Average Error: 14.7 → 0.4
Time: 9.6s
Precision: binary64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
↓
\[\begin{array}{l} t_0 := \sin b \cdot \sin a\\ r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{t_0 \cdot {t_0}^{2}}} \end{array} \]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\begin{array}{l}
t_0 := \sin b \cdot \sin a\\
r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{t_0 \cdot {t_0}^{2}}}
\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))))
   (* r (/ (sin b) (- (* (cos a) (cos b)) (cbrt (* t_0 (pow t_0 2.0))))))))
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 r * (sin(b) / ((cos(a) * cos(b)) - cbrt(t_0 * pow(t_0, 2.0))));
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.7

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

  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. Applied add-cbrt-cube_binary640.4

    \[\leadsto r \cdot \frac{\sin b}{\cos a \cdot \cos b - \color{blue}{\sqrt[3]{\left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)\right) \cdot \left(\sin a \cdot \sin b\right)}}} \]

  4. Applied pow2_binary640.4

    \[\leadsto r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{\color{blue}{{\left(\sin a \cdot \sin b\right)}^{2}} \cdot \left(\sin a \cdot \sin b\right)}} \]

  5. Final simplification0.4

    \[\leadsto r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{\left(\sin b \cdot \sin a\right) \cdot {\left(\sin b \cdot \sin a\right)}^{2}}} \]

Reproduce

herbie shell --seed 2021275 
(FPCore (r a b)
  :name "rsin B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))