2cos (problem 3.3.5)

Average Error: 39.7 → 0.7

Time: 12.2s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -6.415376531290239211363646276709005178418 \cdot 10^{-4} \lor \neg \left(\varepsilon \le 4.575180546003463192540302358762849266327 \cdot 10^{-7}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r36088 = x;
        double r36089 = eps;
        double r36090 = r36088 + r36089;
        double r36091 = cos(r36090);
        double r36092 = cos(r36088);
        double r36093 = r36091 - r36092;
        return r36093;
}

↓

double f(double x, double eps) {
        double r36094 = eps;
        double r36095 = -0.0006415376531290239;
        bool r36096 = r36094 <= r36095;
        double r36097 = 4.575180546003463e-07;
        bool r36098 = r36094 <= r36097;
        double r36099 = !r36098;
        bool r36100 = r36096 || r36099;
        double r36101 = x;
        double r36102 = cos(r36101);
        double r36103 = cos(r36094);
        double r36104 = r36102 * r36103;
        double r36105 = sin(r36101);
        double r36106 = sin(r36094);
        double r36107 = r36105 * r36106;
        double r36108 = r36104 - r36107;
        double r36109 = r36108 - r36102;
        double r36110 = -2.0;
        double r36111 = 2.0;
        double r36112 = r36094 / r36111;
        double r36113 = sin(r36112);
        double r36114 = r36110 * r36113;
        double r36115 = r36101 + r36094;
        double r36116 = r36115 + r36101;
        double r36117 = r36116 / r36111;
        double r36118 = sin(r36117);
        double r36119 = r36114 * r36118;
        double r36120 = r36100 ? r36109 : r36119;
        return r36120;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 2 regimes

2. if eps < -0.0006415376531290239 or 4.575180546003463e-07 < eps

   1. Initial program 30.6

      \[\cos \left(x + \varepsilon\right) - \cos x\]
   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 -0.0006415376531290239 < eps < 4.575180546003463e-07

   1. Initial program 48.9

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

   3. Applied diff-cos 36.8

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

   4. Simplified 0.4

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

   6. Applied associate-*r* 0.4

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6.415376531290239211363646276709005178418 \cdot 10^{-4} \lor \neg \left(\varepsilon \le 4.575180546003463192540302358762849266327 \cdot 10^{-7}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))