Average Error: 0.1 → 0.2
Time: 21.0s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{1}{\frac{\mathsf{fma}\left(\cos v, e, 1\right)}{\sin v}} \cdot e\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{1}{\frac{\mathsf{fma}\left(\cos v, e, 1\right)}{\sin v}} \cdot e
double f(double e, double v) {
        double r1165773 = e;
        double r1165774 = v;
        double r1165775 = sin(r1165774);
        double r1165776 = r1165773 * r1165775;
        double r1165777 = 1.0;
        double r1165778 = cos(r1165774);
        double r1165779 = r1165773 * r1165778;
        double r1165780 = r1165777 + r1165779;
        double r1165781 = r1165776 / r1165780;
        return r1165781;
}
↓
double f(double e, double v) {
        double r1165782 = 1.0;
        double r1165783 = v;
        double r1165784 = cos(r1165783);
        double r1165785 = e;
        double r1165786 = 1.0;
        double r1165787 = fma(r1165784, r1165785, r1165786);
        double r1165788 = sin(r1165783);
        double r1165789 = r1165787 / r1165788;
        double r1165790 = r1165782 / r1165789;
        double r1165791 = r1165790 * r1165785;
        return r1165791;
}

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 clear-num0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0.0 e 1.0)
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))