Average Error: 0.1 → 0.2
Time: 19.8s
Precision: 64
\[0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\left(\frac{e}{\sqrt{\mathsf{fma}\left(\left(\cos v\right), e, 1\right)}} \cdot \sin v\right) \cdot \frac{1}{\sqrt{\mathsf{fma}\left(\left(\cos v\right), e, 1\right)}}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\left(\frac{e}{\sqrt{\mathsf{fma}\left(\left(\cos v\right), e, 1\right)}} \cdot \sin v\right) \cdot \frac{1}{\sqrt{\mathsf{fma}\left(\left(\cos v\right), e, 1\right)}}
double f(double e, double v) {
        double r644746 = e;
        double r644747 = v;
        double r644748 = sin(r644747);
        double r644749 = r644746 * r644748;
        double r644750 = 1.0;
        double r644751 = cos(r644747);
        double r644752 = r644746 * r644751;
        double r644753 = r644750 + r644752;
        double r644754 = r644749 / r644753;
        return r644754;
}
↓
double f(double e, double v) {
        double r644755 = e;
        double r644756 = v;
        double r644757 = cos(r644756);
        double r644758 = 1.0;
        double r644759 = fma(r644757, r644755, r644758);
        double r644760 = sqrt(r644759);
        double r644761 = r644755 / r644760;
        double r644762 = sin(r644756);
        double r644763 = r644761 * r644762;
        double r644764 = r644758 / r644760;
        double r644765 = r644763 * r644764;
        return r644765;
}

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. Simplified0.1

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied *-un-lft-identity0.2

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

  6. Applied times-frac0.2

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

  7. Applied associate-*l*0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))