Average Error: 15.7 → 0.4
Time: 11.1s
Precision: binary64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\cos b \cdot \cos a - \left(\sin b \cdot {\left({\sin a}^{2}\right)}^{0.3333333333333333}\right) \cdot \sqrt[3]{\sin a}}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{\cos b \cdot \cos a - \left(\sin b \cdot {\left({\sin a}^{2}\right)}^{0.3333333333333333}\right) \cdot \sqrt[3]{\sin a}}
(FPCore (r a b) :precision binary64 (/ (* r (sin b)) (cos (+ a b))))
↓
(FPCore (r a b)
 :precision binary64
 (/
  (* r (sin b))
  (-
   (* (cos b) (cos a))
   (* (* (sin b) (pow (pow (sin a) 2.0) 0.3333333333333333)) (cbrt (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(b) * cos(a)) - ((sin(b) * pow(pow(sin(a), 2.0), 0.3333333333333333)) * cbrt(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 15.7

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

  3. Applied cos-sum_binary64_5530.3

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

  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 add-cube-cbrt_binary64_4540.5

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

  8. Applied associate-*r*_binary64_3590.5

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

  9. Taylor expanded around inf 0.4

    \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \left(\sin b \cdot \color{blue}{{\left({\sin a}^{2}\right)}^{0.3333333333333333}}\right) \cdot \sqrt[3]{\sin a}}\]

  10. Final simplification0.4

    \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \left(\sin b \cdot {\left({\sin a}^{2}\right)}^{0.3333333333333333}\right) \cdot \sqrt[3]{\sin a}}\]

Reproduce

herbie shell --seed 2021045 
(FPCore (r a b)
  :name "rsin A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))