Average Error: 15.3 → 0.3
Time: 10.8s
Precision: binary64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
↓
\[\frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, -\sin b \cdot \sin a\right)} \cdot r \]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, -\sin b \cdot \sin a\right)} \cdot r

(FPCore (r a b) :precision binary64 (/ (* r (sin b)) (cos (+ a b))))
↓
(FPCore (r a b)
 :precision binary64
 (* (/ (sin b) (fma (cos a) (cos b) (- (* (sin b) (sin a))))) r))
double code(double r, double a, double b) {
	return (r * sin(b)) / cos(a + b);
}
↓
double code(double r, double a, double b) {
	return (sin(b) / fma(cos(a), cos(b), -(sin(b) * sin(a)))) * r;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.3

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

  2. Applied cos-sum_binary640.4

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

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

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

  4. Applied times-frac_binary640.3

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

  5. Simplified0.3

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

  6. Simplified0.3

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

  7. Applied fma-neg_binary640.3

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

  8. Applied *-commutative_binary640.3

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

  9. Final simplification0.3

    \[\leadsto \frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, -\sin b \cdot \sin a\right)} \cdot r \]

Reproduce

herbie shell --seed 2022077 
(FPCore (r a b)
  :name "rsin A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))