Trigonometry A

Average Error: 0.1 → 0.1

Time: 41.4s

Precision: 64

\[0 \le e \le 1\]

Math

\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]

↓

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

C

double f(double e, double v) {
        double r959091 = e;
        double r959092 = v;
        double r959093 = sin(r959092);
        double r959094 = r959091 * r959093;
        double r959095 = 1.0;
        double r959096 = cos(r959092);
        double r959097 = r959091 * r959096;
        double r959098 = r959095 + r959097;
        double r959099 = r959094 / r959098;
        return r959099;
}

↓

double f(double e, double v) {
        double r959100 = e;
        double r959101 = v;
        double r959102 = cos(r959101);
        double r959103 = r959100 * r959102;
        double r959104 = r959103 * r959103;
        double r959105 = r959104 * r959103;
        double r959106 = r959105 * r959105;
        double r959107 = 1.0;
        double r959108 = r959106 - r959107;
        double r959109 = r959100 / r959108;
        double r959110 = r959105 - r959107;
        double r959111 = sin(r959101);
        double r959112 = r959110 * r959111;
        double r959113 = r959109 * r959112;
        double r959114 = r959104 - r959103;
        double r959115 = r959107 + r959114;
        double r959116 = r959113 * r959115;
        return r959116;
}

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 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. Simplified 0.1

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

7. Applied flip-+ 0.1

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

8. Applied associate-/r/ 0.1

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

9. Applied associate-*l* 0.1

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

10. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019146 
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))