2cos (problem 3.3.5)

Average Error: 39.1 → 0.9

Time: 22.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -3766637.3963067145:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 2.0821597107803942 \cdot 10^{-05}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

C

double f(double x, double eps) {
        double r1500408 = x;
        double r1500409 = eps;
        double r1500410 = r1500408 + r1500409;
        double r1500411 = cos(r1500410);
        double r1500412 = cos(r1500408);
        double r1500413 = r1500411 - r1500412;
        return r1500413;
}

↓

double f(double x, double eps) {
        double r1500414 = eps;
        double r1500415 = -3766637.3963067145;
        bool r1500416 = r1500414 <= r1500415;
        double r1500417 = x;
        double r1500418 = cos(r1500417);
        double r1500419 = cos(r1500414);
        double r1500420 = r1500418 * r1500419;
        double r1500421 = sin(r1500417);
        double r1500422 = sin(r1500414);
        double r1500423 = r1500421 * r1500422;
        double r1500424 = r1500420 - r1500423;
        double r1500425 = r1500424 - r1500418;
        double r1500426 = 2.0821597107803942e-05;
        bool r1500427 = r1500414 <= r1500426;
        double r1500428 = -2.0;
        double r1500429 = 0.5;
        double r1500430 = r1500429 * r1500414;
        double r1500431 = sin(r1500430);
        double r1500432 = 2.0;
        double r1500433 = fma(r1500432, r1500417, r1500414);
        double r1500434 = r1500433 / r1500432;
        double r1500435 = sin(r1500434);
        double r1500436 = r1500431 * r1500435;
        double r1500437 = r1500428 * r1500436;
        double r1500438 = r1500427 ? r1500437 : r1500425;
        double r1500439 = r1500416 ? r1500425 : r1500438;
        return r1500439;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 2 regimes

2. if eps < -3766637.3963067145 or 2.0821597107803942e-05 < eps

   1. Initial program 29.6

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

   3. Applied cos-sum 0.9

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

   if -3766637.3963067145 < eps < 2.0821597107803942e-05

   1. Initial program 48.7

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

   3. Applied diff-cos 37.7

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

   4. Simplified 1.0

      \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3766637.3963067145:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 2.0821597107803942 \cdot 10^{-05}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))