Average Error: 0.1 → 0.1
Time: 3.3m
Precision: 64
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{1 - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)} \cdot \left(1 - e \cdot \cos v\right)\]
double f(double e, double v) {
        double r8074214 = e;
        double r8074215 = v;
        double r8074216 = sin(r8074215);
        double r8074217 = r8074214 * r8074216;
        double r8074218 = 1.0;
        double r8074219 = cos(r8074215);
        double r8074220 = r8074214 * r8074219;
        double r8074221 = r8074218 + r8074220;
        double r8074222 = r8074217 / r8074221;
        return r8074222;
}
↓
double f(double e, double v) {
        double r8074223 = e;
        double r8074224 = v;
        double r8074225 = sin(r8074224);
        double r8074226 = r8074223 * r8074225;
        double r8074227 = 1.0;
        double r8074228 = cos(r8074224);
        double r8074229 = r8074223 * r8074228;
        double r8074230 = r8074229 * r8074229;
        double r8074231 = r8074227 - r8074230;
        double r8074232 = r8074226 / r8074231;
        double r8074233 = r8074227 - r8074229;
        double r8074234 = r8074232 * r8074233;
        return r8074234;
}


\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e \cdot \sin v}{1 - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)} \cdot \left(1 - e \cdot \cos v\right)

Error

Bits error versus e

Bits error versus v

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

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

Reproduce

herbie shell --seed 2019101 
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))