Trigonometry A

Average Error: 0.1 → 0.2

Time: 16.2s

Precision: 64

\[0 \le e \le 1\]

Math

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

↓

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

C

double f(double e, double v) {
        double r927393 = e;
        double r927394 = v;
        double r927395 = sin(r927394);
        double r927396 = r927393 * r927395;
        double r927397 = 1.0;
        double r927398 = cos(r927394);
        double r927399 = r927393 * r927398;
        double r927400 = r927397 + r927399;
        double r927401 = r927396 / r927400;
        return r927401;
}

↓

double f(double e, double v) {
        double r927402 = e;
        double r927403 = v;
        double r927404 = sin(r927403);
        double r927405 = cos(r927403);
        double r927406 = 1.0;
        double r927407 = fma(r927405, r927402, r927406);
        double r927408 = r927404 / r927407;
        double r927409 = expm1(r927408);
        double r927410 = log1p(r927409);
        double r927411 = r927402 * r927410;
        return r927411;
}

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

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

4. Applied log1p-expm1-u 0.2

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

5. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))