Average Error: 0.1 → 0.1
Time: 9.2s
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 r8930 = e;
        double r8931 = v;
        double r8932 = sin(r8931);
        double r8933 = r8930 * r8932;
        double r8934 = 1.0;
        double r8935 = cos(r8931);
        double r8936 = r8930 * r8935;
        double r8937 = r8934 + r8936;
        double r8938 = r8933 / r8937;
        return r8938;
}
↓
double f(double e, double v) {
        double r8939 = e;
        double r8940 = v;
        double r8941 = sin(r8940);
        double r8942 = cos(r8940);
        double r8943 = 1.0;
        double r8944 = fma(r8942, r8939, r8943);
        double r8945 = r8941 / r8944;
        double r8946 = r8939 * r8945;
        return r8946;
}

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 2020042 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))