2cos (problem 3.3.5)

Average Error: 39.4 → 0.7

Time: 19.5s

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 r741510 = x;
        double r741511 = eps;
        double r741512 = r741510 + r741511;
        double r741513 = cos(r741512);
        double r741514 = cos(r741510);
        double r741515 = r741513 - r741514;
        return r741515;
}

↓

double f(double x, double eps) {
        double r741516 = eps;
        double r741517 = -1.347365952010613e-07;
        bool r741518 = r741516 <= r741517;
        double r741519 = x;
        double r741520 = cos(r741519);
        double r741521 = cos(r741516);
        double r741522 = r741520 * r741521;
        double r741523 = sin(r741519);
        double r741524 = sin(r741516);
        double r741525 = r741523 * r741524;
        double r741526 = r741522 - r741525;
        double r741527 = r741526 - r741520;
        double r741528 = 2.2247927857601395e-06;
        bool r741529 = r741516 <= r741528;
        double r741530 = -2.0;
        double r741531 = 2.0;
        double r741532 = fma(r741531, r741519, r741516);
        double r741533 = r741532 / r741531;
        double r741534 = sin(r741533);
        double r741535 = r741516 / r741531;
        double r741536 = sin(r741535);
        double r741537 = r741534 * r741536;
        double r741538 = r741530 * r741537;
        double r741539 = r741529 ? r741538 : r741527;
        double r741540 = r741518 ? r741527 : r741539;
        return r741540;
}

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)))