Average Error: 0.1 → 0.1
Time: 37.4s
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 r1393113 = e;
        double r1393114 = v;
        double r1393115 = sin(r1393114);
        double r1393116 = r1393113 * r1393115;
        double r1393117 = 1.0;
        double r1393118 = cos(r1393114);
        double r1393119 = r1393113 * r1393118;
        double r1393120 = r1393117 + r1393119;
        double r1393121 = r1393116 / r1393120;
        return r1393121;
}
↓
double f(double e, double v) {
        double r1393122 = e;
        double r1393123 = v;
        double r1393124 = sin(r1393123);
        double r1393125 = cos(r1393123);
        double r1393126 = 1.0;
        double r1393127 = fma(r1393125, r1393122, r1393126);
        double r1393128 = r1393124 / r1393127;
        double r1393129 = r1393122 * r1393128;
        return r1393129;
}

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. Final simplification0.1

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0.0 e 1.0)
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))