Trigonometry A

Average Error: 0.1 → 0.1

Time: 38.4s

Precision: 64

\[0.0 \le e \le 1\]

Math

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

↓

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

C

double f(double e, double v) {
        double r25274 = e;
        double r25275 = v;
        double r25276 = sin(r25275);
        double r25277 = r25274 * r25276;
        double r25278 = 1.0;
        double r25279 = cos(r25275);
        double r25280 = r25274 * r25279;
        double r25281 = r25278 + r25280;
        double r25282 = r25277 / r25281;
        return r25282;
}

↓

double f(double e, double v) {
        double r25283 = e;
        double r25284 = v;
        double r25285 = sin(r25284);
        double r25286 = r25283 * r25285;
        double r25287 = cos(r25284);
        double r25288 = 1.0;
        double r25289 = fma(r25287, r25283, r25288);
        double r25290 = r25286 / r25289;
        double r25291 = log1p(r25290);
        double r25292 = expm1(r25291);
        return r25292;
}

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. Using strategy rm

3. Applied expm1-log1p-u 0.1

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

4. Simplified 0.1

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

5. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))