Average Error: 14.7 → 0.4
Time: 17.2s
Precision: binary64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
↓
\[\frac{r \cdot \sin b}{\cos a \cdot \cos b - \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin b}^{2} \cdot {\sin a}^{2}\right)\right) \cdot \left(\sin b \cdot \sin a\right)}} \]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{\cos a \cdot \cos b - \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin b}^{2} \cdot {\sin a}^{2}\right)\right) \cdot \left(\sin b \cdot \sin a\right)}}

(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))
   (cbrt
    (*
     (expm1 (log1p (* (pow (sin b) 2.0) (pow (sin a) 2.0))))
     (* (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)) - cbrt(expm1(log1p(pow(sin(b), 2.0) * pow(sin(a), 2.0))) * (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 14.7

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

  2. Applied cos-sum_binary640.3

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

  3. Applied add-cbrt-cube_binary640.4

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

  4. Applied expm1-log1p-u_binary640.4

    \[\leadsto \frac{r \cdot \sin b}{\cos a \cdot \cos b - \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)\right)\right)} \cdot \left(\sin a \cdot \sin b\right)}} \]

  5. Simplified0.4

    \[\leadsto \frac{r \cdot \sin b}{\cos a \cdot \cos b - \sqrt[3]{\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left({\sin b}^{2} \cdot {\sin a}^{2}\right)}\right) \cdot \left(\sin a \cdot \sin b\right)}} \]

  6. Final simplification0.4

    \[\leadsto \frac{r \cdot \sin b}{\cos a \cdot \cos b - \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin b}^{2} \cdot {\sin a}^{2}\right)\right) \cdot \left(\sin b \cdot \sin a\right)}} \]

Reproduce

herbie shell --seed 2021275 
(FPCore (r a b)
  :name "rsin A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))