Average Error: 0.1 → 0.2
Time: 19.4s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e}{\sqrt{\mathsf{fma}\left(\cos v, e, 1\right)}} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin v}{\sqrt{\mathsf{fma}\left(\cos v, e, 1\right)}}\right)\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e}{\sqrt{\mathsf{fma}\left(\cos v, e, 1\right)}} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin v}{\sqrt{\mathsf{fma}\left(\cos v, e, 1\right)}}\right)\right)
double f(double e, double v) {
        double r29755 = e;
        double r29756 = v;
        double r29757 = sin(r29756);
        double r29758 = r29755 * r29757;
        double r29759 = 1.0;
        double r29760 = cos(r29756);
        double r29761 = r29755 * r29760;
        double r29762 = r29759 + r29761;
        double r29763 = r29758 / r29762;
        return r29763;
}
↓
double f(double e, double v) {
        double r29764 = e;
        double r29765 = v;
        double r29766 = cos(r29765);
        double r29767 = 1.0;
        double r29768 = fma(r29766, r29764, r29767);
        double r29769 = sqrt(r29768);
        double r29770 = r29764 / r29769;
        double r29771 = sin(r29765);
        double r29772 = r29771 / r29769;
        double r29773 = expm1(r29772);
        double r29774 = log1p(r29773);
        double r29775 = r29770 * r29774;
        return r29775;
}

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 add-sqr-sqrt0.2

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

  4. Applied times-frac0.2

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

  5. Simplified0.2

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

  6. Simplified0.2

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

  8. Applied log1p-expm1-u0.2

    \[\leadsto \frac{e}{\sqrt{\mathsf{fma}\left(\cos v, e, 1\right)}} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin v}{\sqrt{\mathsf{fma}\left(\cos v, e, 1\right)}}\right)\right)}\]

  9. Final simplification0.2

    \[\leadsto \frac{e}{\sqrt{\mathsf{fma}\left(\cos v, e, 1\right)}} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin v}{\sqrt{\mathsf{fma}\left(\cos v, e, 1\right)}}\right)\right)\]

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))