Average Error: 0.1 → 0.2
Time: 4.9s
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 r11713 = e;
        double r11714 = v;
        double r11715 = sin(r11714);
        double r11716 = r11713 * r11715;
        double r11717 = 1.0;
        double r11718 = cos(r11714);
        double r11719 = r11713 * r11718;
        double r11720 = r11717 + r11719;
        double r11721 = r11716 / r11720;
        return r11721;
}
↓
double f(double e, double v) {
        double r11722 = e;
        double r11723 = 1.0;
        double r11724 = v;
        double r11725 = cos(r11724);
        double r11726 = r11722 * r11725;
        double r11727 = r11723 + r11726;
        double r11728 = sqrt(r11727);
        double r11729 = r11722 / r11728;
        double r11730 = sin(r11724);
        double r11731 = r11730 / r11728;
        double r11732 = r11729 * r11731;
        return r11732;
}

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 2020003 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))