Average Error: 0.1 → 0.1
Time: 22.0s
Precision: 64
\[0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{\sin v}{\mathsf{fma}\left(e, \cos v, 1\right)} \cdot e\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{\sin v}{\mathsf{fma}\left(e, \cos v, 1\right)} \cdot e
double f(double e, double v) {
        double r931185 = e;
        double r931186 = v;
        double r931187 = sin(r931186);
        double r931188 = r931185 * r931187;
        double r931189 = 1.0;
        double r931190 = cos(r931186);
        double r931191 = r931185 * r931190;
        double r931192 = r931189 + r931191;
        double r931193 = r931188 / r931192;
        return r931193;
}
↓
double f(double e, double v) {
        double r931194 = v;
        double r931195 = sin(r931194);
        double r931196 = e;
        double r931197 = cos(r931194);
        double r931198 = 1.0;
        double r931199 = fma(r931196, r931197, r931198);
        double r931200 = r931195 / r931199;
        double r931201 = r931200 * r931196;
        return r931201;
}

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

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

  5. Applied associate-*r*0.1

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

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))