Average Error: 15.1 → 0.4
Time: 7.8s
Precision: binary64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\sin b \cdot \frac{r}{\cos a \cdot \cos b - \sqrt[3]{{\left(\sin b \cdot \sin a\right)}^{3}}}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\sin b \cdot \frac{r}{\cos a \cdot \cos b - \sqrt[3]{{\left(\sin b \cdot \sin a\right)}^{3}}}
double code(double r, double a, double b) {
	return ((double) (r * ((double) (((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(a)) * ((double) cos(b)))) - ((double) cbrt(((double) pow(((double) (((double) sin(b)) * ((double) sin(a)))), 3.0))))))))));
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.1

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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. Simplified0.3

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

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

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

  9. Applied times-frac0.4

    \[\leadsto \color{blue}{\frac{\sin b}{1} \cdot \frac{r}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]

  10. Simplified0.4

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

  11. Simplified0.4

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

  13. Applied add-cbrt-cube0.4

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

  14. Applied add-cbrt-cube0.4

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

  15. Applied cbrt-unprod0.4

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

  16. Simplified0.4

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

  17. Final simplification0.4

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

Reproduce

herbie shell --seed 2020181 
(FPCore (r a b)
  :name "rsin B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))