Average Error: 0.1 → 0.1
Time: 18.7s
Precision: 64
\[0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\sin v \cdot \left(\frac{1}{\mathsf{fma}\left(\left(\cos v\right), e, 1\right)} \cdot e\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\sin v \cdot \left(\frac{1}{\mathsf{fma}\left(\left(\cos v\right), e, 1\right)} \cdot e\right)
double f(double e, double v) {
        double r423219 = e;
        double r423220 = v;
        double r423221 = sin(r423220);
        double r423222 = r423219 * r423221;
        double r423223 = 1.0;
        double r423224 = cos(r423220);
        double r423225 = r423219 * r423224;
        double r423226 = r423223 + r423225;
        double r423227 = r423222 / r423226;
        return r423227;
}
↓
double f(double e, double v) {
        double r423228 = v;
        double r423229 = sin(r423228);
        double r423230 = 1.0;
        double r423231 = cos(r423228);
        double r423232 = e;
        double r423233 = fma(r423231, r423232, r423230);
        double r423234 = r423230 / r423233;
        double r423235 = r423234 * r423232;
        double r423236 = r423229 * r423235;
        return r423236;
}

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

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

  4. Applied div-inv0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))