Average Error: 15.2 → 0.4
Time: 22.7s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\left(r \cdot \sin b\right) \cdot \frac{1}{\mathsf{fma}\left(\cos a, \cos b, -\sin b \cdot \sin a\right)}\]
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, -\sin b \cdot \sin a\right)}
double f(double r, double a, double b) {
        double r25446 = r;
        double r25447 = b;
        double r25448 = sin(r25447);
        double r25449 = a;
        double r25450 = r25449 + r25447;
        double r25451 = cos(r25450);
        double r25452 = r25448 / r25451;
        double r25453 = r25446 * r25452;
        return r25453;
}
↓
double f(double r, double a, double b) {
        double r25454 = r;
        double r25455 = b;
        double r25456 = sin(r25455);
        double r25457 = r25454 * r25456;
        double r25458 = 1.0;
        double r25459 = a;
        double r25460 = cos(r25459);
        double r25461 = cos(r25455);
        double r25462 = sin(r25459);
        double r25463 = r25456 * r25462;
        double r25464 = -r25463;
        double r25465 = fma(r25460, r25461, r25464);
        double r25466 = r25458 / r25465;
        double r25467 = r25457 * r25466;
        return r25467;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.2

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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 div-inv0.4

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

  6. Applied associate-*r*0.4

    \[\leadsto \color{blue}{\left(r \cdot \sin b\right) \cdot \frac{1}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]
  7. Using strategy rm

  8. Applied fma-neg0.4

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))