Average Error: 0.1 → 0.1
Time: 4.5s
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 r10555 = e;
        double r10556 = v;
        double r10557 = sin(r10556);
        double r10558 = r10555 * r10557;
        double r10559 = 1.0;
        double r10560 = cos(r10556);
        double r10561 = r10555 * r10560;
        double r10562 = r10559 + r10561;
        double r10563 = r10558 / r10562;
        return r10563;
}
↓
double f(double e, double v) {
        double r10564 = e;
        double r10565 = v;
        double r10566 = sin(r10565);
        double r10567 = 1.0;
        double r10568 = r10566 * r10567;
        double r10569 = cos(r10565);
        double r10570 = 1.0;
        double r10571 = fma(r10569, r10564, r10570);
        double r10572 = r10568 / r10571;
        double r10573 = r10564 * r10572;
        return r10573;
}

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 2020046 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))