Average Error: 0.1 → 0.1
Time: 4.4s
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 r11121 = e;
        double r11122 = v;
        double r11123 = sin(r11122);
        double r11124 = r11121 * r11123;
        double r11125 = 1.0;
        double r11126 = cos(r11122);
        double r11127 = r11121 * r11126;
        double r11128 = r11125 + r11127;
        double r11129 = r11124 / r11128;
        return r11129;
}
↓
double f(double e, double v) {
        double r11130 = e;
        double r11131 = v;
        double r11132 = sin(r11131);
        double r11133 = r11130 * r11132;
        double r11134 = 1.0;
        double r11135 = r11134 * r11134;
        double r11136 = cos(r11131);
        double r11137 = r11130 * r11136;
        double r11138 = r11137 * r11137;
        double r11139 = r11135 - r11138;
        double r11140 = r11133 / r11139;
        double r11141 = r11134 - r11137;
        double r11142 = r11140 * r11141;
        return r11142;
}

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 2020047 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))