Average Error: 0.1 → 0.2
Time: 18.8s
Precision: 64
\[0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\right)\right) \cdot e\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\right)\right) \cdot e
double f(double e, double v) {
        double r995240 = e;
        double r995241 = v;
        double r995242 = sin(r995241);
        double r995243 = r995240 * r995242;
        double r995244 = 1.0;
        double r995245 = cos(r995241);
        double r995246 = r995240 * r995245;
        double r995247 = r995244 + r995246;
        double r995248 = r995243 / r995247;
        return r995248;
}
↓
double f(double e, double v) {
        double r995249 = v;
        double r995250 = sin(r995249);
        double r995251 = cos(r995249);
        double r995252 = e;
        double r995253 = 1.0;
        double r995254 = fma(r995251, r995252, r995253);
        double r995255 = r995250 / r995254;
        double r995256 = expm1(r995255);
        double r995257 = log1p(r995256);
        double r995258 = r995257 * r995252;
        return r995258;
}

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 log1p-expm1-u0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))