Average Error: 14.8 → 0.4
Time: 6.7s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos a \cdot \cos b - \sin a \cdot \sin b\right)\right)}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos a \cdot \cos b - \sin a \cdot \sin b\right)\right)}
double code(double r, double a, double b) {
	return (r * (sin(b) / cos((a + b))));
}
↓
double code(double r, double a, double b) {
	return ((r * sin(b)) / expm1(log1p(((cos(a) * cos(b)) - (sin(a) * sin(b))))));
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.8

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

  3. Applied cos-sum0.3

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

  5. Applied associate-*r/0.3

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

  7. Applied expm1-log1p-u0.4

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos a \cdot \cos b - \sin a \cdot \sin b\right)\right)}}\]

  8. Final simplification0.4

    \[\leadsto \frac{r \cdot \sin b}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos a \cdot \cos b - \sin a \cdot \sin b\right)\right)}\]

Reproduce

herbie shell --seed 2020103 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))