Average Error: 0.1 → 0.1
Time: 20.4s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[e \cdot \frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
e \cdot \frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}
double f(double e, double v) {
        double r2200478 = e;
        double r2200479 = v;
        double r2200480 = sin(r2200479);
        double r2200481 = r2200478 * r2200480;
        double r2200482 = 1.0;
        double r2200483 = cos(r2200479);
        double r2200484 = r2200478 * r2200483;
        double r2200485 = r2200482 + r2200484;
        double r2200486 = r2200481 / r2200485;
        return r2200486;
}
↓
double f(double e, double v) {
        double r2200487 = e;
        double r2200488 = v;
        double r2200489 = sin(r2200488);
        double r2200490 = cos(r2200488);
        double r2200491 = 1.0;
        double r2200492 = fma(r2200490, r2200487, r2200491);
        double r2200493 = r2200489 / r2200492;
        double r2200494 = r2200487 * r2200493;
        return r2200494;
}

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 *-un-lft-identity0.1

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

  5. Applied times-frac0.1

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

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0.0 e 1.0)
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))