Average Error: 0.1 → 0.2
Time: 44.5s
Precision: 64
\[0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\right)\right) \cdot e\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\right)\right) \cdot e
double f(double e, double v) {
        double r895241 = e;
        double r895242 = v;
        double r895243 = sin(r895242);
        double r895244 = r895241 * r895243;
        double r895245 = 1.0;
        double r895246 = cos(r895242);
        double r895247 = r895241 * r895246;
        double r895248 = r895245 + r895247;
        double r895249 = r895244 / r895248;
        return r895249;
}
↓
double f(double e, double v) {
        double r895250 = v;
        double r895251 = sin(r895250);
        double r895252 = cos(r895250);
        double r895253 = e;
        double r895254 = 1.0;
        double r895255 = fma(r895252, r895253, r895254);
        double r895256 = r895251 / r895255;
        double r895257 = log1p(r895256);
        double r895258 = expm1(r895257);
        double r895259 = r895258 * r895253;
        return r895259;
}

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. Using strategy rm

  4. Applied expm1-log1p-u0.2

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\right)\right)} \cdot e\]

  5. Final simplification0.2

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\right)\right) \cdot e\]

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))