Trigonometry A

Average Error: 0.1 → 0.1

Time: 4.5s

Precision: 64

\[0.0 \le e \le 1\]

Math

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

↓

\[\left(e \cdot \sin v\right) \cdot \frac{1}{1 + e \cdot \cos v}\]

C

double f(double e, double v) {
        double r10414 = e;
        double r10415 = v;
        double r10416 = sin(r10415);
        double r10417 = r10414 * r10416;
        double r10418 = 1.0;
        double r10419 = cos(r10415);
        double r10420 = r10414 * r10419;
        double r10421 = r10418 + r10420;
        double r10422 = r10417 / r10421;
        return r10422;
}

↓

double f(double e, double v) {
        double r10423 = e;
        double r10424 = v;
        double r10425 = sin(r10424);
        double r10426 = r10423 * r10425;
        double r10427 = 1.0;
        double r10428 = 1.0;
        double r10429 = cos(r10424);
        double r10430 = r10423 * r10429;
        double r10431 = r10428 + r10430;
        double r10432 = r10427 / r10431;
        double r10433 = r10426 * r10432;
        return r10433;
}

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 div-inv 0.1

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

4. Final simplification 0.1

   \[\leadsto \left(e \cdot \sin v\right) \cdot \frac{1}{1 + e \cdot \cos v}\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))