Average Error: 15.1 → 0.3
Time: 8.8s
Precision: binary64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
↓
\[\frac{-\sin b \cdot r}{\mathsf{fma}\left(\cos b, -\cos a, \sin b \cdot \sin a\right)} \]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{-\sin b \cdot r}{\mathsf{fma}\left(\cos b, -\cos a, \sin b \cdot \sin a\right)}

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

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.1

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

  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 expm1-log1p-u_binary640.4

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

  4. Applied frac-2neg_binary640.4

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

  5. Simplified0.4

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

  6. Simplified0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2021313 
(FPCore (r a b)
  :name "rsin A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))