Average Error: 0.1 → 0.2
Time: 5.3s
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 r11895 = e;
        double r11896 = v;
        double r11897 = sin(r11896);
        double r11898 = r11895 * r11897;
        double r11899 = 1.0;
        double r11900 = cos(r11896);
        double r11901 = r11895 * r11900;
        double r11902 = r11899 + r11901;
        double r11903 = r11898 / r11902;
        return r11903;
}
↓
double f(double e, double v) {
        double r11904 = e;
        double r11905 = 1.0;
        double r11906 = v;
        double r11907 = cos(r11906);
        double r11908 = r11904 * r11907;
        double r11909 = r11905 + r11908;
        double r11910 = sqrt(r11909);
        double r11911 = r11904 / r11910;
        double r11912 = sin(r11906);
        double r11913 = r11912 / r11910;
        double r11914 = r11911 * r11913;
        return r11914;
}

Error

Bits error versus e

Bits error versus v

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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 2020081 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))