Average Error: 0.1 → 0.1
Time: 43.9s
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 r843948 = e;
        double r843949 = v;
        double r843950 = sin(r843949);
        double r843951 = r843948 * r843950;
        double r843952 = 1.0;
        double r843953 = cos(r843949);
        double r843954 = r843948 * r843953;
        double r843955 = r843952 + r843954;
        double r843956 = r843951 / r843955;
        return r843956;
}
↓
double f(double e, double v) {
        double r843957 = e;
        double r843958 = v;
        double r843959 = sin(r843958);
        double r843960 = cos(r843958);
        double r843961 = 1.0;
        double r843962 = fma(r843960, r843957, r843961);
        double r843963 = r843959 / r843962;
        double r843964 = r843957 * r843963;
        return r843964;
}

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

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))