Average Error: 15.1 → 0.3
Time: 5.3s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{-r \cdot \sin b}{\mathsf{fma}\left(-\cos b, \cos a, \sin a \cdot \sin b\right)}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{-r \cdot \sin b}{\mathsf{fma}\left(-\cos b, \cos a, \sin a \cdot \sin b\right)}
double code(double r, double a, double b) {
	return ((r * sin(b)) / cos((a + b)));
}
↓
double code(double r, double a, double b) {
	return (-(r * sin(b)) / fma(-cos(b), cos(a), (sin(a) * 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 15.1

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

  3. Applied cos-sum0.3

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

  5. Applied frac-2neg0.3

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

  6. Simplified0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))