Average Error: 0.1 → 0.1
Time: 5.3s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[e \cdot \frac{\sin v \cdot 1}{\mathsf{fma}\left(\cos v, e, 1\right)}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
e \cdot \frac{\sin v \cdot 1}{\mathsf{fma}\left(\cos v, e, 1\right)}
double f(double e, double v) {
        double r12662 = e;
        double r12663 = v;
        double r12664 = sin(r12663);
        double r12665 = r12662 * r12664;
        double r12666 = 1.0;
        double r12667 = cos(r12663);
        double r12668 = r12662 * r12667;
        double r12669 = r12666 + r12668;
        double r12670 = r12665 / r12669;
        return r12670;
}
↓
double f(double e, double v) {
        double r12671 = e;
        double r12672 = v;
        double r12673 = sin(r12672);
        double r12674 = 1.0;
        double r12675 = r12673 * r12674;
        double r12676 = cos(r12672);
        double r12677 = 1.0;
        double r12678 = fma(r12676, r12671, r12677);
        double r12679 = r12675 / r12678;
        double r12680 = r12671 * r12679;
        return r12680;
}

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 *-un-lft-identity0.1

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

  4. Applied times-frac0.1

    \[\leadsto \color{blue}{\frac{e}{1} \cdot \frac{\sin v}{1 + e \cdot \cos v}}\]

  5. Simplified0.1

    \[\leadsto \color{blue}{e} \cdot \frac{\sin v}{1 + e \cdot \cos v}\]

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))