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 r19046 = r;
        double r19047 = b;
        double r19048 = sin(r19047);
        double r19049 = a;
        double r19050 = r19049 + r19047;
        double r19051 = cos(r19050);
        double r19052 = r19048 / r19051;
        double r19053 = r19046 * r19052;
        return r19053;
}
↓
double f(double r, double a, double b) {
        double r19054 = r;
        double r19055 = b;
        double r19056 = sin(r19055);
        double r19057 = a;
        double r19058 = cos(r19057);
        double r19059 = cos(r19055);
        double r19060 = sin(r19057);
        double r19061 = r19060 * r19056;
        double r19062 = log1p(r19061);
        double r19063 = expm1(r19062);
        double r19064 = -r19063;
        double r19065 = fma(r19058, r19059, r19064);
        double r19066 = r19056 / r19065;
        double r19067 = r19054 * r19066;
        return r19067;
}

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)))))