Average Error: 14.8 → 0.3
Time: 11.5s
Precision: binary64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
↓
\[\begin{array}{l} t_0 := \sin b \cdot \sin a\\ \frac{\sin b \cdot r}{\mathsf{fma}\left(\cos b, \cos a, \mathsf{fma}\left(-\sin a, \sin b, t_0\right) - t_0\right)} \end{array} \]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\begin{array}{l}
t_0 := \sin b \cdot \sin a\\
\frac{\sin b \cdot r}{\mathsf{fma}\left(\cos b, \cos a, \mathsf{fma}\left(-\sin a, \sin b, t_0\right) - 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 b) (cos a) (- (fma (- (sin a)) (sin b) t_0) 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(b), cos(a), (fma(-sin(a), sin(b), t_0) - t_0));
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.8

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

  2. Applied egg-rr0.3

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

  3. Taylor expanded in r around 0 0.3

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

  4. Applied egg-rr0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2022125 
(FPCore (r a b)
  :name "rsin B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))