Average Error: 0.1 → 0.1
Time: 14.1s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e \cdot \sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}
double f(double e, double v) {
        double r19261 = e;
        double r19262 = v;
        double r19263 = sin(r19262);
        double r19264 = r19261 * r19263;
        double r19265 = 1.0;
        double r19266 = cos(r19262);
        double r19267 = r19261 * r19266;
        double r19268 = r19265 + r19267;
        double r19269 = r19264 / r19268;
        return r19269;
}
↓
double f(double e, double v) {
        double r19270 = e;
        double r19271 = v;
        double r19272 = sin(r19271);
        double r19273 = r19270 * r19272;
        double r19274 = cos(r19271);
        double r19275 = 1.0;
        double r19276 = fma(r19274, r19270, r19275);
        double r19277 = r19273 / r19276;
        return r19277;
}

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. Final simplification0.1

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

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))