Average Error: 0.1 → 0.1
Time: 4.9s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e \cdot \sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}
double f(double e, double v) {
        double r10657 = e;
        double r10658 = v;
        double r10659 = sin(r10658);
        double r10660 = r10657 * r10659;
        double r10661 = 1.0;
        double r10662 = cos(r10658);
        double r10663 = r10657 * r10662;
        double r10664 = r10661 + r10663;
        double r10665 = r10660 / r10664;
        return r10665;
}
↓
double f(double e, double v) {
        double r10666 = e;
        double r10667 = v;
        double r10668 = sin(r10667);
        double r10669 = r10666 * r10668;
        double r10670 = cos(r10667);
        double r10671 = 1.0;
        double r10672 = fma(r10670, r10666, r10671);
        double r10673 = r10669 / r10672;
        return r10673;
}

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 associate-/l*0.3

    \[\leadsto \color{blue}{\frac{e}{\frac{1 + e \cdot \cos v}{\sin v}}}\]
  4. Using strategy rm

  5. Applied *-un-lft-identity0.3

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

  6. Applied *-un-lft-identity0.3

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

  7. Applied times-frac0.3

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

  8. Applied *-un-lft-identity0.3

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

  9. Applied times-frac0.3

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

  10. Simplified0.3

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))