Average Error: 0.1 → 0.2
Time: 17.3s
Precision: 64
\[0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\left(\frac{\sin v}{\sqrt{\mathsf{fma}\left(\cos v, e, 1\right)}} \cdot e\right) \cdot \frac{1}{\sqrt{\mathsf{fma}\left(\cos v, e, 1\right)}}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\left(\frac{\sin v}{\sqrt{\mathsf{fma}\left(\cos v, e, 1\right)}} \cdot e\right) \cdot \frac{1}{\sqrt{\mathsf{fma}\left(\cos v, e, 1\right)}}
double f(double e, double v) {
        double r442044 = e;
        double r442045 = v;
        double r442046 = sin(r442045);
        double r442047 = r442044 * r442046;
        double r442048 = 1.0;
        double r442049 = cos(r442045);
        double r442050 = r442044 * r442049;
        double r442051 = r442048 + r442050;
        double r442052 = r442047 / r442051;
        return r442052;
}
↓
double f(double e, double v) {
        double r442053 = v;
        double r442054 = sin(r442053);
        double r442055 = cos(r442053);
        double r442056 = e;
        double r442057 = 1.0;
        double r442058 = fma(r442055, r442056, r442057);
        double r442059 = sqrt(r442058);
        double r442060 = r442054 / r442059;
        double r442061 = r442060 * r442056;
        double r442062 = r442057 / r442059;
        double r442063 = r442061 * r442062;
        return r442063;
}

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 add-sqr-sqrt0.2

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

  5. Applied *-un-lft-identity0.2

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

  6. Applied times-frac0.2

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

  7. Applied associate-*l*0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019154 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))