Average Error: 15.4 → 0.4
Time: 6.9s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\frac{\cos a \cdot \cos b - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin a \cdot \sin b\right)\right)}{\sin b}}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{r}{\frac{\cos a \cdot \cos b - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin a \cdot \sin b\right)\right)}{\sin b}}
double f(double r, double a, double b) {
        double r20047 = r;
        double r20048 = b;
        double r20049 = sin(r20048);
        double r20050 = r20047 * r20049;
        double r20051 = a;
        double r20052 = r20051 + r20048;
        double r20053 = cos(r20052);
        double r20054 = r20050 / r20053;
        return r20054;
}
↓
double f(double r, double a, double b) {
        double r20055 = r;
        double r20056 = a;
        double r20057 = cos(r20056);
        double r20058 = b;
        double r20059 = cos(r20058);
        double r20060 = r20057 * r20059;
        double r20061 = sin(r20056);
        double r20062 = sin(r20058);
        double r20063 = r20061 * r20062;
        double r20064 = log1p(r20063);
        double r20065 = expm1(r20064);
        double r20066 = r20060 - r20065;
        double r20067 = r20066 / r20062;
        double r20068 = r20055 / r20067;
        return r20068;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.4

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  2. Using strategy rm

  3. Applied cos-sum0.3

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]
  4. Using strategy rm

  5. Applied associate-/l*0.4

    \[\leadsto \color{blue}{\frac{r}{\frac{\cos a \cdot \cos b - \sin a \cdot \sin b}{\sin b}}}\]
  6. Using strategy rm

  7. Applied expm1-log1p-u0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2020002 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))