Average Error: 15.4 → 0.3
Time: 20.0s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\mathsf{fma}\left(\cos a, \cos b, \left(-\sin b\right) \cdot \sin a\right)}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{\mathsf{fma}\left(\cos a, \cos b, \left(-\sin b\right) \cdot \sin a\right)}
double f(double r, double a, double b) {
        double r912588 = r;
        double r912589 = b;
        double r912590 = sin(r912589);
        double r912591 = a;
        double r912592 = r912591 + r912589;
        double r912593 = cos(r912592);
        double r912594 = r912590 / r912593;
        double r912595 = r912588 * r912594;
        return r912595;
}
↓
double f(double r, double a, double b) {
        double r912596 = r;
        double r912597 = b;
        double r912598 = sin(r912597);
        double r912599 = r912596 * r912598;
        double r912600 = a;
        double r912601 = cos(r912600);
        double r912602 = cos(r912597);
        double r912603 = -r912598;
        double r912604 = sin(r912600);
        double r912605 = r912603 * r912604;
        double r912606 = fma(r912601, r912602, r912605);
        double r912607 = r912599 / r912606;
        return r912607;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

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 associate-*r/0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))