Average Error: 15.2 → 0.4
Time: 6.9s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\sin b \cdot \frac{r}{\cos b \cdot \cos a - \sin a \cdot \sin b}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\sin b \cdot \frac{r}{\cos b \cdot \cos a - \sin a \cdot \sin b}
double code(double r, double a, double b) {
	return ((double) (((double) (r * ((double) sin(b)))) / ((double) cos(((double) (a + b))))));
}
↓
double code(double r, double a, double b) {
	return ((double) (((double) sin(b)) * ((double) (r / ((double) (((double) (((double) cos(b)) * ((double) cos(a)))) - ((double) (((double) sin(a)) * ((double) 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 15.2

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

  3. Applied cos-sum0.3

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

  5. Applied expm1-log1p-u0.3

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

  7. Applied *-un-lft-identity0.3

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{1 \cdot \left(\cos a \cdot \cos b - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin a \cdot \sin b\right)\right)\right)}}\]

  8. Applied *-commutative0.3

    \[\leadsto \frac{\color{blue}{\sin b \cdot r}}{1 \cdot \left(\cos a \cdot \cos b - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin a \cdot \sin b\right)\right)\right)}\]

  9. Applied times-frac0.4

    \[\leadsto \color{blue}{\frac{\sin b}{1} \cdot \frac{r}{\cos a \cdot \cos b - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin a \cdot \sin b\right)\right)}}\]

  10. Simplified0.4

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

  11. Simplified0.4

    \[\leadsto \sin b \cdot \color{blue}{\frac{r}{\cos b \cdot \cos a - \sin a \cdot \sin b}}\]

  12. Final simplification0.4

    \[\leadsto \sin b \cdot \frac{r}{\cos b \cdot \cos a - \sin a \cdot \sin b}\]

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))