Average Error: 0.1 → 0.1
Time: 9.0s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e \cdot \sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}
double f(double e, double v) {
        double r9948 = e;
        double r9949 = v;
        double r9950 = sin(r9949);
        double r9951 = r9948 * r9950;
        double r9952 = 1.0;
        double r9953 = cos(r9949);
        double r9954 = r9948 * r9953;
        double r9955 = r9952 + r9954;
        double r9956 = r9951 / r9955;
        return r9956;
}
↓
double f(double e, double v) {
        double r9957 = e;
        double r9958 = v;
        double r9959 = sin(r9958);
        double r9960 = r9957 * r9959;
        double r9961 = cos(r9958);
        double r9962 = 1.0;
        double r9963 = fma(r9961, r9957, r9962);
        double r9964 = r9960 / r9963;
        return r9964;
}

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. Final simplification0.1

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

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))