Average Error: 0.1 → 0.1
Time: 23.4s
Precision: 64
\[0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\]
\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 r524550 = e;
        double r524551 = v;
        double r524552 = sin(r524551);
        double r524553 = r524550 * r524552;
        double r524554 = 1.0;
        double r524555 = cos(r524551);
        double r524556 = r524550 * r524555;
        double r524557 = r524554 + r524556;
        double r524558 = r524553 / r524557;
        return r524558;
}
↓
double f(double e, double v) {
        double r524559 = e;
        double r524560 = v;
        double r524561 = sin(r524560);
        double r524562 = r524559 * r524561;
        double r524563 = cos(r524560);
        double r524564 = 1.0;
        double r524565 = fma(r524563, r524559, r524564);
        double r524566 = r524562 / r524565;
        return r524566;
}

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-sqrt0.4

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

  5. Applied associate-*r*0.4

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

  7. Applied associate-*l/0.4

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

  8. Applied associate-*l/0.4

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))