Average Error: 14.8 → 0.3
Time: 14.1s
Precision: binary64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
↓
\[\begin{array}{l} t_0 := e^{\mathsf{log1p}\left(\sin b \cdot \sin a\right)}\\ \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, 1 - t_0\right) + \mathsf{fma}\left(-\sin b, \sin a, t_0 - 1\right)} \end{array} \]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\begin{array}{l}
t_0 := e^{\mathsf{log1p}\left(\sin b \cdot \sin a\right)}\\
\frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, 1 - t_0\right) + \mathsf{fma}\left(-\sin b, \sin a, t_0 - 1\right)}
\end{array}

(FPCore (r a b) :precision binary64 (/ (* r (sin b)) (cos (+ a b))))
↓
(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (exp (log1p (* (sin b) (sin a))))))
   (/
    (* r (sin b))
    (+
     (fma (cos b) (cos a) (- 1.0 t_0))
     (fma (- (sin b)) (sin a) (- t_0 1.0))))))
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 = exp(log1p(sin(b) * sin(a)));
	return (r * sin(b)) / (fma(cos(b), cos(a), (1.0 - t_0)) + fma(-sin(b), sin(a), (t_0 - 1.0)));
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.8

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

  2. Applied egg0.3

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

  3. Applied egg0.4

    \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, -\color{blue}{\left(e^{\mathsf{log1p}\left(\sin a \cdot \sin b\right)} - 1\right)}\right) + \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)} \]

  4. Applied egg0.3

    \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, -\left(e^{\mathsf{log1p}\left(\sin a \cdot \sin b\right)} - 1\right)\right) + \mathsf{fma}\left(-\sin b, \sin a, \color{blue}{e^{\mathsf{log1p}\left(\sin a \cdot \sin b\right)} - 1}\right)} \]

  5. Final simplification0.3

    \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, 1 - e^{\mathsf{log1p}\left(\sin b \cdot \sin a\right)}\right) + \mathsf{fma}\left(-\sin b, \sin a, e^{\mathsf{log1p}\left(\sin b \cdot \sin a\right)} - 1\right)} \]

Reproduce

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