Average Error: 0.1 → 0.1
Time: 36.5s
Precision: 64
\[0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\sin v \cdot \left(\frac{1}{\mathsf{fma}\left(\left(\cos v\right), e, 1\right)} \cdot e\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\sin v \cdot \left(\frac{1}{\mathsf{fma}\left(\left(\cos v\right), e, 1\right)} \cdot e\right)
double f(double e, double v) {
        double r1147430 = e;
        double r1147431 = v;
        double r1147432 = sin(r1147431);
        double r1147433 = r1147430 * r1147432;
        double r1147434 = 1.0;
        double r1147435 = cos(r1147431);
        double r1147436 = r1147430 * r1147435;
        double r1147437 = r1147434 + r1147436;
        double r1147438 = r1147433 / r1147437;
        return r1147438;
}
↓
double f(double e, double v) {
        double r1147439 = v;
        double r1147440 = sin(r1147439);
        double r1147441 = 1.0;
        double r1147442 = cos(r1147439);
        double r1147443 = e;
        double r1147444 = fma(r1147442, r1147443, r1147441);
        double r1147445 = r1147441 / r1147444;
        double r1147446 = r1147445 * r1147443;
        double r1147447 = r1147440 * r1147446;
        return r1147447;
}

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 div-inv0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))