Average Error: 0.1 → 0.2
Time: 14.2s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[e \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\right)\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
e \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\right)\right)
double f(double e, double v) {
        double r19248 = e;
        double r19249 = v;
        double r19250 = sin(r19249);
        double r19251 = r19248 * r19250;
        double r19252 = 1.0;
        double r19253 = cos(r19249);
        double r19254 = r19248 * r19253;
        double r19255 = r19252 + r19254;
        double r19256 = r19251 / r19255;
        return r19256;
}
↓
double f(double e, double v) {
        double r19257 = e;
        double r19258 = v;
        double r19259 = sin(r19258);
        double r19260 = cos(r19258);
        double r19261 = 1.0;
        double r19262 = fma(r19260, r19257, r19261);
        double r19263 = r19259 / r19262;
        double r19264 = log1p(r19263);
        double r19265 = expm1(r19264);
        double r19266 = r19257 * r19265;
        return r19266;
}

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 *-un-lft-identity0.1

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

  4. Applied times-frac0.1

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

  5. Simplified0.1

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

  6. Simplified0.1

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

  8. Applied expm1-log1p-u0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))