Average Error: 0.1 → 0.1
Time: 17.0s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e}{\mathsf{fma}\left(e, \cos v, 1\right)} \cdot \sin v\right)\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e}{\mathsf{fma}\left(e, \cos v, 1\right)} \cdot \sin v\right)\right)
double f(double e, double v) {
        double r24086 = e;
        double r24087 = v;
        double r24088 = sin(r24087);
        double r24089 = r24086 * r24088;
        double r24090 = 1.0;
        double r24091 = cos(r24087);
        double r24092 = r24086 * r24091;
        double r24093 = r24090 + r24092;
        double r24094 = r24089 / r24093;
        return r24094;
}
↓
double f(double e, double v) {
        double r24095 = e;
        double r24096 = v;
        double r24097 = cos(r24096);
        double r24098 = 1.0;
        double r24099 = fma(r24095, r24097, r24098);
        double r24100 = r24095 / r24099;
        double r24101 = sin(r24096);
        double r24102 = r24100 * r24101;
        double r24103 = log1p(r24102);
        double r24104 = expm1(r24103);
        return r24104;
}

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 associate-/l*0.3

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

  4. Simplified0.3

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

  6. Applied expm1-log1p-u0.3

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0.0 e 1.0)
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))