Average Error: 0.1 → 0.2
Time: 5.2s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e}{\sqrt{1 + e \cdot \cos v}} \cdot \frac{\sin v}{\sqrt{1 + e \cdot \cos v}}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e}{\sqrt{1 + e \cdot \cos v}} \cdot \frac{\sin v}{\sqrt{1 + e \cdot \cos v}}
double f(double e, double v) {
        double r11962 = e;
        double r11963 = v;
        double r11964 = sin(r11963);
        double r11965 = r11962 * r11964;
        double r11966 = 1.0;
        double r11967 = cos(r11963);
        double r11968 = r11962 * r11967;
        double r11969 = r11966 + r11968;
        double r11970 = r11965 / r11969;
        return r11970;
}
↓
double f(double e, double v) {
        double r11971 = e;
        double r11972 = 1.0;
        double r11973 = v;
        double r11974 = cos(r11973);
        double r11975 = r11971 * r11974;
        double r11976 = r11972 + r11975;
        double r11977 = sqrt(r11976);
        double r11978 = r11971 / r11977;
        double r11979 = sin(r11973);
        double r11980 = r11979 / r11977;
        double r11981 = r11978 * r11980;
        return r11981;
}

Error

Bits error versus e

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.1

    \[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.2

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

  4. Applied times-frac0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2020100 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))