Average Error: 14.9 → 0.3
Time: 7.2s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, -\mathsf{expm1}\left(\mathsf{log1p}\left(\sin a \cdot \sin b\right)\right)\right)}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, -\mathsf{expm1}\left(\mathsf{log1p}\left(\sin a \cdot \sin b\right)\right)\right)}
double f(double r, double a, double b) {
        double r19038 = r;
        double r19039 = b;
        double r19040 = sin(r19039);
        double r19041 = a;
        double r19042 = r19041 + r19039;
        double r19043 = cos(r19042);
        double r19044 = r19040 / r19043;
        double r19045 = r19038 * r19044;
        return r19045;
}
↓
double f(double r, double a, double b) {
        double r19046 = r;
        double r19047 = b;
        double r19048 = sin(r19047);
        double r19049 = a;
        double r19050 = cos(r19049);
        double r19051 = cos(r19047);
        double r19052 = sin(r19049);
        double r19053 = r19052 * r19048;
        double r19054 = log1p(r19053);
        double r19055 = expm1(r19054);
        double r19056 = -r19055;
        double r19057 = fma(r19050, r19051, r19056);
        double r19058 = r19048 / r19057;
        double r19059 = r19046 * r19058;
        return r19059;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.9

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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 expm1-log1p-u0.3

    \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin a \cdot \sin b\right)\right)}\right)}\]

  8. Final simplification0.3

    \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, -\mathsf{expm1}\left(\mathsf{log1p}\left(\sin a \cdot \sin b\right)\right)\right)}\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))