Average Error: 0.1 → 0.1
Time: 4.6s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[e \cdot \frac{\sin v \cdot 1}{\mathsf{fma}\left(\cos v, e, 1\right)}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
e \cdot \frac{\sin v \cdot 1}{\mathsf{fma}\left(\cos v, e, 1\right)}
double f(double e, double v) {
        double r9459 = e;
        double r9460 = v;
        double r9461 = sin(r9460);
        double r9462 = r9459 * r9461;
        double r9463 = 1.0;
        double r9464 = cos(r9460);
        double r9465 = r9459 * r9464;
        double r9466 = r9463 + r9465;
        double r9467 = r9462 / r9466;
        return r9467;
}
↓
double f(double e, double v) {
        double r9468 = e;
        double r9469 = v;
        double r9470 = sin(r9469);
        double r9471 = 1.0;
        double r9472 = r9470 * r9471;
        double r9473 = cos(r9469);
        double r9474 = 1.0;
        double r9475 = fma(r9473, r9468, r9474);
        double r9476 = r9472 / r9475;
        double r9477 = r9468 * r9476;
        return r9477;
}

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. Using strategy rm

  3. Applied *-un-lft-identity0.1

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

  4. Applied times-frac0.1

    \[\leadsto \color{blue}{\frac{e}{1} \cdot \frac{\sin v}{1 + e \cdot \cos v}}\]

  5. Simplified0.1

    \[\leadsto \color{blue}{e} \cdot \frac{\sin v}{1 + e \cdot \cos v}\]

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))