Average Error: 0.1 → 0.1
Time: 27.7s
Precision: 64
\[0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[e \cdot \frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\]
\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 r539400 = e;
        double r539401 = v;
        double r539402 = sin(r539401);
        double r539403 = r539400 * r539402;
        double r539404 = 1.0;
        double r539405 = cos(r539401);
        double r539406 = r539400 * r539405;
        double r539407 = r539404 + r539406;
        double r539408 = r539403 / r539407;
        return r539408;
}
↓
double f(double e, double v) {
        double r539409 = e;
        double r539410 = v;
        double r539411 = sin(r539410);
        double r539412 = cos(r539410);
        double r539413 = 1.0;
        double r539414 = fma(r539412, r539409, r539413);
        double r539415 = r539411 / r539414;
        double r539416 = r539409 * r539415;
        return r539416;
}

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 *-un-lft-identity0.1

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

  5. Applied associate-/r*0.1

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

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))