Average Error: 14.8 → 0.3
Time: 9.9s
Precision: binary64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a}
(FPCore (r a b) :precision binary64 (/ (* r (sin b)) (cos (+ a b))))
↓
(FPCore (r a b)
 :precision binary64
 (* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin b) (sin a))))))
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) / ((cos(a) * cos(b)) - (sin(b) * sin(a))));
}

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

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

  3. Applied cos-sum_binary640.3

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

  4. Simplified0.3

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

  5. Simplified0.3

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

  7. Applied flip--_binary640.4

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

  8. Simplified0.4

    \[\leadsto \frac{r \cdot \sin b}{\frac{\color{blue}{{\cos a}^{2} \cdot {\cos b}^{2} - {\sin a}^{2} \cdot {\sin b}^{2}}}{\cos b \cdot \cos a + \sin b \cdot \sin a}}\]
  9. Using strategy rm

  10. Applied clear-num_binary640.5

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\frac{1}{\frac{\cos b \cdot \cos a + \sin b \cdot \sin a}{{\cos a}^{2} \cdot {\cos b}^{2} - {\sin a}^{2} \cdot {\sin b}^{2}}}}}\]

  11. Simplified0.4

    \[\leadsto \frac{r \cdot \sin b}{\frac{1}{\color{blue}{\frac{1}{\cos b \cdot \cos a - \sin b \cdot \sin a}}}}\]
  12. Using strategy rm

  13. Applied *-un-lft-identity_binary640.4

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

  14. Applied add-cube-cbrt_binary640.4

    \[\leadsto \frac{r \cdot \sin b}{\frac{1}{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot \left(\cos b \cdot \cos a - \sin b \cdot \sin a\right)}}}\]

  15. Applied times-frac_binary640.4

    \[\leadsto \frac{r \cdot \sin b}{\frac{1}{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\cos b \cdot \cos a - \sin b \cdot \sin a}}}}\]

  16. Applied add-cube-cbrt_binary640.4

    \[\leadsto \frac{r \cdot \sin b}{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\cos b \cdot \cos a - \sin b \cdot \sin a}}}\]

  17. Applied times-frac_binary640.4

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{\sqrt[3]{1}}{\cos b \cdot \cos a - \sin b \cdot \sin a}}}}\]

  18. Applied times-frac_binary640.4

    \[\leadsto \color{blue}{\frac{r}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}}} \cdot \frac{\sin b}{\frac{\sqrt[3]{1}}{\frac{\sqrt[3]{1}}{\cos b \cdot \cos a - \sin b \cdot \sin a}}}}\]

  19. Simplified0.4

    \[\leadsto \color{blue}{r} \cdot \frac{\sin b}{\frac{\sqrt[3]{1}}{\frac{\sqrt[3]{1}}{\cos b \cdot \cos a - \sin b \cdot \sin a}}}\]

  20. Simplified0.3

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

  21. Final simplification0.3

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

Alternatives

Reproduce

herbie shell --seed 2021118 
(FPCore (r a b)
  :name "rsin A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))