Average Error: 0.1 → 0.1
Time: 5.0s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\left(\frac{e}{\mathsf{fma}\left(\cos v, e, 1\right)} \cdot \frac{\sin v}{1 - e \cdot \cos v}\right) \cdot \left(1 - e \cdot \cos v\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\left(\frac{e}{\mathsf{fma}\left(\cos v, e, 1\right)} \cdot \frac{\sin v}{1 - e \cdot \cos v}\right) \cdot \left(1 - e \cdot \cos v\right)
double f(double e, double v) {
        double r12106 = e;
        double r12107 = v;
        double r12108 = sin(r12107);
        double r12109 = r12106 * r12108;
        double r12110 = 1.0;
        double r12111 = cos(r12107);
        double r12112 = r12106 * r12111;
        double r12113 = r12110 + r12112;
        double r12114 = r12109 / r12113;
        return r12114;
}
↓
double f(double e, double v) {
        double r12115 = e;
        double r12116 = v;
        double r12117 = cos(r12116);
        double r12118 = 1.0;
        double r12119 = fma(r12117, r12115, r12118);
        double r12120 = r12115 / r12119;
        double r12121 = sin(r12116);
        double r12122 = r12115 * r12117;
        double r12123 = r12118 - r12122;
        double r12124 = r12121 / r12123;
        double r12125 = r12120 * r12124;
        double r12126 = r12125 * r12123;
        return r12126;
}

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. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019352 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))