Average Error: 0.1 → 0.1
Time: 21.0s
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 r29271 = e;
        double r29272 = v;
        double r29273 = sin(r29272);
        double r29274 = r29271 * r29273;
        double r29275 = 1.0;
        double r29276 = cos(r29272);
        double r29277 = r29271 * r29276;
        double r29278 = r29275 + r29277;
        double r29279 = r29274 / r29278;
        return r29279;
}
↓
double f(double e, double v) {
        double r29280 = e;
        double r29281 = v;
        double r29282 = sin(r29281);
        double r29283 = r29280 * r29282;
        double r29284 = cos(r29281);
        double r29285 = 1.0;
        double r29286 = fma(r29284, r29280, r29285);
        double r29287 = r29283 / r29286;
        return r29287;
}

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 2019304 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))