Trigonometry A

Average Error: 0.1 → 0.1

Time: 4.6s

Precision: 64

\[0.0 \le e \le 1\]

Math

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

↓

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

C

double code(double e, double v) {
	return ((e * sin(v)) / (1.0 + (e * cos(v))));
}

↓

double code(double e, double v) {
	return ((e / (1.0 + (e * cos(v)))) * sin(v));
}

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 associate-/l* 0.3

   \[\leadsto \color{blue}{\frac{e}{\frac{1 + e \cdot \cos v}{\sin v}}}\]
4. Using strategy rm

5. Applied associate-/r/ 0.1

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

6. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020060 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))