Average Error: 0.1 → 0.1
Time: 9.0s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e}{\mathsf{fma}\left(\cos v, e, 1\right)} \cdot \sin v\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e}{\mathsf{fma}\left(\cos v, e, 1\right)} \cdot \sin v
double f(double e, double v) {
        double r9272 = e;
        double r9273 = v;
        double r9274 = sin(r9273);
        double r9275 = r9272 * r9274;
        double r9276 = 1.0;
        double r9277 = cos(r9273);
        double r9278 = r9272 * r9277;
        double r9279 = r9276 + r9278;
        double r9280 = r9275 / r9279;
        return r9280;
}
↓
double f(double e, double v) {
        double r9281 = e;
        double r9282 = v;
        double r9283 = cos(r9282);
        double r9284 = 1.0;
        double r9285 = fma(r9283, r9281, r9284);
        double r9286 = r9281 / r9285;
        double r9287 = sin(r9282);
        double r9288 = r9286 * r9287;
        return r9288;
}

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. Simplified0.1

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

  4. Applied associate-/l*0.3

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

  6. Applied associate-/r/0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))