Average Error: 0.1 → 0.2
Time: 5.1s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[e \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin v \cdot 1}{\mathsf{fma}\left(\cos v, e, 1\right)}\right)\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
e \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin v \cdot 1}{\mathsf{fma}\left(\cos v, e, 1\right)}\right)\right)
double f(double e, double v) {
        double r10578 = e;
        double r10579 = v;
        double r10580 = sin(r10579);
        double r10581 = r10578 * r10580;
        double r10582 = 1.0;
        double r10583 = cos(r10579);
        double r10584 = r10578 * r10583;
        double r10585 = r10582 + r10584;
        double r10586 = r10581 / r10585;
        return r10586;
}
↓
double f(double e, double v) {
        double r10587 = e;
        double r10588 = v;
        double r10589 = sin(r10588);
        double r10590 = 1.0;
        double r10591 = r10589 * r10590;
        double r10592 = cos(r10588);
        double r10593 = 1.0;
        double r10594 = fma(r10592, r10587, r10593);
        double r10595 = r10591 / r10594;
        double r10596 = log1p(r10595);
        double r10597 = expm1(r10596);
        double r10598 = r10587 * r10597;
        return r10598;
}

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

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

  8. Applied expm1-log1p-u0.2

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

  9. Final simplification0.2

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

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)))))