rsin A

Average Error: 15.5 → 0.5

Time: 5.4s

Precision: binary64

Math

\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]

↓

\[r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sqrt[3]{\sin a} \cdot \left(\sin b \cdot \left(\sqrt[3]{\sin a} \cdot \sqrt[3]{\sin a}\right)\right)}\]

C

double code(double r, double a, double b) {
	return (((double) (r * ((double) sin(b)))) / ((double) cos(((double) (a + b)))));
}

↓

double code(double r, double a, double b) {
	return ((double) (r * (((double) sin(b)) / ((double) (((double) (((double) cos(b)) * ((double) cos(a)))) - ((double) (((double) cbrt(((double) sin(a)))) * ((double) (((double) sin(b)) * ((double) (((double) cbrt(((double) sin(a)))) * ((double) cbrt(((double) sin(a)))))))))))))));
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.5

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]

2. Simplified 15.4

   \[\leadsto \color{blue}{r \cdot \frac{\sin b}{\cos \left(b + a\right)}}\]
3. Using strategy rm

4. Applied cos-sum 0.3

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}}\]
5. Using strategy rm

6. Applied add-cube-cbrt 0.5

   \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \color{blue}{\left(\left(\sqrt[3]{\sin a} \cdot \sqrt[3]{\sin a}\right) \cdot \sqrt[3]{\sin a}\right)}}\]

7. Applied associate-*r* 0.5

   \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{\left(\sin b \cdot \left(\sqrt[3]{\sin a} \cdot \sqrt[3]{\sin a}\right)\right) \cdot \sqrt[3]{\sin a}}}\]

8. Final simplification 0.5

   \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sqrt[3]{\sin a} \cdot \left(\sin b \cdot \left(\sqrt[3]{\sin a} \cdot \sqrt[3]{\sin a}\right)\right)}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (r a b)
  :name "rsin A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))