Average Error: 15.4 → 0.3
Time: 7.1s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, -\mathsf{expm1}\left(\mathsf{log1p}\left(\sin a \cdot \sin b\right)\right)\right)}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, -\mathsf{expm1}\left(\mathsf{log1p}\left(\sin a \cdot \sin b\right)\right)\right)}
double code(double r, double a, double b) {
	return ((double) (r * ((double) (((double) sin(b)) / ((double) cos(((double) (a + b))))))));
}
↓
double code(double r, double a, double b) {
	return ((double) (r * ((double) (((double) sin(b)) / ((double) fma(((double) cos(a)), ((double) cos(b)), ((double) -(((double) expm1(((double) log1p(((double) (((double) sin(a)) * ((double) sin(b))))))))))))))));
}

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 15.4

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  2. Using strategy rm

  3. Applied cos-sum0.3

    \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]
  4. Using strategy rm

  5. Applied fma-neg0.3

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

  7. Applied expm1-log1p-u0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2020121 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))