Average Error: 15.5 → 0.4
Time: 17.7s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\left(r \cdot \sin b\right) \cdot \frac{1}{\mathsf{fma}\left(\cos a, \cos b, \left(-\sin a\right) \cdot \sin b\right)}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\left(r \cdot \sin b\right) \cdot \frac{1}{\mathsf{fma}\left(\cos a, \cos b, \left(-\sin a\right) \cdot \sin b\right)}
double f(double r, double a, double b) {
        double r26359 = r;
        double r26360 = b;
        double r26361 = sin(r26360);
        double r26362 = a;
        double r26363 = r26362 + r26360;
        double r26364 = cos(r26363);
        double r26365 = r26361 / r26364;
        double r26366 = r26359 * r26365;
        return r26366;
}
↓
double f(double r, double a, double b) {
        double r26367 = r;
        double r26368 = b;
        double r26369 = sin(r26368);
        double r26370 = r26367 * r26369;
        double r26371 = 1.0;
        double r26372 = a;
        double r26373 = cos(r26372);
        double r26374 = cos(r26368);
        double r26375 = sin(r26372);
        double r26376 = -r26375;
        double r26377 = r26376 * r26369;
        double r26378 = fma(r26373, r26374, r26377);
        double r26379 = r26371 / r26378;
        double r26380 = r26370 * r26379;
        return r26380;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.5

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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. Simplified0.3

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

  6. Applied fma-neg0.3

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

  7. Simplified0.3

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

  9. Applied div-inv0.4

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

  10. Applied associate-*r*0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))