Trigonometry A

Average Error: 0.1 → 0.1

Time: 4.4s

Precision: 64

\[0.0 \le e \le 1\]

Math

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

↓

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

C

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

↓

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

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 *-commutative 0.1

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

4. Applied associate-/l* 0.2

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

6. Applied associate-/r/ 0.1

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

7. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020113 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))