Average Error: 15.3 → 0.3
Time: 14.3s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos a \cdot \cos b\right)}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos a \cdot \cos b\right)}
double f(double r, double a, double b) {
        double r18351 = r;
        double r18352 = b;
        double r18353 = sin(r18352);
        double r18354 = a;
        double r18355 = r18354 + r18352;
        double r18356 = cos(r18355);
        double r18357 = r18353 / r18356;
        double r18358 = r18351 * r18357;
        return r18358;
}
↓
double f(double r, double a, double b) {
        double r18359 = r;
        double r18360 = b;
        double r18361 = sin(r18360);
        double r18362 = a;
        double r18363 = sin(r18362);
        double r18364 = -r18363;
        double r18365 = cos(r18362);
        double r18366 = cos(r18360);
        double r18367 = r18365 * r18366;
        double r18368 = fma(r18361, r18364, r18367);
        double r18369 = r18361 / r18368;
        double r18370 = r18359 * r18369;
        return r18370;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.3

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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. Taylor expanded around inf 0.3

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

  5. Simplified0.3

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

  6. Taylor expanded around inf 0.3

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

  7. Simplified0.3

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

  9. Applied *-un-lft-identity0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))