Average Error: 0.1 → 0.2
Time: 33.8s
Precision: 64
\[0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\left(\frac{\sin v}{\sqrt{\mathsf{fma}\left(\cos v, e, 1\right)}} \cdot e\right) \cdot \frac{1}{\sqrt{\mathsf{fma}\left(\cos v, e, 1\right)}}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\left(\frac{\sin v}{\sqrt{\mathsf{fma}\left(\cos v, e, 1\right)}} \cdot e\right) \cdot \frac{1}{\sqrt{\mathsf{fma}\left(\cos v, e, 1\right)}}
double f(double e, double v) {
        double r657443 = e;
        double r657444 = v;
        double r657445 = sin(r657444);
        double r657446 = r657443 * r657445;
        double r657447 = 1.0;
        double r657448 = cos(r657444);
        double r657449 = r657443 * r657448;
        double r657450 = r657447 + r657449;
        double r657451 = r657446 / r657450;
        return r657451;
}
↓
double f(double e, double v) {
        double r657452 = v;
        double r657453 = sin(r657452);
        double r657454 = cos(r657452);
        double r657455 = e;
        double r657456 = 1.0;
        double r657457 = fma(r657454, r657455, r657456);
        double r657458 = sqrt(r657457);
        double r657459 = r657453 / r657458;
        double r657460 = r657459 * r657455;
        double r657461 = r657456 / r657458;
        double r657462 = r657460 * r657461;
        return r657462;
}

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

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

  4. Applied add-sqr-sqrt0.2

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

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

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

  6. Applied times-frac0.2

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

  7. Applied associate-*l*0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))