Average Error: 0.1 → 0.1
Time: 4.0s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\sin v \cdot \frac{e}{\mathsf{fma}\left(\cos v, e, 1\right)}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\sin v \cdot \frac{e}{\mathsf{fma}\left(\cos v, e, 1\right)}
double f(double e, double v) {
        double r7988 = e;
        double r7989 = v;
        double r7990 = sin(r7989);
        double r7991 = r7988 * r7990;
        double r7992 = 1.0;
        double r7993 = cos(r7989);
        double r7994 = r7988 * r7993;
        double r7995 = r7992 + r7994;
        double r7996 = r7991 / r7995;
        return r7996;
}
↓
double f(double e, double v) {
        double r7997 = v;
        double r7998 = sin(r7997);
        double r7999 = e;
        double r8000 = cos(r7997);
        double r8001 = 1.0;
        double r8002 = fma(r8000, r7999, r8001);
        double r8003 = r7999 / r8002;
        double r8004 = r7998 * r8003;
        return r8004;
}

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. Simplified0.3

    \[\leadsto \color{blue}{\frac{\sin v}{\frac{\mathsf{fma}\left(\cos v, e, 1\right)}{e}}}\]
  3. Using strategy rm

  4. Applied div-inv0.2

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))