Average Error: 0.1 → 0.1
Time: 20.5s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{1 \cdot 1 - \left(e \cdot e\right) \cdot {\left(\cos v\right)}^{2}} \cdot \left(1 - e \cdot \cos v\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e \cdot \sin v}{1 \cdot 1 - \left(e \cdot e\right) \cdot {\left(\cos v\right)}^{2}} \cdot \left(1 - e \cdot \cos v\right)
double f(double e, double v) {
        double r19199 = e;
        double r19200 = v;
        double r19201 = sin(r19200);
        double r19202 = r19199 * r19201;
        double r19203 = 1.0;
        double r19204 = cos(r19200);
        double r19205 = r19199 * r19204;
        double r19206 = r19203 + r19205;
        double r19207 = r19202 / r19206;
        return r19207;
}
↓
double f(double e, double v) {
        double r19208 = e;
        double r19209 = v;
        double r19210 = sin(r19209);
        double r19211 = r19208 * r19210;
        double r19212 = 1.0;
        double r19213 = r19212 * r19212;
        double r19214 = r19208 * r19208;
        double r19215 = cos(r19209);
        double r19216 = 2.0;
        double r19217 = pow(r19215, r19216);
        double r19218 = r19214 * r19217;
        double r19219 = r19213 - r19218;
        double r19220 = r19211 / r19219;
        double r19221 = r19208 * r19215;
        double r19222 = r19212 - r19221;
        double r19223 = r19220 * r19222;
        return r19223;
}

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. Simplified0.1

    \[\leadsto \color{blue}{\frac{e \cdot \sin v}{1 \cdot 1 - \left(e \cdot e\right) \cdot {\left(\cos v\right)}^{2}}} \cdot \left(1 - e \cdot \cos v\right)\]

  6. Final simplification0.1

    \[\leadsto \frac{e \cdot \sin v}{1 \cdot 1 - \left(e \cdot e\right) \cdot {\left(\cos v\right)}^{2}} \cdot \left(1 - e \cdot \cos v\right)\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))