = r612677 <= r612678;
        double r612680 = x;
        double r612681 = cos(r612680);
        double r612682 = cos(r612677);
        double r612683 = r612681 * r612682;
        double r612684 = sin(r612680);
        double r612685 = sin(r612677);
        double r612686 = r612684 * r612685;
        double r612687 = r612683 - r612686;
        double r612688 = r612687 - r612681;
        double r612689 = 2.2247927857601395e-06;
        bool r612690 = r612677 <= r612689;
        double r612691 = -2.0;
        double r612692 = 2.0;
        double r612693 = fma(r612692, r612680, r612677);
        double r612694 = r612693 / r612692;
        double r612695 = sin(r612694);
        double r612696 = r612677 / r612692;
        double r612697 = sin(r612696);
        double r612698 = r612695 * r612697;
        double r612699 = r612691 * r612698;
        double r612700 = r612690 ? r612699 : r612688;
        double r612701 = r612679 ? r612688 : r612700;
        return r612701;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 2 regimes

2. if eps < -1.347365952010613e-07 or 2.2247927857601395e-06 < eps

   1. Initial program 30.3

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

   3. Applied cos-sum 1.0

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

   if -1.347365952010613e-07 < eps < 2.2247927857601395e-06

   1. Initial program 48.9

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

   3. Applied diff-cos 38.1

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

   4. Simplified 0.4

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.347365952010613 \cdot 10^{-07}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 2.2247927857601395 \cdot 10^{-06}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))