Average Error: 0.1 → 0.1
Time: 1.1m
Precision: 64
\[0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{\mathsf{fma}\left(\left(\cos v\right), e, 1\right)}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e \cdot \sin v}{\mathsf{fma}\left(\left(\cos v\right), e, 1\right)}
double f(double e, double v) {
        double r1860128 = e;
        double r1860129 = v;
        double r1860130 = sin(r1860129);
        double r1860131 = r1860128 * r1860130;
        double r1860132 = 1.0;
        double r1860133 = cos(r1860129);
        double r1860134 = r1860128 * r1860133;
        double r1860135 = r1860132 + r1860134;
        double r1860136 = r1860131 / r1860135;
        return r1860136;
}
↓
double f(double e, double v) {
        double r1860137 = e;
        double r1860138 = v;
        double r1860139 = sin(r1860138);
        double r1860140 = r1860137 * r1860139;
        double r1860141 = cos(r1860138);
        double r1860142 = 1.0;
        double r1860143 = fma(r1860141, r1860137, r1860142);
        double r1860144 = r1860140 / r1860143;
        return r1860144;
}

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

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

  4. Applied associate-*l/0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))