Trigonometry A

Average Error: 0.1 → 0.2

Time: 18.3s

Precision: 64

\[0 \le e \le 1\]

Math

\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]

↓

\[\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\right)\right) \cdot e\]

C

double f(double e, double v) {
        double r1026441 = e;
        double r1026442 = v;
        double r1026443 = sin(r1026442);
        double r1026444 = r1026441 * r1026443;
        double r1026445 = 1.0;
        double r1026446 = cos(r1026442);
        double r1026447 = r1026441 * r1026446;
        double r1026448 = r1026445 + r1026447;
        double r1026449 = r1026444 / r1026448;
        return r1026449;
}

↓

double f(double e, double v) {
        double r1026450 = v;
        double r1026451 = sin(r1026450);
        double r1026452 = cos(r1026450);
        double r1026453 = e;
        double r1026454 = 1.0;
        double r1026455 = fma(r1026452, r1026453, r1026454);
        double r1026456 = r1026451 / r1026455;
        double r1026457 = log1p(r1026456);
        double r1026458 = expm1(r1026457);
        double r1026459 = r1026458 * r1026453;
        return r1026459;
}

Error

Bits error versus e

Bits error versus v

Derivation

1. Initial program 0.1

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]

2. Simplified 0.1

   \[\leadsto \color{blue}{\frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)} \cdot e}\]
3. Using strategy rm

4. Applied expm1-log1p-u 0.2

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\right)\right)} \cdot e\]

5. Final simplification 0.2

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\right)\right) \cdot e\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))