Average Error: 0.1 → 0.1
Time: 4.9s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{1 \cdot 1 - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)} \cdot \left(1 - e \cdot \cos v\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e \cdot \sin v}{1 \cdot 1 - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)} \cdot \left(1 - e \cdot \cos v\right)
double f(double e, double v) {
        double r12075 = e;
        double r12076 = v;
        double r12077 = sin(r12076);
        double r12078 = r12075 * r12077;
        double r12079 = 1.0;
        double r12080 = cos(r12076);
        double r12081 = r12075 * r12080;
        double r12082 = r12079 + r12081;
        double r12083 = r12078 / r12082;
        return r12083;
}
↓
double f(double e, double v) {
        double r12084 = e;
        double r12085 = v;
        double r12086 = sin(r12085);
        double r12087 = r12084 * r12086;
        double r12088 = 1.0;
        double r12089 = r12088 * r12088;
        double r12090 = cos(r12085);
        double r12091 = r12084 * r12090;
        double r12092 = r12091 * r12091;
        double r12093 = r12089 - r12092;
        double r12094 = r12087 / r12093;
        double r12095 = r12088 - r12091;
        double r12096 = r12094 * r12095;
        return r12096;
}

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 flip-+0.1

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

  4. Applied associate-/r/0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2020064 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))