Average Error: 0.1 → 0.1
Time: 4.4s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e}{\mathsf{fma}\left(\cos v, e, 1\right)} \cdot \sin v\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e}{\mathsf{fma}\left(\cos v, e, 1\right)} \cdot \sin v
double f(double e, double v) {
        double r10118 = e;
        double r10119 = v;
        double r10120 = sin(r10119);
        double r10121 = r10118 * r10120;
        double r10122 = 1.0;
        double r10123 = cos(r10119);
        double r10124 = r10118 * r10123;
        double r10125 = r10122 + r10124;
        double r10126 = r10121 / r10125;
        return r10126;
}
↓
double f(double e, double v) {
        double r10127 = e;
        double r10128 = v;
        double r10129 = cos(r10128);
        double r10130 = 1.0;
        double r10131 = fma(r10129, r10127, r10130);
        double r10132 = r10127 / r10131;
        double r10133 = sin(r10128);
        double r10134 = r10132 * r10133;
        return r10134;
}

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. Using strategy rm

  3. Applied associate-/l*0.3

    \[\leadsto \color{blue}{\frac{e}{\frac{1 + e \cdot \cos v}{\sin v}}}\]
  4. Using strategy rm

  5. Applied associate-/r/0.1

    \[\leadsto \color{blue}{\frac{e}{1 + e \cdot \cos v} \cdot \sin v}\]

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))