Average Error: 15.1 → 0.3
Time: 10.3s
Precision: binary64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
↓
\[\begin{array}{l} t_0 := \sin b \cdot \sin a\\ \sin b \cdot \frac{r}{\mathsf{fma}\left(\cos a, \cos b, -t_0\right) + \mathsf{fma}\left(-\sin b, \sin a, t_0\right)} \end{array} \]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\begin{array}{l}
t_0 := \sin b \cdot \sin a\\
\sin b \cdot \frac{r}{\mathsf{fma}\left(\cos a, \cos b, -t_0\right) + \mathsf{fma}\left(-\sin b, \sin a, t_0\right)}
\end{array}

(FPCore (r a b) :precision binary64 (* r (/ (sin b) (cos (+ a b)))))
↓
(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) (sin a))))
   (*
    (sin b)
    (/ r (+ (fma (cos a) (cos b) (- t_0)) (fma (- (sin b)) (sin a) t_0))))))
double code(double r, double a, double b) {
	return r * (sin(b) / cos(a + b));
}
↓
double code(double r, double a, double b) {
	double t_0 = sin(b) * sin(a);
	return sin(b) * (r / (fma(cos(a), cos(b), -t_0) + fma(-sin(b), sin(a), t_0)));
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.1

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

  2. Applied cos-sum_binary640.3

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

  3. Taylor expanded in r around 0 0.3

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

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

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

  5. Applied times-frac_binary640.3

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

  6. Simplified0.3

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

  7. Applied prod-diff_binary640.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2022104 
(FPCore (r a b)
  :name "rsin B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))