Average Error: 0.1 → 0.1
Time: 16.8s
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 r1301969 = e;
        double r1301970 = v;
        double r1301971 = sin(r1301970);
        double r1301972 = r1301969 * r1301971;
        double r1301973 = 1.0;
        double r1301974 = cos(r1301970);
        double r1301975 = r1301969 * r1301974;
        double r1301976 = r1301973 + r1301975;
        double r1301977 = r1301972 / r1301976;
        return r1301977;
}
↓
double f(double e, double v) {
        double r1301978 = e;
        double r1301979 = v;
        double r1301980 = sin(r1301979);
        double r1301981 = cos(r1301979);
        double r1301982 = 1.0;
        double r1301983 = fma(r1301981, r1301978, r1301982);
        double r1301984 = r1301980 / r1301983;
        double r1301985 = r1301978 * r1301984;
        return r1301985;
}

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 2019174 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0.0 e 1.0)
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))