Average Error: 0.1 → 0.3
Time: 13.5s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{\sin v}{\frac{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}}{e \cdot \left(1 \cdot 1\right) + e \cdot \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v - 1\right)\right)}}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{\sin v}{\frac{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}}{e \cdot \left(1 \cdot 1\right) + e \cdot \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v - 1\right)\right)}}
double f(double e, double v) {
        double r19977 = e;
        double r19978 = v;
        double r19979 = sin(r19978);
        double r19980 = r19977 * r19979;
        double r19981 = 1.0;
        double r19982 = cos(r19978);
        double r19983 = r19977 * r19982;
        double r19984 = r19981 + r19983;
        double r19985 = r19980 / r19984;
        return r19985;
}
↓
double f(double e, double v) {
        double r19986 = v;
        double r19987 = sin(r19986);
        double r19988 = 1.0;
        double r19989 = 3.0;
        double r19990 = pow(r19988, r19989);
        double r19991 = e;
        double r19992 = cos(r19986);
        double r19993 = r19991 * r19992;
        double r19994 = pow(r19993, r19989);
        double r19995 = r19990 + r19994;
        double r19996 = r19988 * r19988;
        double r19997 = r19991 * r19996;
        double r19998 = r19993 - r19988;
        double r19999 = r19993 * r19998;
        double r20000 = r19991 * r19999;
        double r20001 = r19997 + r20000;
        double r20002 = r19995 / r20001;
        double r20003 = r19987 / r20002;
        return r20003;
}

Error

Bits error versus e

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.1

    \[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
  2. Using strategy rm

  3. Applied flip3-+0.1

    \[\leadsto \frac{e \cdot \sin v}{\color{blue}{\frac{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}}{1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot \left(e \cdot \cos v\right)\right)}}}\]

  4. Applied associate-/r/0.1

    \[\leadsto \color{blue}{\frac{e \cdot \sin v}{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot \left(e \cdot \cos v\right)\right)\right)}\]
  5. Using strategy rm

  6. Applied add-log-exp0.1

    \[\leadsto \frac{e \cdot \sin v}{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot \color{blue}{\log \left(e^{e \cdot \cos v}\right)}\right)\right)\]

  7. Final simplification0.3

    \[\leadsto \frac{\sin v}{\frac{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}}{e \cdot \left(1 \cdot 1\right) + e \cdot \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v - 1\right)\right)}}\]

Reproduce

herbie shell --seed 2019291 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))