Trigonometry A

Average Error: 0.1 → 0.1

Time: 19.6s

Precision: 64

\[0.0 \le e \le 1\]

Math

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

↓

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

C

double f(double e, double v) {
        double r21817 = e;
        double r21818 = v;
        double r21819 = sin(r21818);
        double r21820 = r21817 * r21819;
        double r21821 = 1.0;
        double r21822 = cos(r21818);
        double r21823 = r21817 * r21822;
        double r21824 = r21821 + r21823;
        double r21825 = r21820 / r21824;
        return r21825;
}

↓

double f(double e, double v) {
        double r21826 = e;
        double r21827 = v;
        double r21828 = sin(r21827);
        double r21829 = r21826 * r21828;
        double r21830 = 1.0;
        double r21831 = cos(r21827);
        double r21832 = 1.0;
        double r21833 = fma(r21831, r21826, r21832);
        double r21834 = r21830 / r21833;
        double r21835 = r21829 * r21834;
        return r21835;
}

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. Simplified 0.1

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

5. Final simplification 0.1

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

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)))))