Average Error: 0.1 → 0.1
Time: 28.7s
Precision: 64
\[0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e}{\mathsf{fma}\left(\cos v, e, 1\right)} \cdot \sin v\]
\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 r790253 = e;
        double r790254 = v;
        double r790255 = sin(r790254);
        double r790256 = r790253 * r790255;
        double r790257 = 1.0;
        double r790258 = cos(r790254);
        double r790259 = r790253 * r790258;
        double r790260 = r790257 + r790259;
        double r790261 = r790256 / r790260;
        return r790261;
}
↓
double f(double e, double v) {
        double r790262 = e;
        double r790263 = v;
        double r790264 = cos(r790263);
        double r790265 = 1.0;
        double r790266 = fma(r790264, r790262, r790265);
        double r790267 = r790262 / r790266;
        double r790268 = sin(r790263);
        double r790269 = r790267 * r790268;
        return r790269;
}

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

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

  4. Applied div-inv0.1

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

  5. Applied associate-*l*0.1

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

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))